diff --git a/materials/tutorial_07/data/nature_dat.csv b/materials/tutorial_07/data/nature_dat.csv
new file mode 100644
index 0000000..56ccc06
--- /dev/null
+++ b/materials/tutorial_07/data/nature_dat.csv
@@ -0,0 +1,16705 @@
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+"57576","ENSG00000197448","esophagus",0.000125653920654665,0.000120299999905863
+"57577","ENSG00000197451","esophagus",6.57451832181938e-05,7.3599999942804e-05
+"57579","ENSG00000197548","esophagus",3.96216835465798e-05,2.8399999996924e-05
+"57580","ENSG00000197594","esophagus",3.22994218009726e-07,3.19999999764285e-06
+"57583","ENSG00000197694","esophagus",0.00015765077389285,3.56000000022235e-05
+"57584","ENSG00000197702","esophagus",0.000217290201172074,6.36000000514878e-05
+"57588","ENSG00000197746","esophagus",3.06521285324753e-05,0.000295199999897359
+"57591","ENSG00000197766","esophagus",3.68953734012818e-05,6.70000000461545e-06
+"57595","ENSG00000197837","esophagus",0.00017962413965137,3.0000000019365e-06
+"57598","ENSG00000197879","esophagus",7.57152293126742e-05,5.98999999463043e-05
+"57600","ENSG00000197894","esophagus",0.000618523550223868,8.25000000854977e-05
+"57602","ENSG00000197930","esophagus",0.00062739118144533,0.000243200000223605
+"57605","ENSG00000197956","esophagus",0.00286767064050795,0.000339400000014335
+"57612","ENSG00000198034","esophagus",1.13529960042025e-05,0.000411900000041335
+"57616","ENSG00000198130","esophagus",7.6788400616236e-05,3.18999999957999e-05
+"57619","ENSG00000198171","esophagus",1.04244484968569e-05,1.64999999918731e-05
+"57620","ENSG00000198189","esophagus",5.82815694145258e-06,2.22999999975271e-05
+"57626","ENSG00000198356","esophagus",6.16570681810182e-05,4.78999999542762e-05
+"57627","ENSG00000198363","esophagus",1.35924988834871e-05,7.29999999797527e-05
+"57628","ENSG00000198380","esophagus",3.89075041995129e-05,1.6699999994325e-05
+"57633","ENSG00000198431","esophagus",7.2786622267714e-05,3.18999999957999e-05
+"57634","ENSG00000198467","esophagus",0.00229900529751741,0.000612899999503083
+"57639","ENSG00000198561","esophagus",1.09897403335373e-05,0.000137899999944163
+"57640","ENSG00000198585","esophagus",9.12128582478151e-06,7.80000000555902e-06
+"57642","ENSG00000198612","esophagus",3.68326739860786e-05,2.10999999855367e-05
+"57645","ENSG00000198677","esophagus",2.34648653275934e-06,2.42000000010904e-05
+"57647","ENSG00000198700","esophagus",1.5599204695571e-05,1.22000000091368e-05
+"57649","ENSG00000198712","esophagus",0.000429308146818565,0.00429830000186122
+"57653","ENSG00000198752","esophagus",1.11924998697117e-06,3.41999999955795e-05
+"57660","ENSG00000198805","esophagus",0.000136150331781767,7.22000000715461e-05
+"57662","ENSG00000198836","esophagus",5.89307411378461e-06,3.0699999966268e-05
+"57665","ENSG00000198898","esophagus",8.68332499747373e-05,7.69999999694191e-05
+"57666","ENSG00000198899","esophagus",7.84991421973283e-05,0.00417249999662502
+"57669","ENSG00000198931","esophagus",0.000494847249868009,6.51000000647273e-05
+"57672","ENSG00000198951","esophagus",6.57761214008154e-06,1.63999999981988e-05
+"57680","ENSG00000203879","esophagus",8.63194771308981e-05,4.56000000352336e-05
+"57687","ENSG00000204394","esophagus",5.42476393381063e-05,2.61999999807105e-05
+"57689","ENSG00000204525","esophagus",2.56356248781883e-06,0.000205799999797934
+"57690","ENSG00000204574","esophagus",4.92149099725749e-06,2.20000000090065e-05
+"57692","ENSG00000204628","esophagus",2.04681988725189e-05,0.00117150000018078
+"57695","ENSG00000204843","esophagus",3.65601070751088e-05,4.35999999730359e-05
+"57698","ENSG00000205220","esophagus",1.89398429720894e-05,1.62999999848386e-05
+"57699","ENSG00000205302","esophagus",3.68207556481962e-05,4.50000000232781e-05
+"57700","ENSG00000205323","esophagus",4.69203303312898e-05,3.22999999753747e-05
+"57701","ENSG00000205339","esophagus",3.59924250534985e-05,2.90000000067472e-05
+"57702","ENSG00000205403","esophagus",1.67895927122499e-05,1.22000000091368e-05
+"57704","ENSG00000205531","esophagus",9.17791106465757e-05,5.64000000021633e-05
+"57708","ENSG00000205629","esophagus",2.72757744813427e-06,2.05999999824898e-05
+"57717","ENSG00000206503","esophagus",8.134880357488e-06,0.000182799999832556
+"57720","ENSG00000211445","esophagus",0.000345228191351069,0.00024590000014794
+"57723","ENSG00000211456","esophagus",5.95762324039482e-05,2.21000000160238e-05
+"57724","ENSG00000211460","esophagus",9.68199065217735e-05,3.05999999660683e-05
+"57727","ENSG00000213145","esophagus",4.62955588514223e-05,8.27000000852058e-05
+"57730","ENSG00000213281","esophagus",0.000101966910313565,2.12999999784821e-05
+"57733","ENSG00000213366","esophagus",6.96210580357618e-05,5.60999999669104e-05
+"57741","ENSG00000213563","esophagus",8.13577800871385e-06,3.60000000192903e-06
+"57742","ENSG00000213585","esophagus",0.000247229123977689,8.1000000022633e-05
+"57744","ENSG00000213614","esophagus",9.88589528627601e-06,3.28000000217754e-05
+"57745","ENSG00000213619","esophagus",3.10782427779705e-05,5.06000000186651e-05
+"57746","ENSG00000213639","esophagus",0.00022961731802837,0.000220599999947086
+"57752","ENSG00000213949","esophagus",4.88754782469801e-05,2.6000000001747e-05
+"57756","ENSG00000213995","esophagus",2.33947694673244e-05,2.32000000058281e-05
+"57759","ENSG00000214078","esophagus",0.000108380138333863,6.63999999437332e-05
+"57762","ENSG00000214253","esophagus",6.9209945141156e-06,0.000114399999879618
+"57765","ENSG00000214517","esophagus",2.76157179219972e-05,2.43999999809691e-05
+"57768","ENSG00000214753","esophagus",3.01516690100506e-06,2.42000000010904e-05
+"57770","ENSG00000215021","esophagus",6.53753069532168e-05,8.8399999997339e-05
+"57781","ENSG00000221914","esophagus",4.19363045935383e-06,8.41999999031291e-05
+"57795","ENSG00000233276","esophagus",0.000425183096045817,3.47000000167091e-05
+"57797","ENSG00000234745","esophagus",5.24460488335431e-06,0.000348299999632499
+"57800","ENSG00000239264","esophagus",2.57762690369916e-05,4.01000000350699e-05
+"57802","ENSG00000239306","esophagus",8.46710243699234e-07,1.35999999884066e-05
+"57803","ENSG00000239672","esophagus",0.000356931268149867,5.26999999742083e-05
+"57805","ENSG00000239900","esophagus",6.91179643666621e-07,2.50999999721985e-05
+"57807","ENSG00000240344","esophagus",5.04859600156596e-06,2.32999999986041e-05
+"57809","ENSG00000240849","esophagus",0.000159336146255524,2.78000000052441e-05
+"57810","ENSG00000240857","esophagus",8.57108951847866e-06,1.62999999848386e-05
+"57811","ENSG00000240972","esophagus",0.00378047219311175,1.36999999950661e-05
+"57814","ENSG00000241468","esophagus",5.26212958741703e-05,0.000354399999824541
+"57815","ENSG00000241553","esophagus",0.000378532119943893,0.000124300000102565
+"57817","ENSG00000241685","esophagus",3.11310887096447e-05,7.21000000465794e-05
+"57818","ENSG00000241837","esophagus",0.000477394777834167,0.000181499999844479
+"57826","ENSG00000242372","esophagus",0.000237074853030631,0.000115100000098115
+"57827","ENSG00000242498","esophagus",2.49880174179517e-05,1.36999999950661e-05
+"57839","ENSG00000243989","esophagus",1.70044643512278e-05,1.23999999953679e-05
+"57841","ENSG00000244038","esophagus",9.46204155427771e-05,8.89000000060972e-05
+"57847","ENSG00000244687","esophagus",0.000253862321947295,5.99999999469977e-05
+"57848","ENSG00000244734","esophagus",0.0635455039152776,0.000316499999742472
+"57850","ENSG00000247596","esophagus",1.517495004224e-05,2.42000000010904e-05
+"57852","ENSG00000249915","esophagus",8.3524105641233e-05,8.22999999597743e-05
+"57857","ENSG00000253729","esophagus",2.34809801282476e-06,1.38999999918674e-05
+"57859","ENSG00000254505","esophagus",0.000814701995394408,3.94000000158242e-05
+"57861","ENSG00000254709","esophagus",0.000821293820192087,3.88000000362536e-05
+"57862","ENSG00000254772","esophagus",9.28479274779445e-06,0.000188199999960462
+"57864","ENSG00000254986","esophagus",0.000161500719688988,1.94999999837228e-05
+"57874","ENSG00000257727","esophagus",7.59948806016488e-05,2.95000000014833e-05
+"57882","ENSG00000262246","esophagus",3.33790157152527e-05,6.90000000417445e-06
+"57886","ENSG00000269190","esophagus",3.70350513373877e-06,7.80000000555902e-06
+"57888","ENSG00000272325","esophagus",2.57420086555142e-05,1.34999999846128e-05
diff --git a/materials/tutorial_07/img/forward.png b/materials/tutorial_07/img/forward.png
new file mode 100644
index 0000000..e1e12bb
Binary files /dev/null and b/materials/tutorial_07/img/forward.png differ
diff --git a/materials/tutorial_07/tests_tutorial_07.R b/materials/tutorial_07/tests_tutorial_07.R
new file mode 100644
index 0000000..be39a71
--- /dev/null
+++ b/materials/tutorial_07/tests_tutorial_07.R
@@ -0,0 +1,435 @@
+library(digest)
+library(testthat)
+
+# +
+#abstraction templates
+
+check_TF <- function(answerX.X, expectedHash) {
+  var_name <- deparse(substitute(answerX.X))
+  test_that(paste('Did not assign answer to an object called ', var_name), {
+    expect_true(exists(var_name))
+  })
+  
+  
+  test_that('Solution should be "true" or "false"', {
+    expect_match(answerX.X, "true|false", ignore.case = TRUE)
+  })
+  
+  answer_hash <- digest(tolower(answerX.X))
+  #if (answer_hash == "HASH_HERE") {
+  #  print("HINT_HERE")
+  #}
+  
+  test_that("Solution is incorrect", {
+    expect_equal(answer_hash, expectedHash)
+  })
+  
+  print("Success!")
+}
+
+
+is_alphabetical <- function(x, scan=function(...)x){
+          all(diff(utf8ToInt(scan(,''))%%32)>0)
+      }
+
+
+check_MC <- function(answerX.X, choiceList, expectedHash) {
+  var_name <- deparse(substitute(answerX.X))
+  test_that(paste('Did not assign answer to an object called ', var_name), {
+    expect_true(exists(var_name))
+  })
+  
+  
+  test_that(paste('Solution should be a single alphabetical combination of ', toString(choiceList)), {
+
+    expect_true(is_alphabetical(answerX.X) & (tolower(substr(answerX.X,nchar(answerX.X),nchar(answerX.X))) %in% tolower(choiceList)))
+  })
+  
+  answer_hash <- digest(tolower(answerX.X))
+  #if (answer_hash == "HASH_HERE") {
+  #  print("HINT_HERE")
+  #} else if (answer_hash == "HASH_HERE") {
+  #  print("HINT_HERE")
+  #} else if (answer_hash == "HASH_HERE") {
+  #  print("HINT_HERE")
+  #}
+  
+  test_that("Solution is incorrect", {
+    expect_equal(answer_hash, expectedHash)
+  })
+  
+  print("Success!")
+}
+
+
+
+# dataCheckTuples is data.frame(c(colnames), c(scale factor), c(expectedHash))
+check_DF <- function(answerX.X, expected_colnames, hashNRows, cols_to_check, precision_list, expectedHashes) {
+  dataCheckTuples <- data.frame(cols_to_check, precision_list, expectedHashes) 
+  var_name <- deparse(substitute(answerX.X))
+  test_that(paste('Did not assign answer to an object called ', var_name), {
+    expect_true(exists(var_name))
+  })
+  
+  test_that("Solution should be a data frame", {
+    expect_true("data.frame" %in% class(answerX.X))
+  })
+  
+  given_colnames <- colnames(answerX.X)
+  test_that("Data frame does not have the correct columns", {
+    expect_equal(length(setdiff(
+      union(expected_colnames, given_colnames),
+      intersect(expected_colnames, given_colnames)
+    )), 0)
+  })
+  
+  test_that("Data frame does not contain the correct number of rows", {
+    expect_equal(digest(as.integer(nrow(answerX.X))), hashNRows)
+  })
+  
+  
+  
+  apply(dataCheckTuples, 1, function(tuple) {
+    test_that(paste(tuple[[1]], " does not contain the correct data"), {
+      expect_equal(digest(as.integer(sum(answerX.X[tuple[[1]]]) * as.double(tuple[[2]]))),
+                   tuple[[3]])
+    })
+  })
+  
+  
+  
+  
+  print("Success!")
+}
+
+
+
+
+
+# dataCheckTuples is data.frame(c(colnames), c(scale factor), c(expectedHash))
+check_DF_overflow <- function(answerX.X, expected_colnames, hashNRows, cols_to_check, precision_list, expectedHashes) {
+  dataCheckTuples <- data.frame(cols_to_check, precision_list, expectedHashes) 
+  var_name <- deparse(substitute(answerX.X))
+  test_that(paste('Did not assign answer to an object called ', var_name), {
+    expect_true(exists(var_name))
+  })
+  
+  test_that("Solution should be a data frame", {
+    expect_true("data.frame" %in% class(answerX.X))
+  })
+  
+  given_colnames <- colnames(answerX.X)
+  test_that("Data frame does not have the correct columns", {
+    expect_equal(length(setdiff(
+      union(expected_colnames, given_colnames),
+      intersect(expected_colnames, given_colnames)
+    )), 0)
+  })
+  
+  test_that("Data frame does not contain the correct number of rows", {
+    expect_equal(digest(as.integer(nrow(answerX.X))), hashNRows)
+  })
+  
+  
+  
+  apply(dataCheckTuples, 1, function(tuple) {
+    test_that(paste(tuple[[1]], " does not contain the correct data"), {
+      expect_equal(digest(trunc(sum(answerX.X[tuple[[1]]]) * as.double(tuple[[2]]))),
+                   tuple[[3]])
+    })
+  })
+  
+  
+  
+  
+  print("Success!")
+}
+
+
+
+
+check_numeric <- function(answerX.X, precision, expectedHash) {
+  var_name <- deparse(substitute(answerX.X))
+  test_that(paste('Did not assign answer to an object called ', var_name), {
+    expect_true(exists(var_name))
+  })
+  
+  answer_as_numeric <- as.numeric(answerX.X)
+  test_that(paste(var_name, " should be a number"), {
+    expect_false(is.na(answer_as_numeric))
+  })
+  
+  test_that(paste(var_name, " value is incorrect"), {
+    expect_equal(digest(as.integer(answer_as_numeric * precision)), expectedHash)
+  })
+  
+  print("Success!")
+}
+
+
+
+check_numeric_element <- function(answerX.X, precision, expectedHash) {
+  var_name <- deparse(substitute(answerX.X))
+  
+  answer_as_numeric <- as.numeric(answerX.X)
+  test_that(paste(var_name, " should be a number"), {
+    expect_false(is.na(answer_as_numeric))
+  })
+  
+  test_that(paste(var_name, " value is incorrect"), {
+    expect_equal(digest(as.integer(answer_as_numeric * precision)), expectedHash)
+  })
+  
+  print("Success!")
+}
+
+
+
+
+check_plot <- function(answerX.X, x_axis_var, geom_type, hasVline, bin_width_hash, nrow_hash,
+                       x_axis_var_hash, hasTitle, isFactor, shouldRename) {
+  var_name <- deparse(substitute(answerX.X))
+  test_that(paste('Did not assign answer to an object called ', var_name), {
+    expect_true(exists(var_name))
+  })
+  
+  
+  test_that("Solution should be a ggplot object", {
+    expect_true(is.ggplot(answerX.X))
+  })
+  
+  properties <- c(answerX.X$layers[[1]]$mapping, answerX.X$mapping)
+  
+  test_that(paste("Plot should have ", x_axis_var," on the x-axis"), {
+    expect_true(x_axis_var == rlang::get_expr(properties$x))
+  })
+  
+  test_that("Plot does not have the correct layers", {
+    expect_true(geom_type %in% class(answerX.X$layers[[1]]$geom))
+    
+    if(hasVline) {
+      expect_true("GeomVline" %in% class(answerX.X$layers[[2]]$geom))
+    }
+  })
+  
+  test_that("Plot does not have the correct bin width", {
+    expect_equal(
+      digest(as.integer(mget("stat_params", answerX.X$layers[[1]])[["stat_params"]][["binwidth"]])),
+      bin_width_hash)
+  })
+  
+  test_that("Plot does not use the correct data", {
+    expect_equal(digest(nrow(answerX.X$data)), nrow_hash)
+    if (!isFactor) {
+      expect_equal(digest(round(sum(answerX.X$data[x_axis_var]))), x_axis_var_hash)
+    }
+    # If X_AXIS_VAR is not known:
+    # expect_equal(digest(round(sum(pull(answerX.X$data, rlang::get_expr(properties$x))))), "HASH_HERE")
+  })
+  
+  if (shouldRename) {
+    test_that("x-axis label should be descriptive and human readable", {
+      expect_false(answerX.X$labels$x == toString(rlang::get_expr(properties$x)))
+    })
+  }
+  
+  if(hasTitle){
+    
+    test_that("Plot should have a title", {
+      expect_true("title" %in% names(answerX.X$labels))
+    })
+  }
+  
+  
+  print("Success!")
+}
+
+
+
+
+
+
+
+
+
+
+get_plot_params <- function(answerX.X, isFactor) {
+  
+  properties <- c(answerX.X$layers[[1]]$mapping, answerX.X$mapping)
+  x_axis_var <- as.character(rlang::get_expr(properties$x))
+  print(x_axis_var)
+  print(class(answerX.X$layers[[1]]$geom))
+  if (length(answerX.X$layers) > 1) {
+    print("GeomVline" %in% class(answerX.X$layers[[2]]$geom))
+  } else {
+    print(FALSE)
+  }
+  print(digest(as.integer(mget("stat_params", answerX.X$layers[[1]])[["stat_params"]][["binwidth"]])))
+  print(digest(nrow(answerX.X$data)))
+  if(!isFactor) {
+    print(digest(round(sum(pull(answerX.X$data, x_axis_var)))))
+  } else {
+    print("none")
+  }
+  print("title" %in% names(answerX.X$labels))
+  print(isFactor)
+  print(answerX.X$labels$x != toString(rlang::get_expr(properties$x)))
+}
+
+
+get_DF_params <- function(df, precisionList) {
+  cols <- colnames(df)
+  print(cols)
+  print(digest(as.integer(nrow(df))))
+  print(cols)
+  print(precisionList)
+  for (i in 1: length(precisionList)) {
+    print(digest(as.integer(sum(df[cols[i]],na.rm=T) * as.double(precisionList[[i]]))))
+  }
+  
+}
+
+
+get_DF_params_overflow <- function(df, precisionList) {
+  cols <- colnames(df)
+  print(cols)
+  print(digest(as.integer(nrow(df))))
+  print(cols)
+  print(precisionList)
+  for (i in 1: length(precisionList)) {
+    print(digest(trunc(sum(df[cols[i]],na.rm=T) * as.double(precisionList[[i]]))))
+  }
+  
+}
+
+
+get_numeric_element_params <- function(answerX.X, precision) {
+  print(precision)
+  answer_as_numeric <- as.numeric(answerX.X)
+  print(digest(as.integer(answer_as_numeric * precision)))
+}
+
+#-------------
+#--
+#Question 1.0
+
+test_1.0 <- function() {
+  check_numeric_element(Ftest_3genes_add_full$F[2], 
+                        100, 
+                        "cc18eca3d186eb8772b9768990dcd2ea")
+}
+
+#--
+
+#Question 1.1
+
+test_1.1  <- function() {
+  test_that('Did not assign answer to an object called "answer1.1"', {
+    expect_true(exists("answer1.1"))
+  })
+  
+  test_that('Solution should be a single character ("A", "B", "C", "D")', {
+    expect_match(answer1.1, "a|b|c|d", ignore.case = TRUE)
+  })
+  
+  answer_hash <- digest(tolower(answer1.1 ))
+  
+  test_that("Solution is incorrect", {
+    expect_equal(answer_hash, "ddf100612805359cd81fdc5ce3b9fbba")
+  })
+  
+  print("You are doing great!")
+}
+                            
+#----
+# Question 1.2
+                            
+test_1.2.0 <- function() {
+  check_DF(add_mrna_results,
+           c("term","estimate",  "std.error", "statistic", "p.value", "conf.low", "conf.high"),
+           "234a2a5581872457b9fe1187d1616b13",
+           c("estimate",  "std.error", "statistic", "p.value" , "conf.low", "conf.high"),
+           c(1e6,1e6,100 ,10, 1e6,1e6),
+           c("fe063eefeb8dbe2c9ee184361b79093e",
+             "0426791560fff8db010b19ea3af7b77f",
+             "61a5f7bea3f303f179bfedd59bdc6f52",
+             "8eaca7c9b35d05ab15c9125bc92372fa",
+             "523bd10bd34507591ddd319eef7eae74", 
+             "a152274bbc908e5f9ad240243435e40a"
+           ))
+}
+                          
+                            
+test_1.2.1 <- function() {
+  check_numeric_element(Ftest_3genes_add_mrna$F[2], 
+                        100, 
+                        "9487172f214b56a933ed69dff6a5e934")
+}
+                            
+
+# Question 1.4
+
+test_1.4 <- function() {
+    check_MC(answer1.4, LETTERS[1:4], "ddf100612805359cd81fdc5ce3b9fbba")
+}
+
+# +
+
+# Question 2.0
+
+test_2.0 <- function() {
+    check_DF(training_fat,
+            c("brozek", "age", "weight", "height", "adipos",
+              "neck", "chest", "abdom", "hip", "thigh", "knee", 
+              "ankle", "biceps", "forearm", "wrist"),
+ "10ed7b15ca812930d29ea06c2edde7ef",
+ c("brozek", "age", "weight", "height"),
+ c(10, 1, 100, 100),
+ c("37ee509010689796f56d14d2db0400a3", "518b8d2d85113806a4693c85b804cdb1", 
+   "455968a44152801cb167157de7d85536", "1eed1135ffda908402ec8f5527c07f09"))
+    
+    
+        check_DF(second_set_fat,
+            c("brozek", "age", "weight", "height", "adipos",
+              "neck", "chest", "abdom", "hip", "thigh", "knee", 
+              "ankle", "biceps", "forearm", "wrist"),
+ "d56c4f7c4ce276c7e47e1fa52e70d4b6",
+ c("adipos", "neck", "chest", "abdom"),
+ c(10, 10, 10, 10),
+ c("0582f7ac003f2e854d427f0609cdd5eb", "0d39bb97b55d25ead6f7f807c838542e", 
+   "4f4bc94700a77c2f211abd3dce17ef7e", "e4bd80d768c875a2eb5bbde6eced7f8b"))
+}
+
+# +
+
+
+# Question 2.1
+
+test_2.1 <- function() {
+  check_DF(fat_bwd_summary_df,
+           c("n_input_variables","RSQ", "RSS", "ADJ.R2"),
+           "06cd248dd1409b804444bd9ad5533d1d",
+           c("RSQ", "RSS", "ADJ.R2"),
+           c(1000, 1000, 1000),
+           c("dd578c83d1356e2363859f0ca649f95d","8807c4a844c7de0775a0e3120ca60499", "75ff97ee9d21884c2e0f1206b3b3c399"))
+}
+
+# +
+# Question 2.2
+
+test_2.2 <- function() {
+  check_MC(answer2.2, LETTERS[1:14], "524dee37861cc9bab42227df389509af")
+}
+
+# Question 2.3
+
+test_2.3 <- function() {
+  check_numeric_element(sum(fat_bwd_generative$coefficients),1e4,"f7134397f6afa181af608fffc309e902")
+}
+
+
+# Question 2.4
+
+test_2.4 <- function() {
+  check_numeric_element(answer2.4,1e4,"caa4da3fe2cad4ebe9b4929a9d486462")
+}
diff --git a/materials/tutorial_07/tutorial_07.ipynb b/materials/tutorial_07/tutorial_07.ipynb
new file mode 100644
index 0000000..d339f79
--- /dev/null
+++ b/materials/tutorial_07/tutorial_07.ipynb
@@ -0,0 +1,1424 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "04ce4b2a70add7b1b34c1dc9b8329326",
+     "grade": false,
+     "grade_id": "cell-f1e1d845873036f4",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    }
+   },
+   "source": [
+    "# Tutorial 7: Model Evaluation and Model Selection"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "90c5778757b86e6aa2177ebfaab7f614",
+     "grade": false,
+     "grade_id": "cell-82d9926086d47a80",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    }
+   },
+   "source": [
+    "#### Lecture and Tutorial Learning Goals:\n",
+    "After completing this week's lecture and tutorial work, you will be able to:\n",
+    "\n",
+    "1. List model metrics that are suitable for evaluation of a statistical model developed to make inference about the data-generating mechanism (e.g., $R^2$, $\\text{AIC}$, Likelihood ratio test/$F$-test), their strengths and limitations, as well as how they are calculated.\n",
+    "2. Write a computer script to calculate these model metrics. Interpret and communicate the results from that computer script.\n",
+    "3. Explain a variable selection method based on:\n",
+    "    - $F$-test to compare nested models.\n",
+    "    - RSS for models of equal size\n",
+    "    - Adjusted $R^2$ for models of different sizes"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "049f5b181c8cba658955ed573f8c4c58",
+     "grade": false,
+     "grade_id": "cell-a2a153352bc44a68",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# Run this cell before continuing.\n",
+    "library(tidyverse)\n",
+    "library(repr)\n",
+    "library(digest)\n",
+    "library(infer)\n",
+    "library(gridExtra)\n",
+    "library(faraway)\n",
+    "library(broom)\n",
+    "library(leaps)\n",
+    "library(mltools)\n",
+    "source(\"tests_tutorial_07.R\")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "e75543fba6e00d97b98b3fffa5f2c09f",
+     "grade": false,
+     "grade_id": "cell-93ac8765d809417c",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "## Can we predict protein from mRNA?\n",
+    "\n",
+    "In worksheet_07 you studied the significance of `mrna` and analyzed the goodness-of-fit of some models. However, there are other models that can be compared. For example, are interactions terms needed or is it equivalent to consider an additive model? "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "76c9c8ad5318480e67f9d80f4ae641ba",
+     "grade": false,
+     "grade_id": "cell-58724295b3683c74",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "Consider the following models using a dataset with 3 randomly selected genes:\n",
+    "\n",
+    "- model.1: $\\text{prot}_t=\\beta_0 + \\varepsilon_t$ \n",
+    "\n",
+    "- model.2:  $\\text{prot}_t=\\beta_0 + \\beta_1 \\text{mrna}_{t} + \\varepsilon_t$ \n",
+    "\n",
+    "- model.3:  $\\text{prot}_t=\\beta_0 + \\beta_2 \\text{gene2}_{t} + \\beta_3 \\text{gene3}_{t} + \\varepsilon_t$ \n",
+    "\n",
+    "- model.4:  $\\text{prot}_t=\\beta_0 + \\beta_1 \\text{mrna}_{t} + \\beta_2 \\text{gene2}_{t} + \\beta_3 \\text{gene3}_{t} + \\varepsilon_t$ \n",
+    "\n",
+    "- model.5:  $\\text{prot}_t=\\beta_0 + \\beta_1 \\text{mrna}_{t} + \\beta_2 \\text{gene2}_{t} + \\beta_3 \\text{gene3}_{t} + \\beta_4 \\text{gene2}_{t}\\text{mrna}_{t} + \\beta_5 \\text{gene3}_{t}\\text{mrna}_{t} + \\varepsilon_t$ "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "670a8b9ab5427255445e5543e0daf4df",
+     "grade": false,
+     "grade_id": "cell-616b8ab2f1452495",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# Read and take a look at the data.\n",
+    "dat_bio <- read.csv(\"data/nature_dat.csv\", row.names = 1, stringsAsFactors=TRUE)\n",
+    "str(dat_bio)\n",
+    "head(dat_bio,3)\n",
+    "tail(dat_bio,3)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "fd6ba13bf520cc907cda42e60d0cc87e",
+     "grade": false,
+     "grade_id": "cell-e4e36c63c003122c",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "#run this cell\n",
+    "set.seed(561)\n",
+    "dat_3genes <- dat_bio  %>%  \n",
+    "         subset(gene %in% sample(gene,3)) "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "d7826afed0a12b6b00a85c3e9c1d1b6c",
+     "grade": false,
+     "grade_id": "cell-6c722e3a372a3114",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "**Question 1.0**\n",
+    "<br>{points: 1}\n",
+    "\n",
+    "Test if the model with interaction terms (`model.5`) is significantly different from an additive one (`model.4`). Note that both models have `mrna` but the full model assumes that the change in protein levels per unit change in mRNA levels is different for each gene.\n",
+    "\n",
+    "Store your results in an object called `Ftest_3genes_add_full`.\n",
+    "\n",
+    "*Write your own code and run it.*"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "e4f65db77490141a616c042ae8dd4d5c",
+     "grade": false,
+     "grade_id": "cell-2d5e8281caac870d",
+     "locked": false,
+     "schema_version": 3,
+     "solution": true,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "#[write your code here]\n",
+    "#Ftest_3genes_add_full\n",
+    "\n",
+    "# your code here\n",
+    "fail() # No Answer - remove if you provide an answer"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "f1fb11b3274586f0b9d7d887c5c73bb8",
+     "grade": true,
+     "grade_id": "cell-9520504e5a91abed",
+     "locked": true,
+     "points": 1,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "test_1.0()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "43a49f718713b9a7c2c49408b89c68af",
+     "grade": false,
+     "grade_id": "cell-633de4d02e393e13",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "**Question 1.1**\n",
+    "<br>{points: 1}\n",
+    "\n",
+    "Using a significance level $\\alpha = 0.05$ and the results in `Ftest_3genes_add_full`, in plain words, what is the conclusion from the results of the test run in **Question 1.0**?\n",
+    "\n",
+    "**A.** We reject the null hypothesis; thus, the *full* model is significatly better than the *reduced* model.\n",
+    "\n",
+    "**B.** We fail to reject the null hypothesis; thus, there is not enough evidence that the *full* model with additional interaction terms is better than the additive (reduced) model.\n",
+    "\n",
+    "**C.** We accept the alternative hypothesis; thus, the *full* model is significantly better than the *reduced* model.\n",
+    "\n",
+    "**D.** We do not accept the alternative hypothesis; thus, the *full* model with additional interaction terms is not better than the *reduced* model.\n",
+    "\n",
+    "*Assign your answer to an object called `answer1.1`. Your answer should be one of `\"A\"`, `\"B\"`, `\"C\"`, or `\"D\"` surrounded by quotes.*"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "c654b0916955f810231633dfe148c244",
+     "grade": false,
+     "grade_id": "cell-672ea5acf3fcd97f",
+     "locked": false,
+     "schema_version": 3,
+     "solution": true,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# answer1.1 <- \n",
+    "\n",
+    "# your code here\n",
+    "fail() # No Answer - remove if you provide an answer"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "811ec81626a1926884a1ae32ec0524ce",
+     "grade": true,
+     "grade_id": "cell-f510cd4a1479fd83",
+     "locked": true,
+     "points": 1,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "test_1.1()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "80f37c7b193be99f1e6e0e2b6e3d619d",
+     "grade": false,
+     "grade_id": "cell-73ad156d14319434",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "#### Assessing mRNA in the additive model\n",
+    "\n",
+    "As a final test, let's examine the significance of `mrna` in the additive model."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "c01f6dded74091da16a61fbeb0d5633f",
+     "grade": false,
+     "grade_id": "cell-95e74ec3f56b4208",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "**Question 1.2**\n",
+    "<br>{points: 2}\n",
+    "\n",
+    "Compare the additive model with mRNA as an input and distinct intercepts per gene  (`model.4`) with a model without `mrna` and only the categorical variable `gene` as input variable (`model.3`). Note that the second model predicts protein levels with the average protein level within each gene!!\n",
+    "\n",
+    "Use the function `tidy()` to obtain a summary of the additive model. Include the corresponding asymptotic 90% confidence intervals. Store the results in an object called `add_mrna_results`.\n",
+    "\n",
+    "Use the function `anova()` to compare these models and store the results in an object called `Ftest_3genes_add_mrna`.\n",
+    "\n",
+    "*Fill out those parts indicated with `...`, uncomment the corresponding code in the cell below, and run it.*"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "1f1ed5b3084ea542d7c91d7e7a1ed206",
+     "grade": false,
+     "grade_id": "cell-620823d0f6deab4f",
+     "locked": false,
+     "schema_version": 3,
+     "solution": true,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "#[write your code here]\n",
+    "\n",
+    "#add_mrna_results <- ...\n",
+    "#Ftest_3genes_add_mrna <- ...\n",
+    "\n",
+    "# your code here\n",
+    "fail() # No Answer - remove if you provide an answer"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "2700f1df14ee1375ee05dd2f6f323e26",
+     "grade": true,
+     "grade_id": "cell-9b6acf3241aeeb8e",
+     "locked": true,
+     "points": 2,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "test_1.2.0()\n",
+    "test_1.2.1()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "eab9d791c2502a60fcec95ab41a6e6e1",
+     "grade": false,
+     "grade_id": "cell-2278371543391218",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "**Question 1.3**\n",
+    "<br>{points: 2}\n",
+    "\n",
+    "Compare the $p$-value for `mrna` in `add_mrna_results` with that reported in `Ftest_3genes_add_mrna`. What do you observe? Indicate the null hypotheses tested in each case and explain the results."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "f0f24c741ebf551d2ce4b09f3b6a0e1f",
+     "grade": true,
+     "grade_id": "cell-46d78ce68383a2a1",
+     "locked": false,
+     "points": 2,
+     "schema_version": 3,
+     "solution": true,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "> *Your explanation of the results goes here.*\n",
+    "\n",
+    "DOUBLE CLICK TO EDIT **THIS CELL** AND REPLACE THIS TEXT WITH YOUR ANSWER."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "0253b0195de4d848fa6f63bb98de0d1a",
+     "grade": false,
+     "grade_id": "cell-c096bd3acbd91392",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    }
+   },
+   "source": [
+    "**Question 1.4**\n",
+    "<br>{points: 1}\n",
+    "\n",
+    "Using a **significance level $\\alpha = 0.10$** and the results in `add_mrna_results`, which of the following claims is correct? \n",
+    "\n",
+    "**A.** The `model.4` that includes `mrna` is significatly different from `model.3`.\n",
+    "\n",
+    "**B.** There is not enough evidence that the `model.4` that includes `mrna` as a predictor is significatly better than `model.3`.\n",
+    "\n",
+    "**C.** The `model.4` that includes `mrna` as a predictor is equivalent to `model.3` since the coefficient for `mrna` is not significantly different from zero.\n",
+    "\n",
+    "**D.** The variable `mrna` is essencial to predict protein levels.\n",
+    "\n",
+    "*Assign your answer to an object called `answer1.4`. Your answer should be one of `\"A\"`, `\"B\"`, `\"C\"`, or `\"D\"` surrounded by quotes.*"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "f923f45f08e4b5253c508a2fbf301563",
+     "grade": false,
+     "grade_id": "cell-526e7a36d70d7656",
+     "locked": false,
+     "schema_version": 3,
+     "solution": true,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# answer1.4 <- \n",
+    "\n",
+    "# your code here\n",
+    "fail() # No Answer - remove if you provide an answer"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "a19dd2e3dcbac430d9aa8d65e9044296",
+     "grade": true,
+     "grade_id": "cell-eaee9f2280c396a0",
+     "locked": true,
+     "points": 1,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "test_1.4()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "e2766e39287ee322f07995a4504db3f0",
+     "grade": false,
+     "grade_id": "cell-1aacaa45a46be49a",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    }
+   },
+   "source": [
+    "**Question 1.5**\n",
+    "<br>{points: 4}\n",
+    "\n",
+    "a) We can use the adjusted $R^2$ to compare the goodness-of-fit of `model.5` and `model.4` to conclude which one fits the data better. Use the function `glance()` to obtain these values and discuss the results obtained. \n",
+    "\n",
+    "b) Compare the $R^2$ for both models and explain why that of `model.5` is larger."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "798c513b8a42b7ab0cc50bfead4cb95f",
+     "grade": false,
+     "grade_id": "cell-8949d93baadc071a",
+     "locked": false,
+     "schema_version": 3,
+     "solution": true,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# Your code and numerical results go here. We will grade this cell manually\n",
+    "\n",
+    "# your code here\n",
+    "fail() # No Answer - remove if you provide an answer"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "0959dcbbe4b562e5053fff0e9504dfd1",
+     "grade": true,
+     "grade_id": "cell-f810ee382faddccf",
+     "locked": false,
+     "points": 4,
+     "schema_version": 3,
+     "solution": true,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "> *Your explanation of the results goes here.*\n",
+    "\n",
+    "DOUBLE CLICK TO EDIT **THIS CELL** AND REPLACE THIS TEXT WITH YOUR ANSWER."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "e14d0a248c46554caf6cdd33a4274afc",
+     "grade": false,
+     "grade_id": "cell-780f1094bba1bb0f",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "# 2. Model selection\n",
+    "\n",
+    "In this section you will use the **backward selection** algorithm to construct a generative model. \n",
+    "\n",
+    "> Note that the choice of which algorithm to use in each case was arbitrary and you can play with these algorithms and try other choices!"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "23c54e57b3d4bd9e3eee868770609def",
+     "grade": false,
+     "grade_id": "cell-5db1ec8ab7f6c357",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "Let us start by loading the dataset to be used throughout this tutorial. We will use the dataset `fat` from the library `faraway`. You can find detailed information about it in [Johnson (1996)](https://www.tandfonline.com/doi/full/10.1080/10691898.1996.11910505). This dataset contains the percentage of body fat and a whole variety of body measurements (continuous variables) of 252 men. We will use the variable `brozek` as the response variable and a subset 14 variables to build different models. \n",
+    "\n",
+    "The response variable `brozek` is the percent of body fat using Brozek's equation:\n",
+    "\n",
+    "$$\\texttt{brozek} = \\frac{457}{\\texttt{density}} - 414.2,$$\n",
+    "\n",
+    "where body `density` is measured in $\\text{g}/\\text{cm}^3$.\n",
+    "\n",
+    "The 14 input variables are:\n",
+    "\n",
+    "- `age`: Age in $\\text{years}$.\n",
+    "- `weight`: Weight in $\\text{lb}$.\n",
+    "- `height`: Height in $\\text{in}$.\n",
+    "- `adipos`: Adiposity index in $\\text{kg}/\\text{m}^2$.\n",
+    "\n",
+    "$$\\texttt{adipos} = \\frac{\\texttt{weight}}{\\texttt{height}^2}$$\n",
+    "\n",
+    "- `neck`: Neck circumference in $\\text{cm}$.\n",
+    "- `chest`: Chest circumference in $\\text{cm}$.\n",
+    "- `abdom`: Abdomen circumference at the umbilicus and level with the iliac crest in $\\text{cm}$.\n",
+    "- `hip`: Hip circumference in $\\text{cm}$.\n",
+    "- `thigh`: Thigh circumference in $\\text{cm}$.\n",
+    "- `knee`: Knee circumference in $\\text{cm}$.\n",
+    "- `ankle`: Ankle circumference in $\\text{cm}$.\n",
+    "- `biceps`: Extended biceps circumference in $\\text{cm}$.\n",
+    "- `forearm`: Forearm circumference in $\\text{cm}$.\n",
+    "- `wrist`: Wrist circumference distal to the styloid processes in $\\text{cm}$.\n",
+    "\n",
+    "Run the code below to create the working data frame called `fat_sample`."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "7565e74c203a8b63c5e873aec7837799",
+     "grade": false,
+     "grade_id": "cell-03998f31dc8342ba",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# run this cell\n",
+    "\n",
+    "fat_sample <- fat %>%\n",
+    "  select(\n",
+    "    brozek, age, weight, height, adipos, neck, chest, abdom,\n",
+    "    hip, thigh, knee, ankle, biceps, forearm, wrist\n",
+    "  )\n",
+    "\n",
+    "head(fat_sample,3)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "0ff3785d791913bc000a680e1f41a971",
+     "grade": false,
+     "grade_id": "cell-5dcb4d940b3e62e9",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "### Selecting a generative model\n",
+    "\n",
+    "Although many potential input variables are available in the dataset, not all may be relevant to explain the variation of the response variable. The *subset* algorithms learned in the lecture can be used to select a subset of variable to build generative models. \n",
+    "\n",
+    "Generative models are built and trained to examine the association between the response and the input variables. \n",
+    "\n",
+    "Since the same data can not be used to select and to make inference, we need 2 different datasets: a *selection* set and a *training* set. If two independent datasets are not available to select and build a generative model, we can use the data in hand and split it to create these datasets. \n",
+    "\n",
+    "> using the same data to select and estimate violates the assumptions of the analysis and invalidate inference results. This problem is known as a *post-inference* problem and will be further discuss in future lectures. \n",
+    "\n",
+    "In the following questions, we will use the *backward* selection algorithm and the *adjusted* $R^2$ to select a smaller model. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "d97be6b08f10914bbc8b5c1a98910052",
+     "grade": false,
+     "grade_id": "cell-36a56e9d55bdb89f",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "**Question 2.0**\n",
+    "<br>{points: 1}\n",
+    "\n",
+    "Let's start by randomly splitting `fat_sample` in two sets on a 70-30% basis: `training_fat` (70% of the data) and `second_set_fat` (the remaining 30%). The selection will be done using the `second_set_fat` dataset and the model will be built using the `training_fat` data.\n",
+    "\n",
+    "Follow the next 3 steps to complete the code below:\n",
+    "\n",
+    "1. Create an `ID` column in `fat_sample` (i.e., `fat_sample$ID`) with the row number corresponding to each man in the sample.\n",
+    "\n",
+    "2. Use the function `sample_n()` to create `training_fat` (sampling *without* replacement) with 70\\% of the observations coming from `fat_sample`.\n",
+    "\n",
+    "3. Use `anti_join()` with `fat_sample` and `training_fat` to create `second_set_fat` by column `ID`.\n",
+    "\n",
+    "4. Remove the variable `ID` used to split the data\n",
+    "\n",
+    "*Fill out those parts indicated with `...`, uncomment the corresponding code in the cell below, and run it.*"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "5f31f55c1e2cdbc6b3e88c57fdb3857e",
+     "grade": false,
+     "grade_id": "cell-88095657b7ef5e0d",
+     "locked": false,
+     "schema_version": 3,
+     "solution": true,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "set.seed(123) # DO NOT CHANGE!\n",
+    "\n",
+    "# fat_sample$ID <- rownames(fat_sample)\n",
+    "# training_fat <- ...(..., size = nrow(fat_sample) * 0.70,\n",
+    "#   replace = ...\n",
+    "# )\n",
+    "\n",
+    "# second_set_fat <- anti_join(...,\n",
+    "#   ...,\n",
+    "#   by = ...\n",
+    "# )\n",
+    "\n",
+    "# training_fat <- training_fat  %>% select(-\"ID\")\n",
+    "# second_set_fat <- second_set_fat %>% select(-\"ID\")\n",
+    "\n",
+    "# head(training_fat)\n",
+    "# nrow(training_fat)\n",
+    "\n",
+    "# head(second_set_fat)\n",
+    "# nrow(second_set_fat)\n",
+    "\n",
+    "# your code here\n",
+    "fail() # No Answer - remove if you provide an answer"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "205bb8c19193b79ccc376f7458714fdb",
+     "grade": true,
+     "grade_id": "cell-a648a9e5540d72b0",
+     "locked": true,
+     "points": 1,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "test_2.0()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "5046160daadcab1ec6c1647042e78b79",
+     "grade": false,
+     "grade_id": "cell-78e3bba87f64f63a",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "**Question 2.1**\n",
+    "<br>{points: 1}\n",
+    "\n",
+    "Using only the extra data in `second_set_fat`, select a reduced LR using the **backward selection** algorithm. Recall that this method is implemented in the function `regsubsets()` from library `leaps`.\n",
+    "\n",
+    "The function `regsubsets()` identifies various subsets of input variables selected for models of different sizes. The argument `x` of `regsubsets()` is analogous to `formula` in `lm()`. \n",
+    "\n",
+    "Create one object using `regsubsets()`with `second_set_fat` and call it `fat_backward_sel`. We will use `fat_bwd_summary_df` to check the results.\n",
+    "\n",
+    "**Maintain any ordering of columns seen in `second_set_fat`**\n",
+    "\n",
+    "*Fill out those parts indicated with `...`, uncomment the corresponding code in the cell below, and run it.*"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "e620348547a3cedcfd5d90fe3b596bb5",
+     "grade": false,
+     "grade_id": "cell-9dab19aa23c8ddf6",
+     "locked": false,
+     "schema_version": 3,
+     "solution": true,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# fat_backward_sel <- ...(\n",
+    "#   x=..., \n",
+    "#   nvmax=...,\n",
+    "#   data=...,\n",
+    "#   method=...,\n",
+    "# )\n",
+    "# fat_backward_sel\n",
+    "\n",
+    "#fat_bwd_summary <- summary(fat_backward_sel)\n",
+    "\n",
+    "#fat_bwd_summary_df <- data.frame(\n",
+    "#    n_input_variables = 1:14,\n",
+    "#    RSQ = fat_bwd_summary$rsq,\n",
+    "#    RSS = fat_bwd_summary$rss,\n",
+    "#    ADJ.R2 = fat_bwd_summary$adjr2\n",
+    "#)\n",
+    "\n",
+    "# your code here\n",
+    "fail() # No Answer - remove if you provide an answer"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "4050e053469faec8c42a80963fd28c93",
+     "grade": true,
+     "grade_id": "cell-1834e0983d6a1794",
+     "locked": true,
+     "points": 1,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "test_2.1()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "295442c7f109b6b3413fdc4d601f81bf",
+     "grade": false,
+     "grade_id": "cell-afd9a48edcdcaab0",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "**Question 2.2**\n",
+    "<br>{points: 1}\n",
+    "\n",
+    "The *backward* subset algorithm selected the best model of each size. Results of the 14 models selected are stored in `fat_bwd_summary`. \n",
+    "\n",
+    "Use the *adjusted* $R^2$ of these 14 models, stored in `fat_bwd_summary_df`, to select the best generative model and indicate which input variables are in the selected model.\n",
+    "\n",
+    "**A.** `age`.\n",
+    "\n",
+    "**B.** `weight`.\n",
+    "\n",
+    "**C.** `height`.\n",
+    "\n",
+    "**D.** `adipos`.\n",
+    "\n",
+    "**E.**  `neck`.\n",
+    "\n",
+    "**F.**  `chest`.\n",
+    "\n",
+    "**G.**  `abdom`.\n",
+    "\n",
+    "**H.**  `hip`.\n",
+    "\n",
+    "**I.**  `thigh`.\n",
+    "\n",
+    "**J.**  `knee`.\n",
+    "\n",
+    "**K.**  `ankle`.\n",
+    "\n",
+    "**L.**  `biceps`.\n",
+    "\n",
+    "**M.**  `forearm`.\n",
+    "\n",
+    "**N.**  `wrist`.\n",
+    "\n",
+    "*Assign your answers to the object `answer2.2`. Your answers have to be included in a single string indicating the correct options **in alphabetical order** and surrounded by quotes.*"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "7261f0d665286dfc3c98f0b19e5501f3",
+     "grade": false,
+     "grade_id": "cell-45de567764bea01e",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "#Run this cell before continuing to examine the results\n",
+    "\n",
+    "fat_bwd_summary_df\n",
+    "\n",
+    "fat_bwd_summary "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "40740c833fd0404a782c1282b8bffb54",
+     "grade": false,
+     "grade_id": "cell-6cebf1be4cc59565",
+     "locked": false,
+     "schema_version": 3,
+     "solution": true,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# answer2.2 <- \n",
+    "\n",
+    "# your code here\n",
+    "fail() # No Answer - remove if you provide an answer"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "47dbc102d3b3a6b9319f41c24b6985c6",
+     "grade": true,
+     "grade_id": "cell-1cf2ff609e71bfe7",
+     "locked": true,
+     "points": 1,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "test_2.2()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "e5950487d130eddcb387cd4cf99e66bd",
+     "grade": false,
+     "grade_id": "cell-2f834801ab2b709d",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "**Question 2.3**\n",
+    "<br>{points: 1}\n",
+    "\n",
+    "Now that you have selected a subset of input variables, use the independent dataset `training_fat` to build and evaluate a *generative* model. \n",
+    "\n",
+    "Use `lm` to fit the selected model using `training_fat`, and store the results in an object called `fat_bwd_generative`. \n",
+    "\n",
+    "> Enter the selected variables in the **same order** as they are in `training_fat`. This is not statistically needed, it's only needed to autograde this question.\n",
+    "\n",
+    "*Fill out those parts indicated with `...`, uncomment the corresponding code in the cell below, and run it.*"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "f7786bd61b2e160051fe8a5944654e74",
+     "grade": false,
+     "grade_id": "cell-b903605318ff9e7d",
+     "locked": false,
+     "schema_version": 3,
+     "solution": true,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# fat_bwd_generative <- ...(...,\n",
+    "#   ...\n",
+    "# )\n",
+    "\n",
+    "# tidy(fat_bwd_generative)\n",
+    "\n",
+    "# your code here\n",
+    "fail() # No Answer - remove if you provide an answer"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "7359bd255da61317482f710128be1385",
+     "grade": true,
+     "grade_id": "cell-79c41ee8723c1697",
+     "locked": true,
+     "points": 1,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "test_2.3()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "777448920f7a213d68372f574f4e7440",
+     "grade": false,
+     "grade_id": "cell-9df046a01453d22d",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "**Question 2.4**\n",
+    "<br>{points: 1}\n",
+    "\n",
+    "Compute the coefficient of determination $R^2$ to evaluate the goodness of fit of the model.\n",
+    "\n",
+    "> Note that the evaluation is also based on data from `training_fat`\n",
+    "\n",
+    "*Assign your answer to the object `answer2.4`. Your answer is a numeric object*"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "jp-MarkdownHeadingCollapsed": true,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "1bfa4e87c48d0f1663dbe1b57f3f1695",
+     "grade": false,
+     "grade_id": "cell-785ec8fc3e384cfe",
+     "locked": false,
+     "schema_version": 3,
+     "solution": true,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# *Your code goes here.*\n",
+    "\n",
+    "# your code here\n",
+    "fail() # No Answer - remove if you provide an answer"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "a0b74b744901bf21f21445225d749898",
+     "grade": true,
+     "grade_id": "cell-4cd732f72421bbbd",
+     "locked": true,
+     "points": 1,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "test_2.4()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "ddb69f0046c0818f1ca28c8d085e09c9",
+     "grade": false,
+     "grade_id": "cell-e70fc3de3b0c3210",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "**Question 2.5**\n",
+    "<br>{points: 1}\n",
+    "\n",
+    "Interpret the coefficient of determination $R^2$ computed in **Question 2.4** and comment on the goodness-of-fit of the selected model."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "2cafc9bfd902d43f58f936c84a3cd1b7",
+     "grade": true,
+     "grade_id": "cell-eda9e018dea0aa5f",
+     "locked": false,
+     "points": 1,
+     "schema_version": 3,
+     "solution": true,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "> *Your answer goes here.*\n",
+    "\n",
+    "DOUBLE CLICK TO EDIT **THIS CELL** AND REPLACE THIS TEXT WITH YOUR ANSWER."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "61b98872493eb8f50c6ec0d26781f84a",
+     "grade": false,
+     "grade_id": "cell-d9c0ea56110e917e",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "**Question 2.6**\n",
+    "<br>{points: 2}\n",
+    "\n",
+    "Previous research has shown that while weight can be highly variable during the day and even across days, body circumference measurements (e.g., abdominal circumference) are more stable and better predictors of body fat. Using the results from **Question 2.3**, corroborate if the abdominal circumference, `abdom` is statistically (linearly) associated with the percent of body fat measured by `brozek` (at a significance level of 0.01). \n",
+    "\n",
+    "In your answer, include an interpretation of the estimated coefficients as well as the results of the $t$-tests reported using `tidy()`."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "cfd3b746c10549ec946c6a2277889c67",
+     "grade": true,
+     "grade_id": "cell-5ad112504dc133dd",
+     "locked": false,
+     "points": 2,
+     "schema_version": 3,
+     "solution": true,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "> *Your answer goes here.*\n",
+    "\n",
+    "DOUBLE CLICK TO EDIT **THIS CELL** AND REPLACE THIS TEXT WITH YOUR ANSWER."
+   ]
+  }
+ ],
+ "metadata": {
+  "jupytext": {
+   "formats": "ipynb,Rmd"
+  },
+  "kernelspec": {
+   "display_name": "R",
+   "language": "R",
+   "name": "ir"
+  },
+  "language_info": {
+   "codemirror_mode": "r",
+   "file_extension": ".r",
+   "mimetype": "text/x-r-source",
+   "name": "R",
+   "pygments_lexer": "r",
+   "version": "4.2.3"
+  },
+  "latex_envs": {
+   "LaTeX_envs_menu_present": true,
+   "autoclose": false,
+   "autocomplete": true,
+   "bibliofile": "biblio.bib",
+   "cite_by": "apalike",
+   "current_citInitial": 1,
+   "eqLabelWithNumbers": true,
+   "eqNumInitial": 1,
+   "hotkeys": {
+    "equation": "Ctrl-E",
+    "itemize": "Ctrl-I"
+   },
+   "labels_anchors": false,
+   "latex_user_defs": false,
+   "report_style_numbering": false,
+   "user_envs_cfg": false
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}
diff --git a/materials/worksheet_07/data/geneUN.csv b/materials/worksheet_07/data/geneUN.csv
new file mode 100644
index 0000000..05d4c02
--- /dev/null
+++ b/materials/worksheet_07/data/geneUN.csv
@@ -0,0 +1,6105 @@
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diff --git a/materials/worksheet_07/data/nature_dat.csv b/materials/worksheet_07/data/nature_dat.csv
new file mode 100644
index 0000000..56ccc06
--- /dev/null
+++ b/materials/worksheet_07/data/nature_dat.csv
@@ -0,0 +1,16705 @@
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diff --git a/materials/worksheet_07/data/proteinUN.csv b/materials/worksheet_07/data/proteinUN.csv
new file mode 100644
index 0000000..e36762c
--- /dev/null
+++ b/materials/worksheet_07/data/proteinUN.csv
@@ -0,0 +1,6105 @@
+"","uterus","kidney","testis","pancreas","stomach","prostate","ovary","thyroid.gland","adrenal.gland","salivary.gland","spleen","esophagus"
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+"ENSG00000254772",0.000156628676791528,0.000319185808723143,0.00041865079572126,0.000330597471770239,0.00102677555821708,0.000195731001378536,0.000285421262620475,0.000118661736628809,0.000297559642323244,0.000843029532825161,0.000724290703617392,9.28479274779445e-06
+"ENSG00000254901",NA,NA,NA,NA,NA,3.06992982409886e-06,2.95805419830496e-06,NA,9.37485570674261e-06,NA,NA,NA
+"ENSG00000254986",2.67374084167709e-05,8.66226959279835e-05,0.000184368763488906,7.40913633025127e-05,0.000417610378246589,1.61366927160043e-05,4.56299322557483e-05,4.55331317572747e-05,6.57165707779594e-05,0.000132960903322059,0.000105890812095614,0.000161500719688988
+"ENSG00000254999",NA,2.32490010978668e-06,2.13117707095536e-05,1.42936055819661e-05,NA,2.0464602480143e-05,7.85875265354763e-05,NA,NA,3.9454995392336e-05,0.00011166137205437,5.8919717109101e-06
+"ENSG00000255112",2.20983135790905e-05,7.90146496885292e-06,3.75452794382968e-05,1.85011964666097e-05,1.16743690203618e-05,1.90049439453801e-05,8.13823504866144e-05,NA,1.61287033445006e-05,2.79691669961294e-05,8.54055657661417e-05,1.77338622620482e-05
+"ENSG00000255529",NA,NA,NA,NA,NA,1.73139527504922e-06,8.88836114074216e-06,NA,4.90736360858961e-07,NA,NA,NA
+"ENSG00000255854",NA,NA,NA,NA,NA,NA,1.66887973842625e-05,NA,NA,1.17413638906474e-05,NA,NA
+"ENSG00000255974",NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA
+"ENSG00000256269",1.4512584017203e-05,NA,9.98109593721152e-06,7.63568672562721e-06,2.43153692447838e-05,1.34901192883106e-06,1.30684255782893e-05,1.06237703708687e-05,4.68667746930632e-06,2.77614345382498e-05,6.90359751904234e-05,1.17632615922795e-06
+"ENSG00000256671",NA,1.2066978751689e-05,NA,NA,NA,4.12734501360238e-05,1.44205249406342e-05,1.67021127451748e-05,8.13327421735823e-06,NA,NA,NA
+"ENSG00000257017",0.00143697600092269,0.00255239809610619,0.000932680964189982,0.0186458655361366,0.012775182236159,0.00100612216576816,0.00396320769068645,0.00873185175769863,0.00406100145414252,0.00475151259716892,0.00363668634609553,0.00683763863703773
+"ENSG00000257103",NA,NA,NA,NA,4.38389281275032e-07,NA,2.02563881823942e-05,NA,8.03171005428445e-07,4.61771314717171e-06,NA,NA
+"ENSG00000257218",NA,0.000105212744881322,NA,NA,NA,3.18251921895519e-06,2.90499636970046e-05,6.96586098241299e-06,6.33700195343143e-06,3.39227736077322e-05,NA,NA
+"ENSG00000257335",NA,0.000128367585346242,NA,NA,7.52835888153727e-07,NA,NA,1.3190378108988e-07,NA,NA,1.15643050174212e-06,NA
+"ENSG00000257365",4.41142336409771e-06,NA,1.99288636132841e-05,1.05338067824852e-05,1.37881024597006e-05,2.37504807004242e-05,NA,2.20388057475956e-06,7.61401636640987e-06,8.30308960972306e-06,2.46732311825204e-05,8.14289611107177e-06
+"ENSG00000257727",5.96492099717552e-05,8.29511680919271e-05,0.000154747160581474,3.19771792942301e-05,0.000297119363475965,2.19046563922425e-05,0.000259649488268229,2.43421550846027e-05,7.50675109514545e-05,0.000185723999298407,8.50268926775798e-05,7.59948806016488e-05
+"ENSG00000258315",8.01789993747301e-06,NA,1.84835507382508e-05,2.90024303835316e-05,2.20852119752451e-05,2.30753646117495e-07,7.69989879530804e-06,NA,1.27937458755891e-05,1.95530200226809e-05,6.93755597167847e-05,4.9836347910085e-05
+"ENSG00000258429",NA,4.57621557624407e-06,NA,NA,NA,NA,NA,NA,NA,5.14957724428095e-06,NA,NA
+"ENSG00000258790",NA,NA,NA,NA,6.80421258691225e-07,NA,NA,NA,3.24665130289725e-07,2.45436736713928e-07,NA,NA
+"ENSG00000258818",2.2458605988973e-06,2.68462293445416e-06,NA,NA,NA,2.23411018779769e-05,9.43618343983302e-06,5.08063693376031e-06,1.90985638526214e-06,2.07894835609596e-05,NA,NA
+"ENSG00000258947",1.44486735555717e-06,4.27560373715151e-06,0.000136406686669042,3.07879885943712e-06,1.09777923337448e-05,2.53271433236147e-05,1.66644345513782e-05,NA,6.26935834467121e-06,6.07582093547056e-05,NA,NA
+"ENSG00000259207",4.65781086927604e-06,1.89935788482455e-06,7.62715857802504e-07,6.67153987354037e-07,4.51493696169852e-06,1.82039435039137e-05,NA,2.98139174748215e-06,6.88407810375959e-06,NA,7.47362688085018e-05,8.35138370811316e-06
+"ENSG00000259431",4.21608105807867e-06,NA,1.12873927816311e-05,1.82531687139721e-06,NA,2.5071145712708e-05,4.17789187376735e-05,1.37710102327882e-06,9.01974760352885e-06,1.08820121131851e-05,NA,1.05516282998295e-05
+"ENSG00000259494",2.30253628052895e-06,3.61289578997788e-05,3.04786392795198e-06,NA,5.2438054307699e-07,1.15994137779559e-06,NA,NA,2.49928867914286e-05,1.53175563259087e-05,8.29491750087171e-07,5.18767396829048e-06
+"ENSG00000261052",1.77872754291307e-05,3.07366891513161e-06,NA,1.78473497710751e-05,0.0002128157872716,8.60033238206054e-06,1.81850745587321e-05,2.33384572853656e-05,7.74477661883237e-06,6.63662375167941e-05,0.000191368535356497,NA
+"ENSG00000261701",1.3953126223937e-05,1.43997785414367e-05,6.66772990188672e-06,0.00014558512879377,3.64543394283446e-05,4.7456743997269e-05,8.60137778640181e-05,5.33010727711072e-05,3.41806348675436e-05,6.45969239860696e-05,1.66772254968513e-05,5.98189503297597e-06
+"ENSG00000261857",NA,NA,1.57652229183893e-05,NA,NA,NA,NA,NA,NA,NA,NA,8.01445242115044e-07
+"ENSG00000262246",3.37735969299966e-06,6.14303262757448e-06,8.26094432071314e-05,2.32902621562161e-05,4.64283996351534e-05,5.8105094119534e-06,1.19604963813261e-05,3.61875259080628e-06,2.87480517520878e-05,8.30137228969356e-06,0.000213850000200494,3.33790157152527e-05
+"ENSG00000262664",NA,4.33111153652601e-06,3.19412788330745e-06,5.44901409126222e-06,7.85073624581154e-06,8.99246992478615e-06,1.77484969391053e-05,NA,7.37175690065529e-06,1.38898807766264e-05,8.64622301004584e-06,1.06144689020855e-05
+"ENSG00000262814",7.43991380391405e-07,0.000143312843494772,1.69385001769067e-05,NA,5.80011606189205e-06,NA,1.74121416684065e-06,NA,1.22116360413972e-05,3.16955494665727e-05,NA,1.75966771088704e-05
+"ENSG00000266964",NA,NA,NA,NA,NA,1.76762662284035e-05,1.8135786562568e-05,NA,NA,NA,NA,NA
+"ENSG00000267673",NA,1.57426572112334e-05,1.54304450034953e-06,NA,5.09314226095694e-06,2.48912835430519e-06,1.3583770265844e-05,NA,2.21407311285786e-05,9.99502710047527e-06,NA,NA
+"ENSG00000269190",1.80356794523966e-06,1.40251138382704e-05,2.03417755489711e-06,1.09120163490642e-06,1.65004368751854e-05,3.44142086925826e-06,5.64771277829109e-06,6.8948391824609e-07,2.46034142537104e-05,3.08957283551509e-05,1.06762501505787e-05,3.70350513373877e-06
+"ENSG00000271303",4.68585327823435e-06,NA,2.66967547587316e-06,NA,NA,5.63308577126446e-06,6.52708799979872e-06,2.55455367916519e-06,1.87394236677152e-05,NA,NA,1.31218669231929e-05
+"ENSG00000272325",2.00192937882658e-05,6.37646978328792e-06,6.18022600334538e-05,1.40226997452693e-05,5.60527637656231e-05,7.24861347408873e-05,0.000157105430462,3.08995868763351e-05,1.28266571965503e-05,3.65796736039502e-05,1.56644275717775e-05,2.57420086555142e-05
diff --git a/materials/worksheet_07/tests_worksheet_07.R b/materials/worksheet_07/tests_worksheet_07.R
new file mode 100644
index 0000000..2b4ef66
--- /dev/null
+++ b/materials/worksheet_07/tests_worksheet_07.R
@@ -0,0 +1,785 @@
+library(digest)
+library(testthat)
+
+# +
+#abstraction templates
+
+check_TF <- function(answerX.X, expectedHash) {
+  var_name <- deparse(substitute(answerX.X))
+  test_that(paste('Did not assign answer to an object called ', var_name), {
+    expect_true(exists(var_name))
+  })
+  
+  
+  test_that('Solution should be "true" or "false"', {
+    expect_match(answerX.X, "true|false", ignore.case = TRUE)
+  })
+  
+  answer_hash <- digest(tolower(answerX.X))
+  #if (answer_hash == "HASH_HERE") {
+  #  print("HINT_HERE")
+  #}
+  
+  test_that("Solution is incorrect", {
+    expect_equal(answer_hash, expectedHash)
+  })
+  
+  print("Success!")
+}
+
+check_MC <- function(answerX.X, choiceList, expectedHash) {
+  var_name <- deparse(substitute(answerX.X))
+  test_that(paste('Did not assign answer to an object called ', var_name), {
+    expect_true(exists(var_name))
+  })
+  
+  
+  test_that(paste('Solution should be a single character ', toString(choiceList)), {
+    expect_true(tolower(answerX.X) %in% tolower(choiceList))
+  })
+  
+  answer_hash <- digest(tolower(answerX.X))
+  #if (answer_hash == "HASH_HERE") {
+  #  print("HINT_HERE")
+  #} else if (answer_hash == "HASH_HERE") {
+  #  print("HINT_HERE")
+  #} else if (answer_hash == "HASH_HERE") {
+  #  print("HINT_HERE")
+  #}
+  
+  test_that("Solution is incorrect", {
+    expect_equal(answer_hash, expectedHash)
+  })
+  
+  print("Success!")
+}
+
+
+
+# dataCheckTuples is data.frame(c(colnames), c(scale factor), c(expectedHash))
+check_DF <- function(answerX.X, expected_colnames, hashNRows, cols_to_check, precision_list, expectedHashes) {
+  dataCheckTuples <- data.frame(cols_to_check, precision_list, expectedHashes) 
+  var_name <- deparse(substitute(answerX.X))
+  test_that(paste('Did not assign answer to an object called ', var_name), {
+    expect_true(exists(var_name))
+  })
+  
+  test_that("Solution should be a data frame", {
+    expect_true("data.frame" %in% class(answerX.X))
+  })
+  
+  given_colnames <- colnames(answerX.X)
+  test_that("Data frame does not have the correct columns", {
+    expect_equal(length(setdiff(
+      union(expected_colnames, given_colnames),
+      intersect(expected_colnames, given_colnames)
+    )), 0)
+  })
+  
+  test_that("Data frame does not contain the correct number of rows", {
+    expect_equal(digest(as.integer(nrow(answerX.X))), hashNRows)
+  })
+  
+  
+  
+  apply(dataCheckTuples, 1, function(tuple) {
+    test_that(paste(tuple[[1]], " does not contain the correct data"), {
+      expect_equal(digest(as.integer(sum(answerX.X[tuple[[1]]]) * as.double(tuple[[2]]))),
+                   tuple[[3]])
+    })
+  })
+  
+  
+  
+  
+  print("Success!")
+}
+
+
+
+
+
+# dataCheckTuples is data.frame(c(colnames), c(scale factor), c(expectedHash))
+check_DF_overflow <- function(answerX.X, expected_colnames, hashNRows, cols_to_check, precision_list, expectedHashes) {
+  dataCheckTuples <- data.frame(cols_to_check, precision_list, expectedHashes) 
+  var_name <- deparse(substitute(answerX.X))
+  test_that(paste('Did not assign answer to an object called ', var_name), {
+    expect_true(exists(var_name))
+  })
+  
+  test_that("Solution should be a data frame", {
+    expect_true("data.frame" %in% class(answerX.X))
+  })
+  
+  given_colnames <- colnames(answerX.X)
+  test_that("Data frame does not have the correct columns", {
+    expect_equal(length(setdiff(
+      union(expected_colnames, given_colnames),
+      intersect(expected_colnames, given_colnames)
+    )), 0)
+  })
+  
+  test_that("Data frame does not contain the correct number of rows", {
+    expect_equal(digest(as.integer(nrow(answerX.X))), hashNRows)
+  })
+  
+  
+  
+  apply(dataCheckTuples, 1, function(tuple) {
+    test_that(paste(tuple[[1]], " does not contain the correct data"), {
+      expect_equal(digest(trunc(sum(answerX.X[tuple[[1]]]) * as.double(tuple[[2]]))),
+                   tuple[[3]])
+    })
+  })
+  
+  
+  
+  
+  print("Success!")
+}
+
+
+
+
+check_numeric <- function(answerX.X, precision, expectedHash) {
+  var_name <- deparse(substitute(answerX.X))
+  test_that(paste('Did not assign answer to an object called ', var_name), {
+    expect_true(exists(var_name))
+  })
+  
+  answer_as_numeric <- as.numeric(answerX.X)
+  test_that(paste(var_name, " should be a number"), {
+    expect_false(is.na(answer_as_numeric))
+  })
+  
+  test_that(paste(var_name, " value is incorrect"), {
+    expect_equal(digest(as.integer(answer_as_numeric * precision)), expectedHash)
+  })
+  
+  print("Success!")
+}
+
+
+
+check_numeric_element <- function(answerX.X, precision, expectedHash) {
+  var_name <- deparse(substitute(answerX.X))
+  
+  answer_as_numeric <- as.numeric(answerX.X)
+  test_that(paste(var_name, " should be a number"), {
+    expect_false(is.na(answer_as_numeric))
+  })
+  
+  test_that(paste(var_name, " value is incorrect"), {
+    expect_equal(digest(as.integer(answer_as_numeric * precision)), expectedHash)
+  })
+  
+  print("Success!")
+}
+
+
+
+
+check_plot <- function(answerX.X, x_axis_var, geom_type, hasVline, bin_width_hash, nrow_hash,
+                       x_axis_var_hash, hasTitle, isFactor, shouldRename) {
+  var_name <- deparse(substitute(answerX.X))
+  test_that(paste('Did not assign answer to an object called ', var_name), {
+    expect_true(exists(var_name))
+  })
+  
+  
+  test_that("Solution should be a ggplot object", {
+    expect_true(is.ggplot(answerX.X))
+  })
+  
+  properties <- c(answerX.X$layers[[1]]$mapping, answerX.X$mapping)
+  
+  test_that(paste("Plot should have ", x_axis_var," on the x-axis"), {
+    expect_true(x_axis_var == rlang::get_expr(properties$x))
+  })
+  
+  test_that("Plot does not have the correct layers", {
+    expect_true(geom_type %in% class(answerX.X$layers[[1]]$geom))
+    
+    if(hasVline) {
+      expect_true("GeomVline" %in% class(answerX.X$layers[[2]]$geom))
+    }
+  })
+  
+  test_that("Plot does not have the correct bin width", {
+    expect_equal(
+      digest(as.integer(mget("stat_params", answerX.X$layers[[1]])[["stat_params"]][["binwidth"]])),
+      bin_width_hash)
+  })
+  
+  test_that("Plot does not use the correct data", {
+    expect_equal(digest(nrow(answerX.X$data)), nrow_hash)
+    if (!isFactor) {
+      expect_equal(digest(round(sum(answerX.X$data[x_axis_var]))), x_axis_var_hash)
+    }
+    # If X_AXIS_VAR is not known:
+    # expect_equal(digest(round(sum(pull(answerX.X$data, rlang::get_expr(properties$x))))), "HASH_HERE")
+  })
+  
+  if (shouldRename) {
+    test_that("x-axis label should be descriptive and human readable", {
+      expect_false(answerX.X$labels$x == toString(rlang::get_expr(properties$x)))
+    })
+  }
+  
+  if(hasTitle){
+    
+    test_that("Plot should have a title", {
+      expect_true("title" %in% names(answerX.X$labels))
+    })
+  }
+  
+  
+  print("Success!")
+}
+
+
+getPermutations <- function(vec) {
+  rsf <- c()
+  for (i in 1:length(vec)) {
+    for (j in i:length(vec)) {
+      temp <- vec[i:j] 
+      rsf <- c(rsf, paste(temp, collapse= ''))
+      if (i < j) {
+        for (k in i:j) {
+          rsf <- c(rsf, paste(temp[-k], collapse=''))
+        }
+      }
+    }
+  }
+  return(unique(rsf))
+}
+
+
+
+
+get_plot_params <- function(answerX.X, isFactor) {
+  
+  properties <- c(answerX.X$layers[[1]]$mapping, answerX.X$mapping)
+  x_axis_var <- as.character(rlang::get_expr(properties$x))
+  print(x_axis_var)
+  print(class(answerX.X$layers[[1]]$geom))
+  if (length(answerX.X$layers) > 1) {
+    print("GeomVline" %in% class(answerX.X$layers[[2]]$geom))
+  } else {
+    print(FALSE)
+  }
+  print(digest(as.integer(mget("stat_params", answerX.X$layers[[1]])[["stat_params"]][["binwidth"]])))
+  print(digest(nrow(answerX.X$data)))
+  if(!isFactor) {
+    print(digest(round(sum(pull(answerX.X$data, x_axis_var)))))
+  } else {
+    print("none")
+  }
+  print("title" %in% names(answerX.X$labels))
+  print(isFactor)
+  print(answerX.X$labels$x != toString(rlang::get_expr(properties$x)))
+}
+
+
+get_DF_params <- function(df, precisionList) {
+  cols <- colnames(df)
+  print(cols)
+  print(digest(as.integer(nrow(df))))
+  print(cols)
+  print(precisionList)
+  for (i in 1: length(precisionList)) {
+    print(digest(as.integer(sum(df[cols[i]],na.rm=T) * as.double(precisionList[[i]]))))
+  }
+  
+}
+
+
+get_DF_params_overflow <- function(df, precisionList) {
+  cols <- colnames(df)
+  print(cols)
+  print(digest(as.integer(nrow(df))))
+  print(cols)
+  print(precisionList)
+  for (i in 1: length(precisionList)) {
+    print(digest(trunc(sum(df[cols[i]],na.rm=T) * as.double(precisionList[[i]]))))
+  }
+  
+}
+
+
+get_numeric_element_params <- function(answerX.X, precision) {
+  print(precision)
+  answer_as_numeric <- as.numeric(answerX.X)
+  print(digest(as.integer(answer_as_numeric * precision)))
+}
+# -
+
+
+# Question 1.0
+
+test_1.0 <- function() {
+  test_that('Did not assign answer to an object called "SLR_ENSG00000085733_plot"', {
+    expect_true(exists("SLR_ENSG00000085733_plot"))
+  })
+
+  test_that("Solution should be a ggplot object", {
+    expect_true(is.ggplot(SLR_ENSG00000085733_plot))
+  })
+
+  properties <- c(SLR_ENSG00000085733_plot$layers[[1]]$mapping, SLR_ENSG00000085733_plot$mapping)
+
+  test_that("Plot should have mrna on the x-axis", {
+    expect_true("mrna" == rlang::get_expr(properties$x))
+  })
+
+  test_that("Plot does not have the correct layers", {
+    expect_true("GeomPoint" %in% class(SLR_ENSG00000085733_plot$layers[[1]]$geom))
+
+   })
+
+  test_that("Plot does not have the correct bin width", {
+  expect_equal(
+    digest(as.integer(mget("stat_params", SLR_ENSG00000085733_plot$layers[[1]])[["stat_params"]][["binwidth"]])),
+    "3e2e4a08c44d0224de5b7e668c75ace3"
+    )
+  })
+
+  test_that("Plot does not use the correct data", {
+    expect_equal(digest(nrow(SLR_ENSG00000085733_plot$data)), "fa5a4df7ac0f9782037da890557fd8b8")
+    expect_equal(digest(round(sum(SLR_ENSG00000085733_plot$data$mrna))), "908d1fd10b357ed0ceaaec823abf81bc")
+
+  })
+
+  test_that("x-axis label should be descriptive and human readable", {
+    expect_false(SLR_ENSG00000085733_plot$labels$x == toString(rlang::get_expr(properties$x)))
+  })
+
+  test_that("Plot should have a title", {
+    expect_true("title" %in% names(SLR_ENSG00000085733_plot$labels))
+  })
+
+  print("Success!")
+
+}
+# -
+# Question 1.1
+
+test_1.1.0 <- function() {
+    check_numeric_element(SLR_ENSG00000085733$coefficients[1], 1e6, "25e6a154090e35101d7678d6f034353a")
+    check_numeric_element(SLR_ENSG00000085733$coefficients[2], 1e5, "a03a4514784b3aff814a8aa7ab343d16")
+}
+test_1.1.1 <- function() {
+  check_DF(SLR_ENSG00000085733_results,
+           c("term","estimate",  "std.error", "statistic", "p.value" ),
+           "c01f179e4b57ab8bd9de309e6d576c48",
+           c("estimate",  "std.error", "statistic", "p.value" ),
+           c(1e6,1e6,100 ,10),
+           c("2f9f1c2afd9a90172d6e15a009de84b2",
+             "79ba185f068a5011ce65ec801fd55276",
+             "a7f7cec46d9e1398c6bb0b72351f3eab",
+             "fa5a4df7ac0f9782037da890557fd8b8"
+           ))
+}
+
+test_1.1.2 <- function() {
+  check_DF(dat_ENSG00000085733,
+           c(".rownames","prot","mrna",".fitted",".resid",".hat",".sigma",".cooksd",".std.resid"),
+           "fa5a4df7ac0f9782037da890557fd8b8",
+           c(".fitted",".resid"),
+           c(1e6,1e6),
+           c("a95ceee8390cb47bb05410a8d23c76cf",
+             "1473d70e5646a26de3c52aa1abd85b1f"
+           ))
+}
+
+test_1.2.0 <- function() {
+  check_DF(SLR_ENSG00000085733_gof,
+           c("r.squared","adj.r.squared","sigma","statistic","p.value","df","logLik","AIC","BIC","deviance","df.residual","nobs"),
+           "4b5630ee914e848e8d07221556b0a2fb",
+           c("r.squared"),
+           c(1e3),
+           c("6cb0c41728b80235d1532f88c59ce9e6"
+           ))
+}
+
+test_1.2.1 <- function() {
+  test_that('Did not assign answer to an object called "answer1.2.1"', {
+    expect_true(exists("answer1.2.1"))
+  })
+  
+  test_that('Solution should be "true" or "false"', {
+    expect_match(answer1.2.1, "true|false", ignore.case = TRUE)
+  })
+  
+  answer_hash <- digest(tolower(answer1.2.1))
+ 
+  test_that("Solution is incorrect", {
+    expect_equal(answer_hash, "05ca18b596514af73f6880309a21b5dd")
+  })
+  
+  print("Success!")
+}
+
+
+test_1.3 <- function() {
+  test_that('Did not assign answer to an object called "answer1.3"', {
+    expect_true(exists("answer1.3"))
+  })
+  
+  test_that('Solution should be a single character ("A", "B", "C")', {
+    expect_match(answer1.3, "a|b|c", ignore.case = TRUE)
+  })
+  
+  answer_hash <- digest(tolower(answer1.3))
+  
+  test_that("Solution is incorrect", {
+    expect_equal(answer_hash, "127a2ec00989b9f7faf671ed470be7f8")
+  })
+  
+  print("Success!")
+}
+
+test_1.4 <- function() {
+  test_that('Did not assign answer to an object called "answer1.4"', {
+    expect_true(exists("answer1.4"))
+  })
+  
+  test_that('Solution should be a single character ("A", "B", "C")', {
+    expect_match(answer1.4, "a|b|c", ignore.case = TRUE)
+  })
+  
+  answer_hash <- digest(tolower(answer1.4))
+  
+  test_that("Solution is incorrect", {
+    expect_equal(answer_hash, "257ffd0fec52cbf9ff2dba41f66d6b26")
+  })
+  
+  print("Success!")
+}
+
+
+test_1.5 <- function() {
+  test_that('Did not assign answer to an object called "answer1.5"', {
+    expect_true(exists("answer1.5"))
+  })
+  
+  test_that('Solution should be a single character ("A", "B", "C", "D")', {
+    expect_match(answer1.5, "a|b|c|d", ignore.case = TRUE)
+  })
+  
+  answer_hash <- digest(tolower(answer1.5))
+  
+  test_that("Solution is incorrect", {
+    expect_equal(answer_hash, "5569ed730134a270952e469c2be78612")
+  })
+  
+  print("Success!")
+}
+
+#---------------------
+
+# Question 2.0
+
+
+test_2.0 <- function() {
+  check_DF(summary_gof,
+           c("gene","r.squared", "p.value"),
+           "a1af0464b4260e8c6044ddbf20d5607f",
+           c("r.squared", "p.value"),
+           c(1000, 1e6),
+           c("9b3991f9ae292cb9a49b18824cef818e",
+            "4ff574bd8b88dcca0333d28e927ee31a"))
+}
+
+#--
+
+test_2.1 <- function() {
+  test_that('Did not assign answer to an object called "answer2.1"', {
+    expect_true(exists("answer2.1"))
+  })
+  
+  test_that('Solution should be a single character ("A", "B", "C", "D")', {
+    expect_match(answer2.1, "a|b|c|d", ignore.case = TRUE)
+  })
+  
+  answer_hash <- digest(tolower(answer2.1))
+  
+  test_that("Solution is incorrect", {
+    expect_equal(answer_hash, "127a2ec00989b9f7faf671ed470be7f8")
+  })
+  
+  print("Success!")
+}
+
+#--
+
+test_2.2 <- function() {
+  test_that('Did not assign answer to an object called "hist_slr_F"', {
+    expect_true(exists("hist_slr_F"))
+  })
+
+  test_that("Solution should be a ggplot object", {
+    expect_true(is.ggplot(hist_slr_F))
+  })
+
+  properties <- c(hist_slr_F$layers[[1]]$mapping, hist_slr_F$mapping)
+
+  test_that("Plot should have p.value on the x-axis", {
+    expect_true("p.value" == rlang::get_expr(properties$x))
+  })
+
+  test_that("Plot does not have the correct layers", {
+    expect_true("GeomBar" %in% class(hist_slr_F$layers[[1]]$geom))
+
+  })
+
+  test_that("Plot does not have the correct bin width", {
+  expect_equal(
+    digest(as.integer(mget("stat_params", hist_slr_F$layers[[1]])[["stat_params"]][["binwidth"]])),
+    "3e2e4a08c44d0224de5b7e668c75ace3"
+    )
+  })
+
+  test_that("Plot does not use the correct data", {
+    expect_equal(digest(nrow(hist_slr_F$data)), "a1af0464b4260e8c6044ddbf20d5607f")
+    expect_equal(digest(round(sum(hist_slr_F$data$p.value))), "12923c0c32abd92b47a44187f7d6fb75")
+
+  })
+
+  test_that("x-axis label should be descriptive and human readable", {
+    expect_false(hist_slr_F$labels$x == toString(rlang::get_expr(properties$x)))
+  })
+
+  test_that("Plot should have a title", {
+    expect_true("title" %in% names(hist_slr_F$labels))
+  })
+
+  print("Success!")
+}
+
+#----
+
+test_2.3 <- function() {
+  test_that('Did not assign answer to an object called "answer2.3"', {
+    expect_true(exists("answer2.3"))
+  })
+  
+  test_that('Solution should be "true" or "false"', {
+    expect_match(answer2.3, "true|false", ignore.case = TRUE)
+  })
+  
+  answer_hash <- digest(tolower(answer2.3))
+ 
+  test_that("Solution is incorrect", {
+    expect_equal(answer_hash, "d2a90307aac5ae8d0ef58e2fe730d38b")
+  })
+  
+  print("Success!")
+}
+
+#-----
+#Question 3.0
+
+
+test_3.0 <- function() {
+  test_that('"model.1" is incorrect', {
+    expect_equal(digest(tolower(model.1)), "d110f00cfb1b248e835137025804a23b")
+  })
+
+  test_that('"model.2" is incorrect', {
+    expect_equal(digest(tolower(model.2)), "93a9078c6326f37b481d3e99b60ad987")
+  })
+
+  test_that('"model.3" is incorrect', {
+    expect_equal(digest(tolower(model.3)), "ddf100612805359cd81fdc5ce3b9fbba")
+  })
+
+  test_that('"model.4" is incorrect', {
+    expect_equal(digest(tolower(model.4)), "6e7a8c1c098e8817e3df3fd1b21149d1")
+  })
+
+  test_that('"model.5" is incorrect', {
+    expect_equal(digest(tolower(model.5)), "127a2ec00989b9f7faf671ed470be7f8")
+  })
+
+  print("Success!")
+}
+
+#---
+test_3.1.0 <- function() {
+    check_DF(mlr_3genes_int_results,
+             c("term","estimate",  "std.error", "statistic", "p.value"),
+             "25e6a154090e35101d7678d6f034353a",
+             c("estimate",  "std.error", "statistic", "p.value"),
+             c(100,100,100,100),
+             c('7c7124efff5c7039a1b1e7cba65c5379',
+               'e9f8d19c99ed86518e645170bc3facd7',
+               'b84519040eec75b640a1cce9c7f2be51',
+               'ff9690743156cc5cd42197f6dba7e5a1'))
+}
+
+test_3.1.1 <- function() {
+  check_DF(mlr_3genes_int_gof,
+           c("r.squared","adj.r.squared","sigma","statistic","p.value","df","logLik","AIC","BIC","deviance","df.residual","nobs"),
+           "4b5630ee914e848e8d07221556b0a2fb",
+           c("r.squared"),
+           c(1e3),
+           c("40b423b6494baa72a66fe2fdd02b15c4"
+           ))
+}    
+
+#--
+
+test_3.2  <- function() {
+  test_that('Did not assign answer to an object called "answer3.2"', {
+    expect_true(exists("answer3.2"))
+  })
+  
+  test_that('Solution should be a single character ("A", "B", "C", "D","E")', {
+    expect_match(answer3.2, "a|b|c|d|e", ignore.case = TRUE)
+  })
+  
+  answer_hash <- digest(tolower(answer3.2))
+  
+  test_that("Solution is incorrect", {
+    expect_equal(answer_hash, "d110f00cfb1b248e835137025804a23b")
+  })
+  
+  print("Success!")
+}
+
+#----
+#Question 3.3
+
+test_3.3 <- function() {
+  check_numeric_element(Ftest_3genes_interaction[2,5], 
+                        100, 
+                        "1a825441b9be399cbd0568b97e8902de")
+}
+
+
+#-----
+test_3.7 <- function() {
+  test_that('Did not assign answer to an object called "answer3.2"', {
+    expect_true(exists("answer3.2"))
+  })
+  
+  test_that('Solution should be a single character ("A", "B", "C")', {
+    expect_match(answer3.2, "a|b|c", ignore.case = TRUE)
+  })
+  
+  answer_hash <- digest(tolower(answer3.2))
+  
+  test_that("Solution is incorrect", {
+    expect_equal(answer_hash, "ddf100612805359cd81fdc5ce3b9fbba")
+  })
+  
+  print("Success!")
+}
+
+test_3.3.0 <- function() {
+    check_DF(mlr_3genes_int_results,
+             c("term","estimate",  "std.error", "statistic", "p.value"),
+             "25e6a154090e35101d7678d6f034353a",
+             c("estimate",  "std.error", "statistic", "p.value"),
+             c(100,100,100,100),
+             c('7c7124efff5c7039a1b1e7cba65c5379',
+               'e9f8d19c99ed86518e645170bc3facd7',
+               'b84519040eec75b640a1cce9c7f2be51',
+               'ff9690743156cc5cd42197f6dba7e5a1'))
+}
+
+test_3.3.1 <- function() {
+  check_DF(mlr_3genes_int_gof,
+           c("r.squared","adj.r.squared","sigma","statistic","p.value","df","logLik","AIC","BIC","deviance","df.residual","nobs"),
+           "4b5630ee914e848e8d07221556b0a2fb",
+           c("r.squared"),
+           c(1e3),
+           c("40b423b6494baa72a66fe2fdd02b15c4"
+           ))
+}    
+
+#---
+#Question 3.4
+
+test_3.4  <- function() {
+  test_that('Did not assign answer to an object called "answer3.4"', {
+    expect_true(exists("answer3.4"))
+  })
+  
+  test_that('Solution should be a single character ("A", "B", "C")', {
+    expect_match(answer3.4, "a|b|c", ignore.case = TRUE)
+  })
+  
+  answer_hash <- digest(tolower(answer3.4))
+  
+  test_that("Solution is incorrect", {
+    expect_equal(answer_hash, "ddf100612805359cd81fdc5ce3b9fbba")
+  })
+  
+  print("Success!")
+}
+
+
+#--
+#Question 3.5
+
+test_3.5  <- function() {
+  test_that('Did not assign answer to an object called "answer3.5"', {
+    expect_true(exists("answer3.5"))
+  })
+  
+  test_that('Solution should be a single character ("A", "B", "C", "D","E")', {
+    expect_match(answer3.5, "a|b|c|d|e", ignore.case = TRUE)
+  })
+  
+  answer_hash <- digest(tolower(answer3.5))
+  
+  test_that("Solution is incorrect", {
+    expect_equal(answer_hash, "ddf100612805359cd81fdc5ce3b9fbba")
+  })
+  
+  print("Success!")
+}
+
+#----
+
+#Question 3.6
+
+test_3.6  <- function() {
+  test_that('Did not assign answer to an object called "answer3.6"', {
+    expect_true(exists("answer3.6"))
+  })
+  
+  test_that('Solution should be a single character ("A", "B", "C", "D","E")', {
+    expect_match(answer3.6, "a|b|c|d|e", ignore.case = TRUE)
+  })
+  
+  answer_hash <- digest(tolower(answer3.6))
+  
+  test_that("Solution is incorrect", {
+    expect_equal(answer_hash, "93a9078c6326f37b481d3e99b60ad987")
+  })
+  
+  print("Success!")
+}
+
+
+#--
+
+#Question 3.7
+
+test_3.7 <- function() {
+  check_numeric_element(Ftest_3genes_mrna$F[2], 
+                        100, 
+                        "e6c63b6571d445697f2c2e2089fe7fa4")
+}
+
+
+
+
+#--
+
+
+
+
diff --git a/materials/worksheet_07/worksheet_07.ipynb b/materials/worksheet_07/worksheet_07.ipynb
new file mode 100644
index 0000000..ba67c63
--- /dev/null
+++ b/materials/worksheet_07/worksheet_07.ipynb
@@ -0,0 +1,2164 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "32a95fc7dd24d9e80a38f71eb7348506",
+     "grade": false,
+     "grade_id": "cell-f1e1d845873036f4",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    }
+   },
+   "source": [
+    "# Worksheet 7: Model Evaluation and Model Selection"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "4f368f4a802e15bd991322bf16951c73",
+     "grade": false,
+     "grade_id": "cell-82d9926086d47a80",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    }
+   },
+   "source": [
+    "#### Lecture and Tutorial Learning Goals:\n",
+    "After completing this week's lecture and tutorial work, you will be able to:\n",
+    "\n",
+    "1. List model metrics that are suitable for evaluation of a statistical model developed to make inference about the data-generating mechanism (e.g., $R^2$, $\\text{AIC}$, Likelihood ratio test/$F$-test), their strengths and limitations, as well as how they are calculated.\n",
+    "2. Write a computer script to calculate these model metrics. Interpret and communicate the results from that computer script.\n",
+    "3. Explain the algorithms for the following variable selection methods:\n",
+    "\n",
+    "    - $F$-test to compare nested models.\n",
+    "    - Forward selection.\n",
+    "    - Backward selection."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "1b1929b686c9a0d8acade85a115075cc",
+     "grade": false,
+     "grade_id": "cell-a2a153352bc44a68",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# Run this cell before continuing.\n",
+    "library(tidyverse)\n",
+    "library(repr)\n",
+    "library(broom)\n",
+    "library(leaps)\n",
+    "library(moderndive)\n",
+    "source(\"tests_worksheet_07.R\")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "21a8dd22ad3be8c594d51bbde1d3ccc8",
+     "grade": false,
+     "grade_id": "cell-30d72e348848492c",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    }
+   },
+   "source": [
+    "# PART I: Model Evaluation of Generative Models"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "83aa997a555a3c0e7a7a69886b6f8619",
+     "grade": false,
+     "grade_id": "cell-459184ff51dbea84",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    }
+   },
+   "source": [
+    "In this worksheet you will learn different methods to evaluate and select appropriate models to make inference about the data-generating mechanism. In other words, the main goal is to estimate and evaluate generative models."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "86519b01e29e13e3710c3754c064c647",
+     "grade": false,
+     "grade_id": "cell-9be71f65643c5906",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    }
+   },
+   "source": [
+    "## Case Study: protein vs mRNA\n",
+    "\n",
+    "In this section we will continue working with the case study from Biology to learn how to examine the goodness of the fitted model and choose among different nested models. The data and some of the discussions related to this case were published in Nature (see citations below).\n",
+    "\n",
+    "In 2014, a research group claimed to find a \"predictive model\", which can be used to predict protein from mRNA expression!! \n",
+    "\n",
+    "> Wilhelm, M. et al. Mass-spectrometry-based draft of the human proteome. Nature 509, 582–587 (2014)\n",
+    "\n",
+    "Although their hypotheses were funded in the Central Dogma of Biology, most experimental results have shown very low correlation values between protein and mRNA values. \n",
+    "\n",
+    "Further examination of their analysis has shwon that their data do not support their claims:\n",
+    "\n",
+    "> Paper from my research group: Fortelny N, Overall CM, Pavlidis P, Freue GVC. Can we predict protein from mRNA levels? Nature. 2017 Jul 26;547(7664):E19-E20. doi: 10.1038/nature22293.\n",
+    "\n",
+    "We'll use data from this group submitted to the Journal to (re)analyse their data and evalute different models using concepts learned in this course.\n",
+    "\n",
+    "### Data\n",
+    "\n",
+    "Many proteins were not detected by the technology used due to values below limits of detection and the protein dataset contains many missing values!! For simplicity, we will use **a set of 1392 genes** that were measured in **all 12 tissues** and thus contain complete data on both protein and mRNA sets. \n",
+    "\n",
+    "*Run the cell below before continuing to read in the data and take a peek at it.* "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "292b5084366a95ec55f6d34ab90c9e36",
+     "grade": false,
+     "grade_id": "cell-c3a6eba286971086",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# Read and take a look at the data.\n",
+    "dat_bio <- read.csv(\"data/nature_dat.csv\", row.names = 1, stringsAsFactors=TRUE)\n",
+    "str(dat_bio)\n",
+    "head(dat_bio,3)\n",
+    "tail(dat_bio,3)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "1df774dff238548a24ca978a6a5cc4ad",
+     "grade": false,
+     "grade_id": "cell-e6b7bf58e2eb7a77",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    }
+   },
+   "source": [
+    "In this long format dataset, we can think that we have **1392 datasets**, one for each gene!\n",
+    "\n",
+    "> each dataset contains **12 observations**, one per tissue, and 2 variables, `prot` and `mrna`, along with an accession number for the gene (`gene`, an ID for each gene) and the name of the tissue (`tissue`, that works as an ID for each observation).\n",
+    "\n",
+    "*Run the cell below to get the data for gene ENSG00000085733.* "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "6ed8fd2d104ef35339e78069dc2e039c",
+     "grade": false,
+     "grade_id": "cell-28e85de12a1890ed",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "dat_ENSG00000085733 <- dat_bio  %>% subset(gene == \"ENSG00000085733\") \n",
+    "dat_ENSG00000085733"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "a200797af0d235c2390652c2fa70624f",
+     "grade": false,
+     "grade_id": "cell-4135e761a8664bad",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    }
+   },
+   "source": [
+    "### Models and Estimation\n",
+    "\n",
+    "In the paper, the authors used a linear regression to estimate the relation between protein and mRNA levels *per gene*, and used that model to predict protein levels *per gene*.\n",
+    "\n",
+    "> it will be important at a later phase of the analysis to note that models are *gene-specific*\n",
+    "\n",
+    "**Gene-specific models**: for each gene they estimated the following model (for simplicity we do not use a subscript $g$)  \n",
+    "\n",
+    "$$\\text{prot}_{t} = \\beta_1 \\; \\times \\text{mrna}_{t} + \\varepsilon_t$$ \n",
+    "\n",
+    "where $\\hat{\\beta}_1 = median_t(\\text{mrna}_{t}/\\text{prot}_{t})$\n",
+    "\n",
+    "> $prot_{t}$ and $mrna_{t}$ are the protein and mRNA levels of a gene $g$ in tissue $t$, respectively\n",
+    "\n",
+    "<font color=\"blue\"> Note that these models do not contain an intercept! and they were not estimated by LS.</font> \n",
+    "    \n",
+    "While in general different models and estimation methods can be used to analyze the same data, it is important to evaluate the results according to the assumptions and the methodology used. \n",
+    "    \n",
+    "**In this worksheet we will use this dataset to estimate different models and evaluate their fit to the data.**"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "9cdb56ca4b66f4c701038cc35d4ed5aa",
+     "grade": false,
+     "grade_id": "cell-f76d01efb9d4c744",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    }
+   },
+   "source": [
+    "### 1. Gene-specific SLR for one selected gene\n",
+    "\n",
+    "For each gene we will use LS to estimate the following SLR (for simplicity we do not use a subscript $g$)  \n",
+    "\n",
+    "$$prot_{t} = \\beta_0 +\\beta_1 \\; \\times mrna_{t} + \\varepsilon$$ \n",
+    "\n",
+    "where $\\hat{\\beta}_1$ is estimated by LS"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "582ef7625b84736b7daad8d5ffe78213",
+     "grade": false,
+     "grade_id": "cell-b6e85c5927c1dc25",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "**Question 1.0: Visualization of the estimated line**\n",
+    "<br>{points: 1}\n",
+    "\n",
+    "Using data from gene ENSG00000085733 in `dat_ENSG00000085733`, we can plot `prot` versus `mrna` **with points** and add the estimated SLR. The `ggplot()` object's name will be `SLR_ENSG00000085733_plot`\n",
+    "\n",
+    "*Fill out those parts indicated with `...`, uncomment the corresponding code in the cell below, and run it.*"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "7e5c37866bec72275dff358c61b63778",
+     "grade": false,
+     "grade_id": "cell-c23cc58da9842616",
+     "locked": false,
+     "schema_version": 3,
+     "solution": true,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "options(repr.plot.width = 6, repr.plot.height = 3) # Adjust these numbers so the plot looks good in your desktop.\n",
+    "\n",
+    "# SLR_ENSG00000085733_plot <- ggplot(..., aes(..., ...)) +\n",
+    "#   ...() +\n",
+    "#   ...(..., se = FALSE, linewidth = 1.5) +\n",
+    "#   xlab(...) +\n",
+    "#   ylab(...) +\n",
+    "#   ggtitle(\"Sample and Estimated SLR for gene ENSG00000085733\") +\n",
+    "#   theme(\n",
+    "#     text = element_text(size = 10),\n",
+    "#     plot.title = element_text(face = \"bold\"),\n",
+    "#     axis.title = element_text(face = \"bold\")\n",
+    "#   )\n",
+    "\n",
+    "#SLR_ENSG00000085733_plot\n",
+    "\n",
+    "# your code here\n",
+    "fail() # No Answer - remove if you provide an answer"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "4751fc19fd84bf62cacaa3488aad2cab",
+     "grade": true,
+     "grade_id": "cell-f1a69250e199ce6c",
+     "locked": true,
+     "points": 1,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "test_1.0()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "4458a0fc569aa37699422862d7119534",
+     "grade": false,
+     "grade_id": "cell-38b13dc576bc33ab",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "**Question 1.1**\n",
+    "<br>{points: 3}\n",
+    "\n",
+    "For the selected gene:\n",
+    "\n",
+    "- 1.1.0 Use `lm()` to obtain the LS estimated coefficients for the selected gene. Call the model `SLR_ENSG00000085733`. \n",
+    "\n",
+    "- 1.1.1 Use `tidy()` to store all inference quantities from this model in an objected called `SLR_ENSG00000085733_results`.\n",
+    "\n",
+    "- 1.1.2 Use `augment` to obtain residuals and fitted values from the estimated SLR. Add all the output to the original dataset. \n",
+    "\n",
+    "*Fill out those parts indicated with `...`, uncomment the corresponding code in the cell below, and run it.*"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "d68f0d343af4f4aa94693c7fc5c62c1d",
+     "grade": false,
+     "grade_id": "cell-c05f3277a48ee808",
+     "locked": false,
+     "schema_version": 3,
+     "solution": true,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# SLR_ENSG00000085733 <- lm(...,...)\n",
+    "# SLR_ENSG00000085733_results <- ...(...)\n",
+    "# dat_ENSG00000085733 <- ...(SLR_ENSG00000085733)\n",
+    "\n",
+    "# SLR_ENSG00000085733_results\n",
+    "# head(dat_ENSG00000085733,3)\n",
+    "\n",
+    "# your code here\n",
+    "fail() # No Answer - remove if you provide an answer"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "3d9255ea44370a1a850df30a7a94d06e",
+     "grade": true,
+     "grade_id": "cell-5ea5ee0592297e4b",
+     "locked": true,
+     "points": 3,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "test_1.1.0()\n",
+    "test_1.1.1()\n",
+    "test_1.1.2()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "6cd654b4b56edcc8f363b2711ac2d65b",
+     "grade": false,
+     "grade_id": "cell-b0306ce9b39255e8",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "**Question 1.2**\n",
+    "<br>{points: 2}\n",
+    "\n",
+    "1.2.0 For the selected gene, use `glance()` to obtain goodness of fit measurements and tests for the fitted model `SLR_ENSG00000085733`. Store the results in an object called `SLR_ENSG00000085733_gof`.\n",
+    "\n",
+    "1.2.1 Compute the $R^2$, the adjusted $R^2$ and the standard deviation of the error term, $\\sigma$. Note that you will need the residuals and the standard deviation of the observed protein values, as well as the sample size to compute these measures. Are your results the same as those computed by `glance()`?? *Assign your answer to an object called answer1.2.1 Your answer should be either \"true\" or \"false\", surrounded by quotes.*\n",
+    "\n",
+    "*Fill out those parts indicated with `...`, uncomment the corresponding code in the cell below, and run it.*"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "7f2dae5fb38918058b61abd298b6f7f5",
+     "grade": false,
+     "grade_id": "cell-52f593f50e45adde",
+     "locked": false,
+     "schema_version": 3,
+     "solution": true,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# n_tissues <- ...\n",
+    "\n",
+    "# SLR_ENSG00000085733_gof <- ...(...)\n",
+    "# my_gof <- dat_ENSG00000085733 %>% \n",
+    "#  summarize(RSS= sum(...^2),TSS=...(...)^2*(n_tissues - 1),\n",
+    "#            myR2 = 1- .../..., my.adjR2 = 1- (.../(...))/(.../(...)),\n",
+    "#            my.sigma = sqrt(.../(...)))\n",
+    "# answer1.2.1 <- ...\n",
+    "\n",
+    "\n",
+    "#SLR_ENSG00000085733_gof\n",
+    "#my_gof\n",
+    "\n",
+    "# your code here\n",
+    "fail() # No Answer - remove if you provide an answer"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "d478330b9495ba71ac5d2dcec82293b9",
+     "grade": true,
+     "grade_id": "cell-d76ca69750b31b53",
+     "locked": true,
+     "points": 2,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "test_1.2.0()\n",
+    "test_1.2.1()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "0e2179f415844da3905c0056f72414f1",
+     "grade": false,
+     "grade_id": "cell-eaa6c4456b1dcd1b",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "**Question 1.3**\n",
+    "<br>{points: 1}\n",
+    "\n",
+    "The output of `glance()` includes a $p$-value corresponding to a hypothesis test that compares the proposed SLR $\\text{prot}_t=\\beta_0 + \\beta_1 \\text{mrna}_{t} + \\varepsilon_t$ with a reduced (null) model. Which of the following models is the null model in this comparison?\n",
+    "\n",
+    "**A.** $\\text{prot}_t=\\beta_0 + \\varepsilon_t$ \n",
+    "\n",
+    "**B.** $\\text{prot}_t=\\beta_0 + \\beta_1 \\text{gene}_{t} + \\varepsilon_t$ \n",
+    "\n",
+    "**C.** $\\text{prot}_t=\\beta_0 + \\beta_1 \\text{gene}_{t} + \\beta_2 \\text{mrna}_{t} + \\varepsilon_t$ \n",
+    "\n",
+    "*Assign your answer to an object called `answer1.3`. Your answer should be one of `\"A\"`, `\"B\"`, or `\"C\"` surrounded by quotes.*"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "11143cdba8a0594ae5b7d8e0d0988e99",
+     "grade": false,
+     "grade_id": "cell-7ff1d5b6dd3bb3ca",
+     "locked": false,
+     "schema_version": 3,
+     "solution": true,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# answer1.3 <-\n",
+    "\n",
+    "# your code here\n",
+    "fail() # No Answer - remove if you provide an answer"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "cbe32f9a2dfff992cb423d0fbf8030cd",
+     "grade": true,
+     "grade_id": "cell-20e11a935946c142",
+     "locked": true,
+     "points": 1,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "test_1.3()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "cee41168900f3c0907cc1c82db76477a",
+     "grade": false,
+     "grade_id": "cell-c02c0296af3d1722",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "**Question 1.4**\n",
+    "<br>{points: 1}\n",
+    "\n",
+    "Select the code(s) that can be used to compute the same test as that included in the output of `glance()`:\n",
+    "\n",
+    "**A.** `anova(lm(prot~ 1, dat_ENSG00000085733))`\n",
+    "\n",
+    "**B.** `anova(lm(prot~1, dat_ENSG00000085733), lm(prot~ mrna, dat_ENSG00000085733))`\n",
+    "\n",
+    "**C.** `tidy(lm(prot~ mrna, dat_ENSG00000085733)) %>% filter(term == 'mrna') %>% pull(p.value)`\n",
+    "\n",
+    "*Assign your answer to an object called `answer1.4`. Your answers have to be included in a single string indicating the correct options in alphabetical order and surrounded by quotes (e.g., `\"ABC\"` indicates you are selecting the three options).*"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "728382fc4ac5538f845c1ff17683b97a",
+     "grade": false,
+     "grade_id": "cell-1cfade5ebf2dbb30",
+     "locked": false,
+     "schema_version": 3,
+     "solution": true,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# answer1.4 <-\n",
+    "\n",
+    "# your code here\n",
+    "fail() # No Answer - remove if you provide an answer"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "fa867f05809ea5adcfc0855dd5532830",
+     "grade": true,
+     "grade_id": "cell-10712f928dbb5402",
+     "locked": true,
+     "points": 1,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "test_1.4()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "930566279f8eabd5330607d43759cbf2",
+     "grade": false,
+     "grade_id": "cell-ffbd231cc330cd79",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "**Question 1.5**\n",
+    "<br>{points: 1}\n",
+    "\n",
+    "Based on the results of the goodness-of-fit measurements obtained for the estimated SLR for gene dat_ENSG00000085733, select all the claims that are true:\n",
+    "\n",
+    "**A.** These results suggest that for the selected gene, a gene-specific model explains less than 6% of the observed variation in protein abundance. \n",
+    "\n",
+    "**B.** A gene-specific model fits the data well. \n",
+    "\n",
+    "**C.** It now becomes possible to predict protein abundance in any given tissue with good accuracy from the measured mRNA for any gene.  \n",
+    "\n",
+    "**D.** There is not enough evidence that the gene-specific SLR using `mrna` as an input is better than using the average protein level to predict protein abundances.\n",
+    "\n",
+    "*Assign your answer to an object called `answer1.5`. Your answers have to be included in a single string indicating the correct options in alphabetical order and surrounded by quotes (e.g., `\"ABCD\"` indicates you are selecting the four options).*"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "8aad39343967ae2bb280ba3ed5f64c55",
+     "grade": false,
+     "grade_id": "cell-ab28b7843d179481",
+     "locked": false,
+     "schema_version": 3,
+     "solution": true,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# answer1.5 <-\n",
+    "\n",
+    "# your code here\n",
+    "fail() # No Answer - remove if you provide an answer"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "d841f78852406c4b6a15dc5db922e815",
+     "grade": true,
+     "grade_id": "cell-afa5092f5e23c8f0",
+     "locked": true,
+     "points": 1,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "test_1.5()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "83d6fdde21a1b29b0de5b48aacd31c96",
+     "grade": false,
+     "grade_id": "cell-68e9eecc55019cda",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    }
+   },
+   "source": [
+    "### F-test and t-test for a SLR\n",
+    "\n",
+    "Note that in this simple case, the only difference between the reduced and the full model is the term with $\\beta_1$\n",
+    "\n",
+    "> we are testing if $\\beta_1$ is different from zero!! $H_0: \\beta_1 = 0$\n",
+    "\n",
+    "Same as before!! \n",
+    "\n",
+    "> Note the $p$-values are the same. This is not a coincidence! \n",
+    "\n",
+    "When we test only one parameter: $t^2 = F$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "908564d57570382d4767d01327f93183",
+     "grade": false,
+     "grade_id": "cell-4166dc30d7557b4f",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    }
+   },
+   "source": [
+    "### 2. Many gene-specific SLR models\n",
+    "\n",
+    "**Remember that this is *only 1* gene in the dataset but there are 1391 more genes!!** Results may be different for other genes."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "75d1437fcb399b7b080cd5b5d583c709",
+     "grade": false,
+     "grade_id": "cell-79247febad83752b",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    }
+   },
+   "source": [
+    "**Question 2.0**\n",
+    "<br>{points: 1}\n",
+    "\n",
+    "In this problem you will: \n",
+    "\n",
+    "- fit 1392 SLR, one for each gene in the dataset using `lm()` and `group_by()`\n",
+    "\n",
+    "- compute the $R^2$ for each gene and estimated SLR using the function `glance()`\n",
+    "\n",
+    "- select the columns `gene`, `r.squared` and `p.value`. Store them in an object called `summary_gof`\n",
+    "\n",
+    "*Fill out those parts indicated with `...`, uncomment the corresponding code in the cell below, and run it.*"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "7fb65b02b72986b01d775ea2fbb9cdb3",
+     "grade": false,
+     "grade_id": "cell-30b40631b6494c74",
+     "locked": false,
+     "schema_version": 3,
+     "solution": true,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "#dat_glance <- dat_bio  %>% \n",
+    "#  group_by(...)  %>% \n",
+    "#  do(glance(...(... ~ ..., data = .))) \n",
+    "\n",
+    "#summary_gof <- dat_glance %>% \n",
+    "#  select(..., ..., ...)\n",
+    "\n",
+    "#tail(summary_gof, 3)\n",
+    "\n",
+    "# your code here\n",
+    "fail() # No Answer - remove if you provide an answer"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "2ade93ad6cb031a3b2a438d3bc9f1bf3",
+     "grade": true,
+     "grade_id": "cell-b3ad10a8f196443d",
+     "locked": true,
+     "points": 1,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "test_2.0()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "f0340da0db2dab543befffc93b207ff4",
+     "grade": false,
+     "grade_id": "cell-dfaabf410163976a",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    }
+   },
+   "source": [
+    "#### Visualization of results \n",
+    "\n",
+    "We have computed 1392 coefficients of determination, one for each *gene-specific* fitted model. Let's visualize the results using a histogram of the coefficients of determination"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "f2387f1139267b98a1fcfb4d13b7d3a5",
+     "grade": false,
+     "grade_id": "cell-0dd8c3a8d46da2f2",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "hist_slr_r2 <- summary_gof  %>% \n",
+    "  ggplot(aes(x=r.squared)) + \n",
+    "  geom_histogram(color=\"navy\", bins=20) + \n",
+    "  geom_vline(xintercept=median(summary_gof$r.squared, na.rm=T),color=\"red\")+\n",
+    "  labs(\n",
+    "    title = \"Rsq of gene-specific LS models\",\n",
+    "    x = \"Rsq\",\n",
+    "    y = \"Count\")+\n",
+    "  xlim(0, 1)\n",
+    "\n",
+    "hist_slr_r2"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "255c30f672974a6c5da04649537677f8",
+     "grade": false,
+     "grade_id": "cell-d9d98e89fc0d5560",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    }
+   },
+   "source": [
+    "**Question 2.1**\n",
+    "<br>{points: 1}\n",
+    "\n",
+    "In the following claims, a \"gene-specific model\" refers to a SLR using `mrna` as an input variable and estimated by LS for each gene.\n",
+    "\n",
+    "Select all the claims that are true based on the $R^2$ computed for each gene.\n",
+    "\n",
+    "**A.** These results suggest that the quality of the proposed models vary greatly across genes. For some genes, a gene-specific model explains more than 80% of the observed variation in protein abundance. However, for more than half of the genes it explains less than 15% of the observed variation in protein levels.\n",
+    "\n",
+    "**B.** For the majority of the genes, a gene-specific model fits the data well. \n",
+    "\n",
+    "**C.** With the suggested gene-specific models, it now becomes possible to predict protein abundance in any given tissue with good accuracy from the measured mRNA for any gene.  \n",
+    "\n",
+    "**D.** For gene ENSG00000262246, the gene-specific model explains approximately 87% of the observed variation in protein abundance, making `mrna` statistically significant.\n",
+    "\n",
+    "*Assign your answer to an object called `answer2.1`. Your answer should be one of `\"A\"`, `\"B\"`, `\"C\"`, or `\"D\"` surrounded by quotes.*"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "aee3be7ba3ee12297ac4dfdd3e19c5de",
+     "grade": false,
+     "grade_id": "cell-6f09b9a6725554ff",
+     "locked": false,
+     "schema_version": 3,
+     "solution": true,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# answer2.1 <- \n",
+    "\n",
+    "# your code here\n",
+    "fail() # No Answer - remove if you provide an answer"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "54a6eb0b45ed4f1fd8bd0ccd0d107bbe",
+     "grade": true,
+     "grade_id": "cell-7c63e0b50f9f1aca",
+     "locked": true,
+     "points": 1,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "test_2.1()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "4a4295d29c89c93b84a23de22d3b2acd",
+     "grade": false,
+     "grade_id": "cell-e0a944bf45f4d345",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "**Question 2.2**\n",
+    "<br>{points: 1}\n",
+    "\n",
+    "The $R^2$ provides a measure for the goodness-of-fit for each model but is not a statistical test. In this exercise, you will examine the $p$-values from the $F$-tests for each gene. Each of these tests compares the full model with an intercept-only model. For these simple particular models, the null hypothesis is:\n",
+    "\n",
+    "$H_0: \\beta_1 = 0$\n",
+    "\n",
+    "Plot a histogram of the $p$-values from all gene-specific SLR. Add a solid vertical line with the median of the $p$-values displayed and a dotted line for a threshold value of 0.05.\n",
+    "\n",
+    "*Fill out those parts indicated with `...`, uncomment the corresponding code in the cell below, and run it.*"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "b5dbd96fb6fb0f001649ad0672d89843",
+     "grade": false,
+     "grade_id": "cell-44d9b6ea19ad3cb8",
+     "locked": false,
+     "schema_version": 3,
+     "solution": true,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "#hist_slr_F <- summary_gof  %>% \n",
+    "#  ggplot(aes(x = ...)) + \n",
+    "#  geom_...(color = \"navy\", bins = 20) + \n",
+    "#  geom_...(xintercept = median(summary_gof$..., na.rm = T), color = \"red\") +\n",
+    "#  ...(xintercept = ..., linetype = 2, color = \"green\") +\n",
+    "#  labs(\n",
+    "#    title = \"F-test of gene-specific LS models\",\n",
+    "#    x = \"pvalue\",\n",
+    "#    y = \"Count\")+\n",
+    "#  xlim(0, 1)\n",
+    "\n",
+    "#hist_slr_F\n",
+    "\n",
+    "# your code here\n",
+    "fail() # No Answer - remove if you provide an answer"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "9ba91a3dd74528012f8537d6b435421d",
+     "grade": true,
+     "grade_id": "cell-0232ba9327c6a3f6",
+     "locked": true,
+     "points": 1,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "test_2.2()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "c8f46d89ab776749d8de6ebf24c8a366",
+     "grade": false,
+     "grade_id": "cell-c580661242549a2e",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "**Question 2.3**\n",
+    "<br>{points: 1}\n",
+    "\n",
+    "Looking at the histogram in **Question 2.2** we conclude that for the majority of the genes, a gene-specific model with `mrna` as an input is significantly better than an intercept-only model.\n",
+    "\n",
+    "**TRUE or FALSE??** \n",
+    "\n",
+    "*Assign your answer to an object called answer2.3 Your answer should be either \"true\" or \"false\", surrounded by quotes.*"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "68e9645270ed37b30ad3c6d22d789577",
+     "grade": false,
+     "grade_id": "cell-8af3f1c68b0e62a7",
+     "locked": false,
+     "schema_version": 3,
+     "solution": true,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# answer2.3 <- \n",
+    "\n",
+    "# your code here\n",
+    "fail() # No Answer - remove if you provide an answer"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "f28686ab5939b130b9995dd5e535c520",
+     "grade": true,
+     "grade_id": "cell-f2bf502e0b4969cb",
+     "locked": true,
+     "points": 1,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "test_2.3()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "2ede91b8bd2933d19e7e7ef698318129",
+     "grade": false,
+     "grade_id": "cell-5a3e26705d4f1410",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "### 3. Other LR models\n",
+    "\n",
+    "The authors claimed that it is fundamental to consider different models for different genes and in the previous exercises you fit different models grouping the data by levels of `gene`. However, in previous worksheets we noticed that we can fit different LRs for each level of a categorical variable, simultaneously, adding dummy variables in the model!!. For simplicity, we will use a dataset with only 3 genes, called `dat_3genes`.  \n",
+    "\n",
+    "Let's first outline different models that can be considered to model data from different genes."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "2a6b3fdf0bffe6e6cd11b4a6d2379c78",
+     "grade": false,
+     "grade_id": "cell-0e0703b41e37a2f6",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "**Question 3.0**\n",
+    "<br>{points: 1}\n",
+    "\n",
+    "Consider the following models:\n",
+    "\n",
+    "- model.1: $\\text{prot}_t=\\beta_0 + \\varepsilon_t$ \n",
+    "\n",
+    "- model.2:  $\\text{prot}_t=\\beta_0 + \\beta_1 \\text{mrna}_{t} + \\varepsilon_t$ \n",
+    "\n",
+    "- model.3:  $\\text{prot}_t=\\beta_0 + \\beta_2 \\text{gene2}_{t} + \\beta_3 \\text{gene3}_{t} + \\varepsilon_t$ \n",
+    "\n",
+    "- model.4:  $\\text{prot}_t=\\beta_0 + \\beta_1 \\text{mrna}_{t} + \\beta_2 \\text{gene2}_{t} + \\beta_3 \\text{gene3}_{t} + \\varepsilon_t$ \n",
+    "\n",
+    "- model.5:  $\\text{prot}_t=\\beta_0 + \\beta_1 \\text{mrna}_{t} + \\beta_2 \\text{gene2}_{t} + \\beta_3 \\text{gene3}_{t} + \\beta_4 \\text{gene2}_{t}\\text{mrna}_{t} + \\beta_5 \\text{gene3}_{t}\\text{mrna}_{t} + \\varepsilon_t$ \n",
+    "\n",
+    "In previous worksheets, you've learned how to fit and interpret these models. Match the equations to the codes you can use to estimate these models using `dat_3genes`:\n",
+    "\n",
+    "**A.** `lm(prot ~ mrna * gene, dat_3genes)`\n",
+    "\n",
+    "**B.** `lm(prot ~ gene, dat_3genes)`\n",
+    "\n",
+    "**C.** `lm(prot ~ mrna + gene, dat_3genes)`\n",
+    "\n",
+    "**D.** `lm(prot ~ 1, dat_3genes)`\n",
+    "\n",
+    "**E.** `lm(prot ~ mrna, dat_3genes)`\n",
+    "\n",
+    "\n",
+    "*Assign your answers to the objects `model.1`, `model.2`, `model.3`, `model.4`, and `model.5`. Your answer should each be a single character (`\"A\"`, `\"B\"`, `\"C\"`, `\"D\"`, or `\"E\"`) surrounded by quotes.*"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "04a5d3f7772844189eb56a46f23e6b2e",
+     "grade": false,
+     "grade_id": "cell-a9eead27577f67a5",
+     "locked": false,
+     "schema_version": 3,
+     "solution": true,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# model.1 <- ...\n",
+    "# model.2 <- ...\n",
+    "# model.3 <- ...\n",
+    "# model.4 <- ...\n",
+    "# model.5 <- ...\n",
+    "\n",
+    "# your code here\n",
+    "fail() # No Answer - remove if you provide an answer"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "ae4191e0c3cc7c166e54ff67bfb7447c",
+     "grade": true,
+     "grade_id": "cell-eb9a5dd6f9b891ce",
+     "locked": true,
+     "points": 1,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "test_3.0()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "2f4676e25d7acf0b37a867d08634d825",
+     "grade": false,
+     "grade_id": "cell-47b313837db7eed1",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "Let's randomly select 3 genes to run some analyses:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "aa87ab5b2fb674429d7b064c5a19562a",
+     "grade": false,
+     "grade_id": "cell-5ca588daa7120836",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "#run this cell\n",
+    "set.seed(561)\n",
+    "dat_3genes <- dat_bio  %>%  \n",
+    "         subset(gene %in% sample(gene,3)) "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "b26d2d473bc583dbd39b4fe2c2352225",
+     "grade": false,
+     "grade_id": "cell-b274ebba90dde81b",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "**Question 3.1**\n",
+    "<br>{points: 2}\n",
+    "\n",
+    "Using the data from the 3 selected genes in `dat_3genes`, fit a model with interaction terms (`model.5`). Store the results in an object called `mlr_3genes_int`.\n",
+    "\n",
+    "Use `tidy()` to obtain a table with results from the LS estimation and inference, call the output `mlr_3genes_int_results`.\n",
+    "\n",
+    "Use `glance()` to obtain goodness-of-fit  for this model, call the output `mlr_3genes_int_gof`. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "3a8c537cd3627f73c1cdd3e9d0f31966",
+     "grade": false,
+     "grade_id": "cell-5f1240b71f992980",
+     "locked": false,
+     "schema_version": 3,
+     "solution": true,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "#mlr_3genes_int <- ...(...~ ..., data = ...)\n",
+    "#mlr_3genes_int_results <- tidy(...)\n",
+    "#mlr_3genes_int_gof <- glance(...) %>% round(3)\n",
+    "\n",
+    "#mlr_3genes_int_results\n",
+    "#mlr_3genes_int_gof\n",
+    "\n",
+    "# your code here\n",
+    "fail() # No Answer - remove if you provide an answer"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "811028de20b0990dde2b4836a137c3fd",
+     "grade": true,
+     "grade_id": "cell-17675f9b1e991ece",
+     "locked": true,
+     "points": 2,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "test_3.1.0()\n",
+    "test_3.1.1()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "e9ae0fa15d617762d0212b055858a116",
+     "grade": false,
+     "grade_id": "cell-0b072a63d9248e64",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "**Question 3.2**\n",
+    "<br>{points: 1}\n",
+    "\n",
+    "The output of `glance()` for the model with interaction includes the $p$-value of a test of hypothesis. Which null hypothesis is tested?\n",
+    "\n",
+    "**A.** $H_0: \\beta_2 = \\beta_3 = 0$ vs. $H_1: \\text{at least one } \\beta_j \\neq 0 \\text{ (for } $j = 2, 3$ \\text{)}$\n",
+    "\n",
+    "**B.** $H_0: \\beta_4 = \\beta_5 = 0$ vs. $H_1: \\text{at least one } \\beta_j \\neq 0 \\text{ (for } $j = 4, 5$ \\text{)}$\n",
+    "\n",
+    "**C.** $H_0: \\beta_0 = \\beta_1 = \\beta_2 = \\beta_3 = \\beta_4 = \\beta_5 = 0$ vs. $H_1: \\text{at least one } \\beta_j \\neq 0 \\text{ (for } $j = 0, 1, 2, 3, 4, 5$ \\text{)}$\n",
+    "\n",
+    "**D.** $H_0: \\beta_1 = \\beta_2 = \\beta_3 = \\beta_4 = \\beta_5 = 0$ vs. $H_1: \\text{at least one } \\beta_j \\neq 0 \\text{ (for } $j = 1, 2, 3, 4, 5$ \\text{)}$\n",
+    "\n",
+    "**E.** $H_0: \\hat{\\beta}_2 = \\hat{\\beta}_3 = 0$ vs. $H_1: \\text{at least one } \\hat{\\beta}_j \\neq 0 \\text{ (for } $j = 2, 3$ \\text{)}$\n",
+    "\n",
+    "*Assign your answer to an object called `answer3.2`. Your answer should be one of `\"A\"`, `\"B\"`, `\"C\"`, `\"D\"` or `\"E\"` surrounded by quotes.*"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "c2c82524372c48166aca419fcf19ef78",
+     "grade": false,
+     "grade_id": "cell-477c4ef09e58a3e4",
+     "locked": false,
+     "schema_version": 3,
+     "solution": true,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# answer3.2 <- \n",
+    "\n",
+    "# your code here\n",
+    "fail() # No Answer - remove if you provide an answer"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "b6a96cf61fd9f1d639a27b1da4c439c2",
+     "grade": true,
+     "grade_id": "cell-3ea3c3b5606f7047",
+     "locked": true,
+     "points": 1,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "test_3.2()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "71ee43e989a847a0fdefb0fb38b1bf7e",
+     "grade": false,
+     "grade_id": "cell-0fbccf4edd998dda",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    }
+   },
+   "source": [
+    "**Question 3.3**\n",
+    "<br>{points: 1}\n",
+    "\n",
+    "Use the function `anova()` to reproduce this test using `dat_3genes`. \n",
+    "\n",
+    "Store $F$-test and its correspoinding $p$-value in an object called `Ftest_3genes_interaction`.\n",
+    "\n",
+    "*Fill out those parts indicated with `...`, uncomment the corresponding code in the cell below, and run it.*"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "66c6d43c59e00bed57f0aa34cc381aed",
+     "grade": false,
+     "grade_id": "cell-20ab328a3e81d820",
+     "locked": false,
+     "schema_version": 3,
+     "solution": true,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# Ftest_3genes_interaction  <- ...(..., ...)  \n",
+    "\n",
+    "# Ftest_3genes_interaction \n",
+    "\n",
+    "# your code here\n",
+    "fail() # No Answer - remove if you provide an answer"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "3b5e90460e339d7911a7904361823c0d",
+     "grade": true,
+     "grade_id": "cell-727820da4a4d0b48",
+     "locked": true,
+     "points": 1,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "test_3.3()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "ba3c2b620a634433ad4964d5bec9b3ae",
+     "grade": false,
+     "grade_id": "cell-533bd8c35b5c0851",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "**Question 3.4**\n",
+    "<br>{points: 1}\n",
+    "\n",
+    "Which of the following claims is correct based on the results from  **Question 3.3**.\n",
+    "\n",
+    "**A.** The three linear regressions are statistically significant.\n",
+    "\n",
+    "**B.** There is enough evidence to reject the null hypothesis that the model with interaction terms (`model.5`) is equivalent to an intercept-only model (`model.1`)\n",
+    "\n",
+    "**C.** `mrna` is statistically significant.\n",
+    "\n",
+    "\n",
+    "*Assign your answer to an object called `answer3.4`. Your answer should be one of `\"A\"`, `\"B\"`, or `\"C\"` surrounded by quotes.*"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "3dad10375d65566579fcfc8e6964e067",
+     "grade": false,
+     "grade_id": "cell-a0b0accd3b2e1b8a",
+     "locked": false,
+     "schema_version": 3,
+     "solution": true,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# answer3.4 <- \n",
+    "\n",
+    "# your code here\n",
+    "fail() # No Answer - remove if you provide an answer"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "993db9c546703951bea6de80add12f76",
+     "grade": true,
+     "grade_id": "cell-a3ca37511dc2b2ad",
+     "locked": true,
+     "points": 1,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "test_3.4()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "d31dc70f79ae385b73aabab993ab9656",
+     "grade": false,
+     "grade_id": "cell-b7a372995c317f97",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "### How does the interaction model fit the data?\n",
+    "\n",
+    "Results from **Question 3.1** suggests that `model.5` does not fit the data well, even if it is better than the intercept-only model. Only 41% of the variation observed in protein levels is explained by the model! While the $p$-value from `glance()` indicates that there is enough evidence to reject $H_0$, it still doesn't mean that `mrna` is a relevant variable. Let's compare this model with other models. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "3c05f05189ab6b6dad0e965eac25b5d1",
+     "grade": false,
+     "grade_id": "cell-b46cc7938531182c",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "To examine that question let's compare the following nested models:\n",
+    "\n",
+    "$$\\textbf{reduced}:\\text{prot}_t=\\beta_0 + \\beta_2 \\text{gene2}_{t} + \\beta_3 \\text{gene3}_{t} + \\varepsilon_t$$\n",
+    "\n",
+    "$$\\textbf{full}:\\text{prot}_t = \\beta_0 + \\beta_1 \\text{mrna}_{t} + \\beta_2 \\text{gene2}_{t}  + \\beta_3 \\text{gene3}_{t}  + \\beta_4 \\text{gene2}_{t}  \\text{mrna}_{t} + \\beta_5 \\text{gene3}_{t}  \\text{mrna}_{t} +\\varepsilon_t$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "e09c706c3b6bfd4bfe1170e1ba6f4572",
+     "grade": false,
+     "grade_id": "cell-c52a18cc8af85c36",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    }
+   },
+   "source": [
+    "**Question 3.5**\n",
+    "<br>{points: 1}\n",
+    "\n",
+    "An $F$-test can be used to test *simultaneously* whether the additional parameters in the full model are zero. Which hypotheses need to be tested using an $F$-test?\n",
+    "\n",
+    "**A.** $H_0: \\beta_2 = \\beta_3 = 0$ vs. $H_1: \\text{at least one } \\beta_j \\neq 0 \\text{ (for } $j = 2, 3$ \\text{)}$\n",
+    "\n",
+    "**B.** $H_0: \\beta_1 = \\beta_4 = \\beta_5 = 0$ vs. $H_1: \\text{at least one } \\beta_j \\neq 0 \\text{ (for } $j = 1, 4, 5$ \\text{)}$\n",
+    "\n",
+    "**C.** $H_0: \\beta_0 = \\beta_1 = \\beta_2 = \\beta_3 = \\beta_4 = \\beta_5 = 0$ vs. $H_1: \\text{at least one } \\beta_j \\neq 0 \\text{ (for } $j = 0, 1, 2, 3, 4, 5$ \\text{)}$\n",
+    "\n",
+    "**D.** $H_0: \\beta_1 = \\beta_2 = \\beta_3 = \\beta_4 = \\beta_5 = 0$ vs. $H_1: \\text{at least one } \\beta_j \\neq 0 \\text{ (for } $j = 1, 2, 3, 4, 5$ \\text{)}$\n",
+    "\n",
+    "**E.** $H_0: \\hat{\\beta}_2 = \\hat{\\beta}_3 = 0$ vs. $H_1: \\text{at least one } \\hat{\\beta}_j \\neq 0 \\text{ (for } $j = 2, 3$ \\text{)}$\n",
+    "\n",
+    "*Assign your answer to an object called `answer3.5`. Your answer should be one of `\"A\"`, `\"B\"`, `\"C\"`, \"D\"` or `\"E\"` surrounded by quotes.*"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "9c58fcaf117a23cfa106fd30bc723ca7",
+     "grade": false,
+     "grade_id": "cell-b5723b1752ac1294",
+     "locked": false,
+     "schema_version": 3,
+     "solution": true,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# answer3.5 <- \n",
+    "\n",
+    "# your code here\n",
+    "fail() # No Answer - remove if you provide an answer"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "f5b516f9593a0157f21221e5917ca6fa",
+     "grade": true,
+     "grade_id": "cell-c1bf7a0bc0c2ef25",
+     "locked": true,
+     "points": 1,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "test_3.5()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "14f71cf4fae75856318f9fc112fc43c7",
+     "grade": false,
+     "grade_id": "cell-377ceb2f92921288",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    }
+   },
+   "source": [
+    "**Question 3.6**\n",
+    "<br>{points: 1}\n",
+    "\n",
+    "The following hypotheses compare the reduced (`model.3` without `mrna`) and the full model (`model.5` with interaction) stated above.\n",
+    "\n",
+    "Select the code that you can use to test these hypotheses.\n",
+    "\n",
+    "**A.** `tidy(lm(prot ~ gene * mrna, dat_3genes))`\n",
+    "\n",
+    "**B.** `tidy(lm(prot ~ gene + mrna, dat_3genes))`\n",
+    "\n",
+    "**C.** `glance(lm(prot ~ gene * mrna, dat_3genes))`\n",
+    "\n",
+    "**D.** `glance(lm(prot ~ gene + mrna, dat_3genes))`\n",
+    "\n",
+    "**E.** `anova(lm(prot ~ gene, dat_3genes), lm(prot ~ gene * mrna, dat_3genes))`\n",
+    "\n",
+    "\n",
+    "*Assign your answer to an object called `answer3.6`. Your answer should be one of `\"A\"`, `\"B\"`, `\"C\"`, `\"D\"`, or `\"E\"` surrounded by quotes.*"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "8c0e14f1843980216c0e8460cbc48e6a",
+     "grade": false,
+     "grade_id": "cell-9d0d4fac52a22f63",
+     "locked": false,
+     "schema_version": 3,
+     "solution": true,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# answer3.6 <- \n",
+    "\n",
+    "# your code here\n",
+    "fail() # No Answer - remove if you provide an answer"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "6a7fa961cc60b98871472411c53a1766",
+     "grade": true,
+     "grade_id": "cell-6c817e4c8dcc41ee",
+     "locked": true,
+     "points": 1,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "test_3.6()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "835cdcfc245c9055cecd8b6860f9a9b7",
+     "grade": false,
+     "grade_id": "cell-2876dc9f0b8a015a",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "**Question 3.7**\n",
+    "<br>{points: 1}\n",
+    "\n",
+    "Use the function `anova()` to run the test from **Question 3.5**. Store your results in an object called `Ftest_3genes_mrna`. Your output should be a table of size (2 x 6)\n",
+    "\n",
+    "*Fill out those parts indicated with `...`, uncomment the corresponding code in the cell below, and run it.*"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "9863449eade2d72a694a5d825427d552",
+     "grade": false,
+     "grade_id": "cell-a290ce6bc1a11890",
+     "locked": false,
+     "schema_version": 3,
+     "solution": true,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "#[write your code here]\n",
+    "#Ftest_3genes_mrna\n",
+    "\n",
+    "# your code here\n",
+    "fail() # No Answer - remove if you provide an answer"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "code",
+     "checksum": "a1586b962e6c9c7009c6fcd76f6d83a1",
+     "grade": true,
+     "grade_id": "cell-4f1cde48cac3b95a",
+     "locked": true,
+     "points": 1,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "test_3.7()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "4e1eb1439f737a1675cba60edaff47ba",
+     "grade": false,
+     "grade_id": "cell-1de0636f42b700c8",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    },
+    "tags": []
+   },
+   "source": [
+    "#### Conclusions from the analysis\n",
+    "\n",
+    "While the authors claimed that \"***it now becomes possible to predict protein abundance in any given tissue with good accuracy from the measured mRNA abundance***\" as long as we consider different models for different genes, the test above does not provide enough evidence to support this hypothesis (at least for the 3 genes selected). Note that the complex model with different LR for each genes is not significantly different than a model without mRNA!! Most of the variation in the data seems to be explained by the variable `gene`!"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": false,
+    "editable": false,
+    "nbgrader": {
+     "cell_type": "markdown",
+     "checksum": "00ecdf3d78cd4b12c4017eb80f675ad7",
+     "grade": false,
+     "grade_id": "cell-a7f0358b7a8d93f7",
+     "locked": true,
+     "schema_version": 3,
+     "solution": false,
+     "task": false
+    }
+   },
+   "source": [
+    "## Conclusions of PART I\n",
+    "\n",
+    "#### Evaluation of Models when the main goal is estimation and inference:\n",
+    "\n",
+    "- The $R^2$, coefficient of determination, can be used to compared the sum of squares of the residuals of the fitted model with that of the null model\n",
+    "\n",
+    "\n",
+    "- The $R^2$ is usually interpreted as the part of the variation in the response explained by the model\n",
+    "\n",
+    "\n",
+    "- Many definitions and interpretations of the $R^2$ are for LS estimators of LR containing an intercept\n",
+    "\n",
+    "\n",
+    "- The $R^2$ is not a test and it does not provide a probabilistic result and it's distribution is unknown!\n",
+    "\n",
+    "\n",
+    "- Instead, we can use an $F$ test, also refer as ANOVA, to compare nested models\n",
+    "\n",
+    "    - tests the simultaneous significance of additional coefficients of the full model (not in the reduced model)\n",
+    "    \n",
+    "    - in particular, we can use it to test the significance of the fitted model over the null model\n",
+    "    \n",
+    "\n",
+    "- These $F$ tests can be used to select variables!! since we are comparing and testing how the fit changes as we select variables\n",
+    "\n",
+    "#### Evaluation of Models when the main goal is prediction:\n",
+    "\n",
+    "- The test MSE is a natural measure to compare new responses from a test set with the predicted values $\\hat{y}_i$ from the LR estimated with training data\n",
+    "\n",
+    "- The $R^2$ based on test data can also be used but it should not be called a \"coefficient of determination\" "
+   ]
+  }
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