Multiple linear regression
Grading the professor
Many college courses conclude by giving students the opportunity to evaluate the course and the instructor anonymously. However, the use of these student evaluations as an indicator of course quality and teaching effectiveness is often criticized because these measures may reflect the influence of non-teaching related characteristics, such as the physical appearance of the instructor. The article titled, “Beauty in the classroom: instructors’ pulchritude and putative pedagogical productivity” (Hamermesh and Parker, 2005) found that instructors who are viewed to be better looking receive higher instructional ratings. (Daniel S. Hamermesh, Amy Parker, Beauty in the classroom: instructors pulchritude and putative pedagogical productivity, Economics of Education Review, Volume 24, Issue 4, August 2005, Pages 369-376, ISSN 0272-7757, 10.1016/j.econedurev.2004.07.013. http://www.sciencedirect.com/science/article/pii/S0272775704001165.)
In this lab we will analyze the data from this study in order to learn what goes into a positive professor evaluation.
The data
The data were gathered from end of semester student evaluations for a large sample of professors from the University of Texas at Austin. In addition, six students rated the professors’ physical appearance. (This is aslightly modified version of the original data set that was released as part of the replication data for Data Analysis Using Regression and Multilevel/Hierarchical Models (Gelman and Hill, 2007).) The result is a data frame where each row contains a different course and columns represent variables about the courses and professors.
load("more/evals.RData")| variable | description |
|---|---|
score |
average professor evaluation score: (1) very unsatisfactory - (5) excellent. |
rank |
rank of professor: teaching, tenure track, tenured. |
ethnicity |
ethnicity of professor: not minority, minority. |
gender |
gender of professor: female, male. |
language |
language of school where professor received education: english or non-english. |
age |
age of professor. |
cls_perc_eval |
percent of students in class who completed evaluation. |
cls_did_eval |
number of students in class who completed evaluation. |
cls_students |
total number of students in class. |
cls_level |
class level: lower, upper. |
cls_profs |
number of professors teaching sections in course in sample: single, multiple. |
cls_credits |
number of credits of class: one credit (lab, PE, etc.), multi credit. |
bty_f1lower |
beauty rating of professor from lower level female: (1) lowest - (10) highest. |
bty_f1upper |
beauty rating of professor from upper level female: (1) lowest - (10) highest. |
bty_f2upper |
beauty rating of professor from second upper level female: (1) lowest - (10) highest. |
bty_m1lower |
beauty rating of professor from lower level male: (1) lowest - (10) highest. |
bty_m1upper |
beauty rating of professor from upper level male: (1) lowest - (10) highest. |
bty_m2upper |
beauty rating of professor from second upper level male: (1) lowest - (10) highest. |
bty_avg |
average beauty rating of professor. |
pic_outfit |
outfit of professor in picture: not formal, formal. |
pic_color |
color of professor’s picture: color, black & white. |
Exploring the data
- Is this an observational study or an experiment? The original research question posed in the paper is whether beauty leads directly to the differences in course evaluations. Given the study design, is it possible to answer this question as it is phrased? If not, rephrase the question.
Answer :- Observational study.
- Describe the distribution of
score. Is the distribution skewed? What does that tell you about how students rate courses? Is this what you expected to see? Why, or why not?
Answer:-As distribution is left skewed it shows that the students are less likely to rate for courses.
hist(evals$score)
- Excluding
score, select two other variables and describe their relationship using an appropriate visualization (scatterplot, side-by-side boxplots, or mosaic plot).
Answer :-This show that average beauty per professor diminshes by age.
evals## score rank ethnicity gender language age cls_perc_eval
## 1 4.7 tenure track minority female english 36 55.81395
## 2 4.1 tenure track minority female english 36 68.80000
## 3 3.9 tenure track minority female english 36 60.80000
## 4 4.8 tenure track minority female english 36 62.60163
## 5 4.6 tenured not minority male english 59 85.00000
## 6 4.3 tenured not minority male english 59 87.50000
## 7 2.8 tenured not minority male english 59 88.63636
## 8 4.1 tenured not minority male english 51 100.00000
## 9 3.4 tenured not minority male english 51 56.92308
## 10 4.5 tenured not minority female english 40 86.95652
## 11 3.8 tenured not minority female english 40 88.88889
## 12 4.5 tenured not minority female english 40 96.00000
## 13 4.6 tenured not minority female english 40 85.00000
## 14 3.9 tenured not minority female english 40 56.00000
## 15 3.9 tenured not minority female english 40 88.09524
## 16 4.3 tenured not minority female english 40 90.00000
## 17 4.5 tenured not minority female english 40 83.33334
## 18 4.8 tenure track not minority female english 31 87.50000
## 19 4.6 tenure track not minority female english 31 90.90909
## 20 4.6 tenure track not minority female english 31 79.16666
## 21 4.9 tenure track not minority female english 31 88.88889
## 22 4.6 tenure track not minority female english 31 88.13559
## 23 4.5 tenure track not minority female english 31 56.32184
## 24 4.4 tenured not minority male english 62 64.53901
## 25 4.6 tenured not minority male english 62 54.79452
## 26 4.7 tenured not minority male english 62 60.76923
## 27 4.5 tenured not minority male english 62 61.75439
## 28 4.8 tenured not minority male english 62 56.98529
## 29 4.9 tenured not minority male english 62 58.04196
## 30 4.5 tenured not minority male english 62 61.58941
## 31 4.4 tenure track not minority female english 33 80.48781
## 32 4.3 tenure track not minority female english 33 85.29412
## 33 4.1 tenure track not minority female english 33 90.24390
## 34 4.2 tenure track not minority female english 33 70.73170
## 35 3.5 tenure track not minority female english 33 82.35294
## 36 3.4 tenured not minority female english 51 60.97561
## 37 4.5 tenured not minority female english 51 95.45454
## 38 4.4 tenured not minority female english 51 61.90476
## 39 4.4 tenured not minority female english 51 94.11765
## 40 2.5 tenured not minority female english 51 80.00000
## 41 4.3 tenured not minority female english 51 100.00000
## 42 4.5 tenured not minority female english 51 100.00000
## 43 4.8 tenure track not minority female english 33 80.00000
## 44 4.8 tenure track not minority female english 33 87.87878
## 45 4.4 tenure track not minority female english 33 25.00000
## 46 4.7 tenure track not minority female english 33 59.18367
## 47 4.4 tenure track not minority female english 33 86.20689
## 48 4.7 tenure track not minority female english 33 87.50000
## 49 4.5 tenure track not minority female english 33 85.00000
## 50 4.0 teaching not minority male english 47 84.21053
## 51 4.3 teaching not minority male english 47 75.00000
## 52 4.4 teaching not minority male english 47 93.33334
## 53 4.5 teaching not minority male english 47 95.65218
## 54 5.0 teaching not minority male english 47 90.90909
## 55 4.9 teaching not minority male english 47 58.62069
## 56 4.6 teaching not minority male english 47 76.19048
## 57 5.0 teaching not minority male english 47 83.33334
## 58 4.7 teaching not minority male english 47 84.21053
## 59 5.0 teaching not minority male english 47 80.00000
## 60 3.6 tenure track minority male non-english 35 72.00000
## 61 3.7 tenure track minority male non-english 35 90.90909
## 62 4.3 tenure track minority male non-english 35 95.83334
## 63 4.1 teaching not minority male english 37 88.23529
## 64 4.2 teaching not minority male english 37 61.90476
## 65 4.7 teaching not minority male english 37 80.00000
## 66 4.7 teaching not minority male english 37 96.00000
## 67 3.5 teaching not minority male english 37 71.42857
## 68 4.1 tenured not minority male english 42 70.00000
## 69 4.2 tenured not minority male english 42 60.00000
## 70 4.0 tenured not minority male english 42 66.66666
## 71 4.0 tenured not minority male english 42 43.85965
## 72 3.9 tenured not minority male english 42 70.17544
## 73 4.4 tenured not minority male english 42 78.43137
## 74 3.8 tenured not minority male english 42 60.00000
## 75 3.5 tenured not minority male non-english 49 83.33334
## 76 4.2 tenured not minority male non-english 49 83.78378
## 77 3.5 tenured not minority male non-english 49 51.72414
## 78 3.6 tenured not minority male non-english 49 85.18519
## 79 2.9 tenure track not minority female english 37 82.14286
## 80 3.3 tenure track not minority female english 37 65.38461
## 81 3.3 tenure track not minority female english 37 80.76923
## 82 3.2 tenure track not minority female english 37 96.66666
## 83 4.6 tenured not minority male english 45 69.69697
## 84 4.2 tenured not minority male english 45 25.42373
## 85 4.3 tenured not minority male english 45 45.22613
## 86 4.4 tenured not minority male english 45 84.37500
## 87 4.1 tenured not minority male english 45 94.59460
## 88 4.6 tenured not minority male english 45 74.53416
## 89 4.4 teaching not minority female english 56 65.85366
## 90 4.8 teaching not minority female english 56 88.63636
## 91 4.3 teaching not minority female english 56 66.03773
## 92 3.6 teaching not minority female english 56 69.38776
## 93 4.3 teaching not minority female english 56 84.37500
## 94 4.0 teaching not minority male english 48 74.07407
## 95 4.2 teaching not minority male english 48 60.60606
## 96 4.1 teaching not minority male english 48 73.68421
## 97 4.1 teaching not minority male english 48 58.55856
## 98 4.4 teaching not minority male english 48 63.75839
## 99 4.3 teaching not minority male english 48 66.66666
## 100 4.4 teaching not minority male english 48 62.50000
## 101 4.4 teaching not minority male english 48 80.71429
## 102 4.9 tenured not minority female english 46 80.64516
## 103 5.0 tenured not minority female english 46 93.33334
## 104 4.4 tenured not minority female english 46 79.31035
## 105 4.8 tenured not minority female english 46 92.00000
## 106 4.9 tenured not minority female english 46 50.00000
## 107 4.3 tenured not minority female english 46 66.66666
## 108 5.0 tenured not minority female english 46 100.00000
## 109 4.7 tenured not minority female english 46 81.57895
## 110 4.5 tenured not minority female english 46 80.00000
## 111 3.5 teaching not minority female english 57 60.71429
## 112 3.9 teaching not minority female english 57 43.47826
## 113 4.0 teaching not minority female english 57 78.94736
## 114 4.0 teaching not minority female english 57 30.43478
## 115 3.7 teaching not minority female english 57 31.81818
## 116 3.4 teaching not minority female english 57 70.00000
## 117 3.3 teaching not minority female english 57 42.10526
## 118 3.8 teaching not minority female english 57 73.91304
## 119 3.9 teaching not minority female english 57 45.45454
## 120 3.4 teaching not minority female english 57 80.00000
## 121 3.7 teaching not minority female english 52 86.36364
## 122 4.1 teaching not minority female english 52 87.09677
## 123 3.7 teaching not minority female english 52 95.23810
## 124 3.5 teaching not minority female english 52 86.11111
## 125 3.5 teaching not minority female english 52 89.47369
## 126 4.4 teaching not minority female english 52 62.16216
## 127 3.4 tenure track minority female non-english 29 73.07692
## 128 4.3 tenured not minority male english 62 76.92308
## 129 3.7 tenured not minority male english 62 51.08696
## 130 4.7 tenured not minority male english 62 92.00000
## 131 3.9 tenured not minority male english 62 50.95541
## 132 3.6 tenured not minority male english 62 37.19512
## 133 4.5 tenured not minority male english 64 62.50000
## 134 4.5 tenured not minority male english 64 75.00000
## 135 4.8 tenured not minority male english 64 46.80851
## 136 4.8 tenured not minority male english 64 71.42857
## 137 4.7 tenured not minority male english 64 73.33334
## 138 4.5 tenured not minority male english 64 62.50000
## 139 4.3 tenured not minority male english 64 53.84615
## 140 4.8 tenure track not minority female english 34 76.92308
## 141 4.1 tenure track not minority female english 34 82.50000
## 142 4.4 tenured not minority male english 58 52.83019
## 143 4.3 tenured not minority male english 58 47.01987
## 144 3.6 tenured not minority male english 58 76.59574
## 145 4.5 tenured not minority male english 58 59.83607
## 146 4.3 tenured not minority male english 58 68.88889
## 147 4.4 tenured minority male non-english 52 81.25000
## 148 4.7 tenured minority male non-english 52 100.00000
## 149 4.8 tenured minority male non-english 52 75.00000
## 150 3.5 tenured minority male non-english 52 83.33334
## 151 3.8 tenured minority male non-english 52 68.75000
## 152 3.6 tenured minority male non-english 52 80.00000
## 153 4.2 tenured minority male non-english 52 64.28571
## 154 3.6 tenured not minority male english 73 70.58823
## 155 4.4 tenured not minority male english 73 76.92308
## 156 3.7 tenured not minority male english 73 76.19048
## 157 4.3 tenured not minority male english 73 76.47059
## 158 4.6 tenured not minority male english 70 35.07463
## 159 4.6 tenured not minority male english 70 10.41667
## 160 4.1 tenured not minority male english 70 53.12500
## 161 3.6 tenured not minority male english 70 34.78261
## 162 2.3 tenure track not minority female english 41 83.33334
## 163 4.3 teaching not minority male english 63 65.11628
## 164 4.4 teaching not minority male english 63 78.57143
## 165 3.6 teaching not minority male english 63 66.66666
## 166 4.4 teaching not minority male english 63 77.77778
## 167 3.9 teaching not minority male english 63 75.00000
## 168 3.8 teaching not minority male english 63 80.00000
## 169 3.4 teaching not minority male english 63 48.93617
## 170 4.9 tenured not minority male english 47 66.66666
## 171 4.1 tenured not minority male english 47 71.42857
## 172 3.2 tenured not minority male english 47 83.33334
## 173 4.2 tenured not minority male english 39 34.95935
## 174 3.9 tenured not minority male english 39 48.73418
## 175 4.9 tenured not minority male english 39 80.00000
## 176 4.7 tenured not minority male english 39 93.33334
## 177 4.4 tenured not minority male english 39 93.10345
## 178 4.2 tenure track minority female english 47 100.00000
## 179 4.0 tenure track minority female english 47 100.00000
## 180 4.4 tenure track minority female english 47 100.00000
## 181 3.9 tenure track minority female english 47 92.30769
## 182 4.4 tenure track minority female english 47 70.00000
## 183 3.0 tenure track minority female english 47 96.15385
## 184 3.5 tenure track minority female english 47 92.30769
## 185 2.8 tenure track minority female english 47 92.30769
## 186 4.6 tenure track minority female english 47 95.23810
## 187 4.3 tenure track minority female english 47 100.00000
## 188 3.4 tenure track minority female english 47 92.59259
## 189 3.0 tenure track minority female english 47 96.29630
## 190 4.2 tenure track minority female english 47 96.00000
## 191 4.3 tenured not minority male english 54 100.00000
## 192 4.1 tenured not minority male english 54 66.66666
## 193 4.6 tenured not minority male english 54 94.11765
## 194 3.9 tenured minority female english 44 54.54546
## 195 3.5 tenured minority female english 44 72.91666
## 196 4.0 tenured minority female english 44 100.00000
## 197 4.0 tenured minority female english 44 89.74359
## 198 3.9 tenured minority male english 47 74.07407
## 199 3.3 tenured minority male english 47 64.28571
## 200 4.0 tenured minority male english 47 80.76923
## 201 3.8 tenured minority male english 47 81.25000
## 202 4.2 tenured minority male english 47 81.25000
## 203 4.0 tenured minority male english 47 92.30769
## 204 3.8 tenured minority male english 47 100.00000
## 205 3.3 tenured minority male english 47 88.23529
## 206 4.1 tenured not minority male english 62 92.30769
## 207 4.7 tenured not minority male english 62 86.66666
## 208 4.4 tenured not minority male english 62 70.00000
## 209 4.8 tenured not minority male english 60 73.52941
## 210 4.8 tenured not minority male english 60 75.00000
## 211 4.6 tenured not minority male english 60 78.57143
## 212 4.6 tenured not minority male english 60 66.66666
## 213 4.8 tenured not minority male english 60 61.53846
## 214 4.4 tenured not minority male english 60 42.85714
## 215 4.7 tenured not minority male english 60 68.88889
## 216 4.7 tenured not minority male english 60 80.00000
## 217 3.3 tenure track not minority male english 37 88.23529
## 218 4.4 tenure track not minority male english 42 92.85714
## 219 4.3 tenure track not minority male english 42 100.00000
## 220 4.9 tenure track not minority male english 42 92.85714
## 221 4.4 tenure track not minority male english 42 100.00000
## 222 4.7 tenure track not minority male english 42 86.66666
## 223 4.3 tenured not minority male english 35 82.35294
## 224 4.8 tenured not minority male english 35 91.30434
## 225 4.5 tenured not minority male english 35 85.96491
## 226 4.7 tenured not minority male english 35 88.00000
## 227 3.3 teaching not minority female english 39 91.66666
## 228 4.7 teaching not minority female english 39 78.26087
## 229 4.6 teaching not minority female english 39 86.95652
## 230 3.6 teaching not minority female english 39 96.42857
## 231 4.0 tenured not minority male english 49 62.22222
## 232 4.1 tenured not minority male english 49 64.28571
## 233 4.0 tenured not minority male english 49 85.96491
## 234 4.5 tenured not minority male english 61 81.48148
## 235 4.6 tenured not minority male english 61 81.57895
## 236 4.8 tenured not minority male english 61 86.36364
## 237 4.6 tenured not minority male english 61 62.79070
## 238 4.9 tenure track not minority male english 33 96.77419
## 239 3.1 tenure track not minority male english 33 92.30769
## 240 3.7 tenure track not minority male english 33 86.66666
## 241 3.7 tenured not minority female english 58 76.47059
## 242 3.9 tenured not minority female english 56 63.15789
## 243 3.9 tenured not minority female english 56 80.00000
## 244 3.2 tenured not minority female english 56 73.91304
## 245 4.4 teaching not minority female english 50 81.48148
## 246 4.2 teaching not minority female english 50 75.00000
## 247 4.7 teaching not minority female english 50 66.66666
## 248 3.9 teaching not minority female english 50 95.83334
## 249 3.6 teaching not minority female english 50 90.47619
## 250 3.4 teaching not minority female english 50 64.28571
## 251 4.4 teaching not minority female english 50 79.31035
## 252 4.4 tenured not minority male english 52 85.07462
## 253 4.1 tenured not minority male english 52 91.01124
## 254 3.6 tenured not minority male english 52 90.24390
## 255 3.5 tenured not minority male english 52 83.60656
## 256 4.1 tenured not minority male english 52 71.75572
## 257 3.8 tenured not minority male english 52 78.07018
## 258 4.0 tenured not minority male english 52 89.26174
## 259 4.8 tenured not minority male english 52 95.65218
## 260 4.2 tenured not minority male english 52 79.59184
## 261 4.6 tenured not minority male english 52 81.48148
## 262 4.3 tenured not minority male english 52 90.00000
## 263 4.8 tenured not minority male english 52 90.00000
## 264 3.8 tenured not minority male english 52 86.95652
## 265 4.5 tenure track not minority female english 33 66.66666
## 266 4.9 tenure track not minority female english 33 100.00000
## 267 4.9 tenure track not minority female english 33 90.90909
## 268 4.8 tenure track not minority female english 33 71.42857
## 269 4.7 tenure track not minority female english 33 63.63636
## 270 4.6 tenure track not minority female english 33 100.00000
## 271 4.3 tenured not minority male english 57 89.61039
## 272 4.4 tenured not minority male english 57 87.80488
## 273 4.5 tenured not minority male english 57 81.81818
## 274 4.2 tenured not minority male english 57 80.76923
## 275 4.8 tenured not minority female english 38 70.76923
## 276 4.6 tenured not minority female english 38 69.42675
## 277 4.9 tenured not minority female english 38 79.41177
## 278 4.8 tenured not minority female english 38 76.11940
## 279 4.8 tenured not minority female english 38 76.25000
## 280 4.6 tenured not minority female english 38 74.45255
## 281 4.7 tenured not minority female english 38 84.05797
## 282 4.1 tenure track not minority female english 34 59.34066
## 283 3.8 tenure track not minority female english 34 57.50000
## 284 4.0 tenure track not minority female english 34 58.88889
## 285 4.1 tenure track not minority female english 34 55.88235
## 286 4.0 tenure track not minority female english 34 56.16438
## 287 4.1 tenure track not minority female english 34 56.81818
## 288 3.5 tenure track not minority male english 34 66.66666
## 289 4.1 tenure track not minority male english 34 80.00000
## 290 3.6 tenure track not minority male english 34 57.14286
## 291 4.0 tenure track not minority male english 32 34.27419
## 292 3.9 tenure track not minority male english 32 41.07143
## 293 3.8 tenure track not minority male english 32 39.67611
## 294 4.4 tenure track not minority male english 32 90.90909
## 295 4.7 tenure track not minority male english 32 69.90292
## 296 3.8 tenured not minority female english 42 70.96774
## 297 4.1 tenured not minority female english 42 62.19512
## 298 4.1 tenured not minority female english 43 68.62745
## 299 4.7 tenured not minority female english 43 51.42857
## 300 4.3 tenured not minority female english 43 76.47059
## 301 4.4 tenured not minority female english 43 75.67567
## 302 4.5 tenured not minority female english 43 92.85714
## 303 3.1 tenured not minority female english 43 65.41354
## 304 3.7 tenured not minority female english 43 67.32284
## 305 4.5 tenured not minority female english 43 76.92308
## 306 3.0 tenured not minority female english 43 58.86525
## 307 4.6 tenured not minority female english 43 100.00000
## 308 3.7 tenure track not minority male non-english 35 57.89474
## 309 3.6 tenure track not minority male non-english 35 52.38095
## 310 3.2 tenured not minority female english 62 81.48148
## 311 3.3 tenured not minority female english 62 56.25000
## 312 2.9 tenured not minority female english 62 73.68421
## 313 4.2 tenured not minority male english 42 52.32558
## 314 4.5 tenured not minority male english 39 75.86207
## 315 3.8 tenured not minority female english 52 72.72727
## 316 3.7 tenured not minority female english 52 71.42857
## 317 3.7 tenured not minority female english 52 70.45454
## 318 4.0 tenured not minority female english 52 75.38461
## 319 3.7 tenured not minority female english 52 50.79365
## 320 4.5 teaching not minority female english 52 78.66666
## 321 3.8 teaching not minority female english 52 81.39535
## 322 3.9 teaching not minority female english 52 73.75000
## 323 4.6 teaching not minority female english 52 86.53846
## 324 4.5 teaching not minority female english 52 70.83334
## 325 4.2 teaching not minority female english 52 75.75758
## 326 4.0 teaching not minority female english 52 67.00000
## 327 3.8 tenured not minority male english 64 81.81818
## 328 3.5 tenured not minority male english 64 50.00000
## 329 2.7 tenured not minority male english 64 81.81818
## 330 4.0 tenured not minority male english 64 45.45454
## 331 4.6 tenured not minority male english 64 50.00000
## 332 3.9 tenured not minority male english 64 100.00000
## 333 4.5 tenured not minority male english 50 93.75000
## 334 3.7 tenured not minority male english 50 90.00000
## 335 2.4 tenured not minority male english 60 71.87500
## 336 3.1 tenured not minority male english 60 70.00000
## 337 2.5 tenured not minority male english 60 62.50000
## 338 3.0 tenured not minority female english 51 70.14925
## 339 4.5 tenure track not minority male english 43 86.36364
## 340 4.8 tenure track not minority male english 43 53.57143
## 341 4.9 tenure track not minority male english 43 60.00000
## 342 4.5 tenure track not minority male english 43 73.33334
## 343 4.6 tenure track not minority male english 43 76.92308
## 344 4.5 tenure track not minority male english 43 94.44444
## 345 4.9 tenure track not minority male english 43 84.61539
## 346 4.4 tenure track not minority male english 43 60.00000
## 347 4.6 tenure track not minority male english 43 100.00000
## 348 4.6 teaching minority male english 50 70.83334
## 349 5.0 teaching minority male english 50 90.90909
## 350 4.9 teaching minority male english 50 84.00000
## 351 4.6 teaching minority male english 50 88.46154
## 352 4.8 teaching minority male english 50 86.36364
## 353 4.9 teaching minority male english 50 76.92308
## 354 4.9 teaching minority male english 50 85.00000
## 355 4.9 teaching minority male english 50 81.81818
## 356 5.0 teaching minority male english 50 95.23810
## 357 4.5 teaching minority male english 50 90.47619
## 358 3.5 tenured not minority male english 52 66.66666
## 359 3.8 tenured not minority male english 52 70.76923
## 360 3.9 tenured not minority male english 52 62.90322
## 361 3.9 tenured not minority male english 52 50.74627
## 362 4.2 tenured not minority male english 52 72.50000
## 363 4.1 tenured not minority male english 52 60.00000
## 364 4.8 tenured not minority male english 51 60.62718
## 365 4.8 tenured not minority male english 51 64.24870
## 366 4.8 tenured not minority male english 51 63.87337
## 367 4.8 tenured not minority male english 51 65.40447
## 368 4.9 tenured not minority male english 51 61.10057
## 369 4.2 tenured not minority male english 38 50.57471
## 370 4.5 tenured not minority male english 38 78.57143
## 371 3.9 tenured not minority male english 38 50.63291
## 372 4.4 tenured not minority male english 38 65.21739
## 373 4.0 tenured not minority female english 47 79.16666
## 374 3.6 tenured not minority female english 47 56.71642
## 375 3.7 tenured minority female english 43 45.63107
## 376 2.7 tenured minority female english 43 48.94737
## 377 4.5 teaching not minority female english 38 48.52941
## 378 4.4 teaching not minority female english 38 60.00000
## 379 3.9 teaching not minority female english 38 45.31250
## 380 3.6 teaching not minority female english 38 74.19355
## 381 4.4 teaching not minority female english 38 58.06452
## 382 4.4 teaching not minority female english 38 81.08108
## 383 4.7 tenured not minority male english 43 76.92308
## 384 4.5 tenured not minority male english 43 61.53846
## 385 4.1 tenured not minority male english 43 80.00000
## 386 3.7 tenured not minority male english 43 82.27848
## 387 4.3 tenured not minority male english 57 53.84615
## 388 3.5 tenured not minority male english 57 54.08163
## 389 3.7 tenured not minority male english 57 59.79382
## 390 4.0 tenured not minority female english 51 81.81818
## 391 4.0 tenured not minority female english 51 85.89744
## 392 3.1 tenured not minority female english 51 64.28571
## 393 4.5 tenured not minority female english 51 85.00000
## 394 4.8 teaching not minority male english 45 70.58823
## 395 4.2 teaching not minority male english 45 85.00000
## 396 4.9 teaching not minority male english 45 68.42105
## 397 4.8 teaching not minority male english 45 73.07692
## 398 3.5 tenured not minority male english 57 92.85714
## 399 3.6 tenured not minority male english 57 66.66666
## 400 4.4 tenured not minority male english 57 100.00000
## 401 3.4 tenured not minority male english 57 84.21053
## 402 3.9 tenured not minority male english 57 75.00000
## 403 3.8 tenured not minority male english 57 87.50000
## 404 4.8 tenured not minority male english 57 91.66666
## 405 4.6 tenured not minority male english 57 88.23529
## 406 5.0 tenured not minority male english 57 40.00000
## 407 3.8 tenured not minority male english 57 93.75000
## 408 4.2 tenured not minority male english 57 70.58823
## 409 3.3 teaching not minority female english 47 76.19048
## 410 4.7 teaching not minority female english 47 88.23529
## 411 4.6 teaching not minority female english 47 100.00000
## 412 4.6 teaching not minority female english 47 94.11765
## 413 4.0 teaching not minority female english 47 88.23529
## 414 4.2 tenured minority female english 54 100.00000
## 415 4.9 tenured minority female english 54 93.75000
## 416 4.5 tenured minority female english 54 80.76923
## 417 4.8 tenured minority female english 54 77.77778
## 418 3.8 tenured minority female english 54 80.00000
## 419 4.8 teaching not minority male english 58 94.11765
## 420 5.0 teaching not minority male english 58 100.00000
## 421 5.0 teaching not minority male english 58 85.71429
## 422 4.9 teaching not minority male english 58 85.00000
## 423 4.6 teaching not minority male english 58 95.00000
## 424 5.0 teaching not minority male english 58 84.61539
## 425 4.8 teaching not minority male english 58 87.50000
## 426 4.9 teaching not minority male english 58 94.11765
## 427 4.9 tenured not minority male english 42 83.33334
## 428 3.9 tenured not minority male english 42 91.66666
## 429 3.9 tenured not minority male english 42 85.00000
## 430 4.5 tenure track not minority male english 33 70.83334
## 431 4.5 tenure track not minority male english 33 43.22581
## 432 3.3 tenured not minority male english 62 28.94737
## 433 3.1 tenured not minority male english 62 40.00000
## 434 2.8 tenured not minority male english 62 40.93960
## 435 3.1 tenured not minority male english 62 35.76642
## 436 4.2 tenured not minority male english 62 44.82759
## 437 3.4 tenured not minority male english 62 50.90909
## 438 3.0 tenured not minority male english 62 49.26471
## 439 3.3 tenure track minority female english 35 62.50000
## 440 3.6 tenure track minority female english 35 33.33333
## 441 3.7 tenure track minority female english 35 39.81482
## 442 3.6 tenured not minority male english 61 69.23077
## 443 4.3 tenured not minority male english 61 86.66666
## 444 4.1 tenured not minority female english 52 54.95496
## 445 4.9 tenured not minority female english 52 82.35294
## 446 4.8 tenured not minority female english 52 100.00000
## 447 3.7 tenure track not minority female non-english 60 85.18519
## 448 3.9 tenure track not minority female non-english 60 94.73684
## 449 4.5 tenure track not minority female non-english 60 84.61539
## 450 3.6 tenure track not minority female non-english 60 94.73684
## 451 4.4 tenure track not minority female non-english 60 50.00000
## 452 3.4 tenure track not minority female non-english 60 35.00000
## 453 4.4 tenure track not minority female non-english 60 88.88889
## 454 4.5 tenure track not minority male english 32 74.24242
## 455 4.5 tenure track not minority male english 32 87.40157
## 456 4.5 tenure track not minority male english 32 72.94118
## 457 4.6 tenure track not minority male english 32 75.24753
## 458 4.1 tenure track not minority male english 32 42.85714
## 459 4.5 tenure track not minority male english 32 60.46511
## 460 3.5 tenure track minority female non-english 42 57.14286
## 461 4.4 tenure track minority female non-english 42 77.61194
## 462 4.4 tenure track minority female non-english 42 81.81818
## 463 4.1 tenure track minority female non-english 42 80.00000
## cls_did_eval cls_students cls_level cls_profs cls_credits bty_f1lower
## 1 24 43 upper single multi credit 5
## 2 86 125 upper single multi credit 5
## 3 76 125 upper single multi credit 5
## 4 77 123 upper single multi credit 5
## 5 17 20 upper multiple multi credit 4
## 6 35 40 upper multiple multi credit 4
## 7 39 44 upper multiple multi credit 4
## 8 55 55 upper single multi credit 5
## 9 111 195 upper single multi credit 5
## 10 40 46 upper single multi credit 2
## 11 24 27 upper single multi credit 2
## 12 24 25 upper multiple multi credit 2
## 13 17 20 upper single multi credit 2
## 14 14 25 upper single multi credit 2
## 15 37 42 upper single multi credit 2
## 16 18 20 upper multiple multi credit 2
## 17 15 18 upper multiple multi credit 2
## 18 42 48 upper multiple multi credit 7
## 19 40 44 upper multiple multi credit 7
## 20 38 48 upper multiple multi credit 7
## 21 40 45 upper multiple multi credit 7
## 22 52 59 upper multiple multi credit 7
## 23 49 87 upper single multi credit 7
## 24 182 282 upper multiple multi credit 6
## 25 160 292 upper multiple multi credit 6
## 26 79 130 upper multiple multi credit 6
## 27 176 285 upper single multi credit 6
## 28 155 272 upper multiple multi credit 6
## 29 166 286 upper multiple multi credit 6
## 30 186 302 upper single multi credit 6
## 31 33 41 upper single multi credit 4
## 32 29 34 upper single multi credit 4
## 33 37 41 upper single multi credit 4
## 34 29 41 upper single multi credit 4
## 35 28 34 upper single multi credit 4
## 36 25 41 upper multiple multi credit 4
## 37 21 22 upper multiple multi credit 4
## 38 13 21 upper multiple multi credit 4
## 39 16 17 upper multiple multi credit 4
## 40 24 30 lower multiple multi credit 4
## 41 23 23 upper multiple multi credit 4
## 42 20 20 upper multiple multi credit 4
## 43 48 60 upper multiple multi credit 3
## 44 29 33 upper single multi credit 3
## 45 11 44 upper multiple multi credit 3
## 46 29 49 upper single multi credit 3
## 47 25 29 upper single multi credit 3
## 48 42 48 upper multiple multi credit 3
## 49 34 40 upper multiple multi credit 3
## 50 16 19 upper single multi credit 5
## 51 12 16 upper multiple multi credit 5
## 52 14 15 upper single multi credit 5
## 53 22 23 upper single multi credit 5
## 54 10 11 lower multiple multi credit 5
## 55 17 29 lower multiple multi credit 5
## 56 16 21 upper multiple multi credit 5
## 57 15 18 lower multiple multi credit 5
## 58 16 19 upper single multi credit 5
## 59 16 20 lower multiple multi credit 5
## 60 18 25 upper single multi credit 5
## 61 30 33 upper single multi credit 5
## 62 23 24 upper single multi credit 5
## 63 30 34 upper multiple multi credit 4
## 64 13 21 upper single multi credit 4
## 65 24 30 upper multiple multi credit 4
## 66 24 25 upper multiple multi credit 4
## 67 25 35 upper multiple multi credit 4
## 68 28 40 upper multiple multi credit 4
## 69 18 30 upper multiple multi credit 4
## 70 28 42 upper multiple multi credit 4
## 71 25 57 upper single multi credit 4
## 72 40 57 upper single multi credit 4
## 73 40 51 upper single multi credit 4
## 74 18 30 upper multiple multi credit 4
## 75 30 36 upper multiple multi credit 3
## 76 31 37 upper single multi credit 3
## 77 15 29 upper multiple multi credit 3
## 78 23 27 upper single multi credit 3
## 79 23 28 upper single multi credit 6
## 80 34 52 upper multiple multi credit 6
## 81 21 26 upper single multi credit 6
## 82 29 30 upper single multi credit 6
## 83 23 33 upper single multi credit 2
## 84 45 177 lower single multi credit 2
## 85 90 199 lower single multi credit 2
## 86 27 32 upper single multi credit 2
## 87 35 37 upper single multi credit 2
## 88 120 161 lower single multi credit 2
## 89 27 41 upper single multi credit 3
## 90 39 44 upper single multi credit 3
## 91 35 53 upper multiple multi credit 3
## 92 34 49 upper multiple multi credit 3
## 93 27 32 upper single multi credit 3
## 94 100 135 lower multiple multi credit 3
## 95 20 33 upper multiple multi credit 3
## 96 14 19 upper multiple multi credit 3
## 97 65 111 lower multiple multi credit 3
## 98 95 149 lower multiple multi credit 3
## 99 18 27 upper multiple multi credit 3
## 100 85 136 lower multiple multi credit 3
## 101 113 140 lower multiple multi credit 3
## 102 25 31 upper multiple multi credit 4
## 103 14 15 lower multiple multi credit 4
## 104 23 29 upper multiple multi credit 4
## 105 23 25 upper multiple multi credit 4
## 106 9 18 upper multiple multi credit 4
## 107 30 45 upper multiple multi credit 4
## 108 15 15 lower multiple multi credit 4
## 109 31 38 upper multiple multi credit 4
## 110 12 15 lower multiple multi credit 4
## 111 17 28 upper multiple multi credit 5
## 112 10 23 upper multiple multi credit 5
## 113 15 19 upper multiple multi credit 5
## 114 7 23 upper multiple multi credit 5
## 115 7 22 upper multiple multi credit 5
## 116 14 20 upper multiple multi credit 5
## 117 8 19 upper multiple multi credit 5
## 118 17 23 upper multiple multi credit 5
## 119 10 22 upper multiple multi credit 5
## 120 12 15 upper multiple multi credit 5
## 121 19 22 upper single multi credit 6
## 122 27 31 upper multiple multi credit 6
## 123 20 21 upper multiple multi credit 6
## 124 31 36 upper multiple multi credit 6
## 125 17 19 upper multiple one credit 6
## 126 23 37 upper single multi credit 6
## 127 19 26 upper single multi credit 3
## 128 30 39 upper single multi credit 4
## 129 94 184 lower single multi credit 4
## 130 46 50 upper single multi credit 4
## 131 80 157 lower single multi credit 4
## 132 61 164 lower single multi credit 4
## 133 15 24 upper multiple multi credit 5
## 134 51 68 upper multiple multi credit 5
## 135 22 47 upper multiple multi credit 5
## 136 10 14 upper multiple multi credit 5
## 137 11 15 upper multiple multi credit 5
## 138 15 24 upper multiple multi credit 5
## 139 21 39 upper multiple multi credit 5
## 140 20 26 upper multiple multi credit 8
## 141 33 40 upper multiple multi credit 8
## 142 84 159 upper single multi credit 4
## 143 71 151 upper single multi credit 4
## 144 36 47 lower single multi credit 4
## 145 73 122 upper single multi credit 4
## 146 31 45 lower single multi credit 4
## 147 13 16 upper single multi credit 4
## 148 23 23 upper single multi credit 4
## 149 12 16 upper single multi credit 4
## 150 15 18 upper single multi credit 4
## 151 11 16 upper single multi credit 4
## 152 12 15 upper single multi credit 4
## 153 18 28 upper multiple multi credit 4
## 154 12 17 upper single multi credit 1
## 155 10 13 upper single multi credit 1
## 156 16 21 upper multiple multi credit 1
## 157 13 17 upper single multi credit 1
## 158 47 134 upper single multi credit 5
## 159 5 48 upper single multi credit 5
## 160 34 64 upper single multi credit 5
## 161 24 69 upper single multi credit 5
## 162 10 12 upper single multi credit 3
## 163 28 43 upper single multi credit 5
## 164 11 14 upper multiple multi credit 5
## 165 10 15 upper single multi credit 5
## 166 14 18 upper multiple multi credit 5
## 167 12 16 upper multiple multi credit 5
## 168 8 10 upper multiple multi credit 5
## 169 23 47 upper multiple multi credit 5
## 170 10 15 upper multiple multi credit 1
## 171 10 14 upper multiple multi credit 1
## 172 10 12 upper single multi credit 1
## 173 86 246 lower multiple multi credit 5
## 174 154 316 lower multiple multi credit 5
## 175 12 15 upper multiple multi credit 5
## 176 14 15 upper multiple multi credit 5
## 177 27 29 upper single multi credit 5
## 178 21 21 lower single multi credit 2
## 179 8 8 upper multiple multi credit 2
## 180 16 16 lower single one credit 2
## 181 24 26 lower single multi credit 2
## 182 7 10 upper multiple multi credit 2
## 183 25 26 lower single multi credit 2
## 184 24 26 lower single multi credit 2
## 185 24 26 lower single multi credit 2
## 186 20 21 lower single one credit 2
## 187 12 12 lower single multi credit 2
## 188 25 27 lower single multi credit 2
## 189 26 27 lower single multi credit 2
## 190 24 25 lower single multi credit 2
## 191 15 15 upper single multi credit 1
## 192 10 15 upper single multi credit 1
## 193 16 17 upper single multi credit 1
## 194 30 55 upper multiple multi credit 6
## 195 35 48 upper single multi credit 6
## 196 21 21 upper multiple multi credit 6
## 197 35 39 upper multiple multi credit 6
## 198 20 27 upper single multi credit 2
## 199 9 14 upper single multi credit 2
## 200 21 26 upper single multi credit 2
## 201 13 16 upper multiple multi credit 2
## 202 13 16 upper multiple multi credit 2
## 203 12 13 upper multiple multi credit 2
## 204 14 14 upper single multi credit 2
## 205 15 17 upper multiple multi credit 2
## 206 12 13 upper multiple multi credit 3
## 207 13 15 upper single multi credit 3
## 208 7 10 upper multiple multi credit 3
## 209 25 34 upper multiple multi credit 4
## 210 12 16 upper single multi credit 4
## 211 11 14 upper single multi credit 4
## 212 8 12 upper single multi credit 4
## 213 24 39 upper single multi credit 4
## 214 15 35 upper single multi credit 4
## 215 31 45 upper multiple multi credit 4
## 216 36 45 lower multiple multi credit 4
## 217 15 17 upper single multi credit 4
## 218 13 14 upper multiple multi credit 3
## 219 14 14 upper multiple multi credit 3
## 220 13 14 upper multiple multi credit 3
## 221 12 12 upper multiple multi credit 3
## 222 13 15 upper multiple multi credit 3
## 223 42 51 upper single multi credit 4
## 224 21 23 lower multiple multi credit 4
## 225 49 57 upper single multi credit 4
## 226 44 50 lower multiple multi credit 4
## 227 22 24 lower single multi credit 8
## 228 18 23 upper multiple multi credit 8
## 229 20 23 upper multiple multi credit 8
## 230 27 28 lower single multi credit 8
## 231 28 45 upper single multi credit 6
## 232 27 42 upper single multi credit 6
## 233 49 57 upper single multi credit 6
## 234 22 27 lower single multi credit 5
## 235 31 38 upper multiple multi credit 5
## 236 19 22 lower single multi credit 5
## 237 27 43 upper multiple multi credit 5
## 238 30 31 upper single multi credit 7
## 239 12 13 upper single multi credit 7
## 240 13 15 upper single multi credit 7
## 241 26 34 upper single multi credit 3
## 242 12 19 upper single multi credit 2
## 243 16 20 upper single multi credit 2
## 244 17 23 upper single multi credit 2
## 245 22 27 lower single multi credit 2
## 246 24 32 lower single one credit 2
## 247 14 21 lower single one credit 2
## 248 23 24 upper multiple multi credit 2
## 249 19 21 upper multiple multi credit 2
## 250 18 28 upper multiple multi credit 2
## 251 23 29 upper multiple multi credit 2
## 252 57 67 upper multiple multi credit 4
## 253 81 89 upper multiple multi credit 4
## 254 74 82 upper multiple multi credit 4
## 255 102 122 upper multiple multi credit 4
## 256 94 131 upper multiple multi credit 4
## 257 89 114 upper multiple multi credit 4
## 258 133 149 upper multiple multi credit 4
## 259 22 23 upper single multi credit 4
## 260 78 98 upper multiple multi credit 4
## 261 22 27 upper single multi credit 4
## 262 27 30 upper single multi credit 4
## 263 27 30 upper single multi credit 4
## 264 60 69 upper multiple multi credit 4
## 265 10 15 upper multiple multi credit 3
## 266 10 10 upper multiple multi credit 3
## 267 10 11 upper multiple multi credit 3
## 268 10 14 upper multiple multi credit 3
## 269 7 11 upper multiple multi credit 3
## 270 14 14 upper multiple multi credit 3
## 271 69 77 upper multiple multi credit 5
## 272 36 41 upper single multi credit 5
## 273 72 88 upper multiple multi credit 5
## 274 63 78 upper multiple multi credit 5
## 275 46 65 upper single multi credit 6
## 276 109 157 lower multiple multi credit 6
## 277 54 68 upper single multi credit 6
## 278 51 67 upper single multi credit 6
## 279 61 80 upper single multi credit 6
## 280 102 137 lower multiple multi credit 6
## 281 58 69 upper multiple multi credit 6
## 282 54 91 lower multiple multi credit 2
## 283 46 80 lower multiple multi credit 2
## 284 53 90 lower multiple multi credit 2
## 285 19 34 upper multiple multi credit 2
## 286 41 73 upper multiple multi credit 2
## 287 25 44 upper multiple multi credit 2
## 288 24 36 upper multiple multi credit 6
## 289 16 20 upper single multi credit 6
## 290 20 35 lower multiple multi credit 6
## 291 85 248 lower multiple multi credit 3
## 292 69 168 lower multiple multi credit 3
## 293 98 247 lower multiple multi credit 3
## 294 20 22 upper single multi credit 3
## 295 72 103 upper multiple multi credit 3
## 296 44 62 lower multiple multi credit 5
## 297 51 82 upper multiple multi credit 5
## 298 35 51 lower multiple multi credit 2
## 299 18 35 upper multiple multi credit 2
## 300 26 34 lower multiple multi credit 2
## 301 28 37 lower multiple multi credit 2
## 302 13 14 lower multiple multi credit 2
## 303 174 266 lower multiple multi credit 2
## 304 171 254 lower multiple multi credit 2
## 305 10 13 upper multiple multi credit 2
## 306 166 282 lower multiple multi credit 2
## 307 17 17 upper multiple multi credit 2
## 308 11 19 upper single multi credit 4
## 309 22 42 lower multiple multi credit 4
## 310 22 27 upper multiple multi credit 1
## 311 9 16 upper multiple multi credit 1
## 312 14 19 upper multiple multi credit 1
## 313 45 86 upper multiple multi credit 3
## 314 22 29 upper multiple multi credit 5
## 315 64 88 upper multiple multi credit 5
## 316 70 98 upper multiple multi credit 5
## 317 31 44 upper multiple multi credit 6
## 318 49 65 upper multiple multi credit 6
## 319 32 63 upper multiple multi credit 6
## 320 59 75 upper multiple multi credit 2
## 321 35 43 upper multiple multi credit 2
## 322 59 80 upper multiple multi credit 2
## 323 45 52 upper multiple multi credit 2
## 324 34 48 upper multiple multi credit 2
## 325 50 66 upper multiple multi credit 2
## 326 67 100 upper multiple multi credit 2
## 327 9 11 upper multiple multi credit 2
## 328 8 16 upper multiple multi credit 2
## 329 18 22 upper multiple multi credit 2
## 330 5 11 upper multiple multi credit 2
## 331 5 10 upper multiple multi credit 2
## 332 16 16 upper multiple multi credit 2
## 333 15 16 upper multiple multi credit 6
## 334 9 10 upper multiple multi credit 6
## 335 23 32 upper multiple multi credit 1
## 336 7 10 upper multiple multi credit 1
## 337 10 16 upper multiple multi credit 1
## 338 47 67 upper multiple multi credit 4
## 339 19 22 lower multiple multi credit 3
## 340 15 28 lower multiple one credit 3
## 341 18 30 lower multiple one credit 3
## 342 11 15 lower multiple multi credit 3
## 343 10 13 lower multiple multi credit 3
## 344 17 18 lower multiple one credit 3
## 345 22 26 lower multiple one credit 3
## 346 18 30 lower multiple one credit 3
## 347 14 14 lower multiple multi credit 3
## 348 17 24 lower multiple one credit 1
## 349 20 22 lower multiple one credit 1
## 350 21 25 lower multiple one credit 1
## 351 23 26 lower multiple one credit 1
## 352 19 22 lower multiple one credit 1
## 353 20 26 lower multiple one credit 1
## 354 17 20 lower multiple one credit 1
## 355 18 22 lower multiple one credit 1
## 356 20 21 lower multiple one credit 1
## 357 19 21 lower multiple one credit 1
## 358 46 69 upper multiple multi credit 7
## 359 46 65 upper multiple multi credit 7
## 360 39 62 upper multiple multi credit 7
## 361 34 67 upper multiple multi credit 7
## 362 29 40 upper multiple multi credit 7
## 363 27 45 upper multiple multi credit 7
## 364 348 574 lower multiple multi credit 6
## 365 372 579 lower multiple multi credit 6
## 366 343 537 lower multiple multi credit 6
## 367 380 581 lower multiple multi credit 6
## 368 322 527 lower multiple multi credit 6
## 369 44 87 lower multiple multi credit 3
## 370 66 84 lower multiple multi credit 3
## 371 40 79 lower multiple multi credit 3
## 372 60 92 lower multiple multi credit 3
## 373 19 24 lower single multi credit 5
## 374 38 67 lower single multi credit 5
## 375 47 103 lower multiple multi credit 5
## 376 93 190 lower multiple multi credit 5
## 377 33 68 lower multiple multi credit 3
## 378 36 60 lower single multi credit 3
## 379 29 64 lower single multi credit 3
## 380 23 31 lower single multi credit 3
## 381 36 62 lower multiple multi credit 3
## 382 30 37 lower multiple multi credit 3
## 383 10 13 lower multiple multi credit 4
## 384 8 13 lower multiple multi credit 4
## 385 12 15 upper single multi credit 4
## 386 65 79 upper single multi credit 4
## 387 7 13 upper multiple multi credit 3
## 388 53 98 upper multiple multi credit 3
## 389 58 97 upper multiple multi credit 3
## 390 9 11 upper multiple multi credit 5
## 391 67 78 upper multiple multi credit 5
## 392 36 56 upper multiple multi credit 5
## 393 17 20 upper multiple multi credit 5
## 394 12 17 lower single one credit 1
## 395 17 20 lower multiple one credit 1
## 396 13 19 lower multiple multi credit 1
## 397 19 26 lower multiple one credit 1
## 398 13 14 lower multiple multi credit 2
## 399 12 18 lower single multi credit 2
## 400 12 12 lower single multi credit 2
## 401 16 19 lower multiple multi credit 2
## 402 12 16 lower multiple multi credit 2
## 403 14 16 lower multiple multi credit 2
## 404 11 12 lower multiple multi credit 2
## 405 15 17 lower multiple multi credit 2
## 406 6 15 upper multiple multi credit 2
## 407 15 16 lower multiple multi credit 2
## 408 12 17 lower multiple multi credit 2
## 409 16 21 lower multiple multi credit 8
## 410 15 17 lower multiple one credit 8
## 411 10 10 lower multiple one credit 8
## 412 16 17 lower multiple one credit 8
## 413 15 17 lower multiple multi credit 8
## 414 18 18 lower single multi credit 8
## 415 15 16 lower single multi credit 8
## 416 21 26 lower single multi credit 8
## 417 14 18 lower single multi credit 8
## 418 16 20 lower single multi credit 8
## 419 16 17 lower single multi credit 8
## 420 21 21 lower single multi credit 8
## 421 18 21 lower single multi credit 8
## 422 17 20 lower single multi credit 8
## 423 19 20 lower single multi credit 8
## 424 11 13 lower single multi credit 8
## 425 14 16 lower single multi credit 8
## 426 16 17 lower multiple multi credit 8
## 427 15 18 lower multiple multi credit 8
## 428 22 24 lower multiple multi credit 8
## 429 17 20 upper multiple multi credit 8
## 430 85 120 lower multiple multi credit 6
## 431 67 155 lower multiple multi credit 6
## 432 11 38 lower multiple multi credit 1
## 433 28 70 lower multiple multi credit 1
## 434 61 149 lower multiple multi credit 1
## 435 49 137 lower multiple multi credit 1
## 436 13 29 upper multiple multi credit 1
## 437 28 55 lower multiple multi credit 1
## 438 67 136 lower multiple multi credit 1
## 439 60 96 lower multiple multi credit 7
## 440 20 60 lower multiple multi credit 7
## 441 43 108 lower multiple multi credit 7
## 442 27 39 lower multiple multi credit 3
## 443 13 15 lower multiple multi credit 3
## 444 61 111 lower multiple multi credit 4
## 445 14 17 lower multiple multi credit 4
## 446 19 19 lower multiple multi credit 4
## 447 23 27 upper multiple multi credit 4
## 448 18 19 upper multiple multi credit 4
## 449 11 13 upper multiple multi credit 4
## 450 18 19 upper multiple multi credit 4
## 451 11 22 upper multiple multi credit 4
## 452 7 20 upper multiple multi credit 4
## 453 24 27 upper multiple multi credit 4
## 454 98 132 lower multiple multi credit 6
## 455 111 127 lower multiple multi credit 6
## 456 62 85 upper multiple multi credit 6
## 457 76 101 lower multiple multi credit 6
## 458 9 21 lower multiple multi credit 6
## 459 52 86 upper multiple multi credit 6
## 460 48 84 upper multiple multi credit 3
## 461 52 67 upper multiple multi credit 3
## 462 54 66 upper multiple multi credit 3
## 463 28 35 lower multiple one credit 3
## bty_f1upper bty_f2upper bty_m1lower bty_m1upper bty_m2upper bty_avg
## 1 7 6 2 4 6 5.000
## 2 7 6 2 4 6 5.000
## 3 7 6 2 4 6 5.000
## 4 7 6 2 4 6 5.000
## 5 4 2 2 3 3 3.000
## 6 4 2 2 3 3 3.000
## 7 4 2 2 3 3 3.000
## 8 2 5 2 3 3 3.333
## 9 2 5 2 3 3 3.333
## 10 5 4 3 3 2 3.167
## 11 5 4 3 3 2 3.167
## 12 5 4 3 3 2 3.167
## 13 5 4 3 3 2 3.167
## 14 5 4 3 3 2 3.167
## 15 5 4 3 3 2 3.167
## 16 5 4 3 3 2 3.167
## 17 5 4 3 3 2 3.167
## 18 9 9 7 6 6 7.333
## 19 9 9 7 6 6 7.333
## 20 9 9 7 6 6 7.333
## 21 9 9 7 6 6 7.333
## 22 9 9 7 6 6 7.333
## 23 9 9 7 6 6 7.333
## 24 6 5 5 5 6 5.500
## 25 6 5 5 5 6 5.500
## 26 6 5 5 5 6 5.500
## 27 6 5 5 5 6 5.500
## 28 6 5 5 5 6 5.500
## 29 6 5 5 5 6 5.500
## 30 6 5 5 5 6 5.500
## 31 4 5 4 4 4 4.167
## 32 4 5 4 4 4 4.167
## 33 4 5 4 4 4 4.167
## 34 4 5 4 4 4 4.167
## 35 4 5 4 4 4 4.167
## 36 6 6 2 3 3 4.000
## 37 6 6 2 3 3 4.000
## 38 6 6 2 3 3 4.000
## 39 6 6 2 3 3 4.000
## 40 6 6 2 3 3 4.000
## 41 6 6 2 3 3 4.000
## 42 6 6 2 3 3 4.000
## 43 7 5 5 3 5 4.667
## 44 7 5 5 3 5 4.667
## 45 7 5 5 3 5 4.667
## 46 7 5 5 3 5 4.667
## 47 7 5 5 3 5 4.667
## 48 7 5 5 3 5 4.667
## 49 7 5 5 3 5 4.667
## 50 7 6 3 6 6 5.500
## 51 7 6 3 6 6 5.500
## 52 7 6 3 6 6 5.500
## 53 7 6 3 6 6 5.500
## 54 7 6 3 6 6 5.500
## 55 7 6 3 6 6 5.500
## 56 7 6 3 6 6 5.500
## 57 7 6 3 6 6 5.500
## 58 7 6 3 6 6 5.500
## 59 7 6 3 6 6 5.500
## 60 7 4 2 4 7 4.833
## 61 7 4 2 4 7 4.833
## 62 7 4 2 4 7 4.833
## 63 5 5 3 5 4 4.333
## 64 5 5 3 5 4 4.333
## 65 5 5 3 5 4 4.333
## 66 5 5 3 5 4 4.333
## 67 5 5 3 5 4 4.333
## 68 7 5 4 4 5 4.833
## 69 7 5 4 4 5 4.833
## 70 7 5 4 4 5 4.833
## 71 7 5 4 4 5 4.833
## 72 7 5 4 4 5 4.833
## 73 7 5 4 4 5 4.833
## 74 7 5 4 4 5 4.833
## 75 3 5 1 7 5 4.000
## 76 3 5 1 7 5 4.000
## 77 3 5 1 7 5 4.000
## 78 3 5 1 7 5 4.000
## 79 7 7 5 4 4 5.500
## 80 7 7 5 4 4 5.500
## 81 7 7 5 4 4 5.500
## 82 7 7 5 4 4 5.500
## 83 4 3 5 4 7 4.167
## 84 4 3 5 4 7 4.167
## 85 4 3 5 4 7 4.167
## 86 4 3 5 4 7 4.167
## 87 4 3 5 4 7 4.167
## 88 4 3 5 4 7 4.167
## 89 4 1 2 2 3 2.500
## 90 4 1 2 2 3 2.500
## 91 4 1 2 2 3 2.500
## 92 4 1 2 2 3 2.500
## 93 4 1 2 2 3 2.500
## 94 5 6 4 4 4 4.333
## 95 5 6 4 4 4 4.333
## 96 5 6 4 4 4 4.333
## 97 5 6 4 4 4 4.333
## 98 5 6 4 4 4 4.333
## 99 5 6 4 4 4 4.333
## 100 5 6 4 4 4 4.333
## 101 5 6 4 4 4 4.333
## 102 4 5 2 6 5 4.333
## 103 4 5 2 6 5 4.333
## 104 4 5 2 6 5 4.333
## 105 4 5 2 6 5 4.333
## 106 4 5 2 6 5 4.333
## 107 4 5 2 6 5 4.333
## 108 4 5 2 6 5 4.333
## 109 4 5 2 6 5 4.333
## 110 4 5 2 6 5 4.333
## 111 4 4 5 2 6 4.333
## 112 4 4 5 2 6 4.333
## 113 4 4 5 2 6 4.333
## 114 4 4 5 2 6 4.333
## 115 4 4 5 2 6 4.333
## 116 4 4 5 2 6 4.333
## 117 4 4 5 2 6 4.333
## 118 4 4 5 2 6 4.333
## 119 4 4 5 2 6 4.333
## 120 4 4 5 2 6 4.333
## 121 6 4 2 4 7 4.833
## 122 6 4 2 4 7 4.833
## 123 6 4 2 4 7 4.833
## 124 6 4 2 4 7 4.833
## 125 6 4 2 4 7 4.833
## 126 6 4 2 4 7 4.833
## 127 3 4 2 3 2 2.833
## 128 3 4 1 2 4 3.000
## 129 3 4 1 2 4 3.000
## 130 3 4 1 2 4 3.000
## 131 3 4 1 2 4 3.000
## 132 3 4 1 2 4 3.000
## 133 4 5 3 3 5 4.167
## 134 4 5 3 3 5 4.167
## 135 4 5 3 3 5 4.167
## 136 4 5 3 3 5 4.167
## 137 4 5 3 3 5 4.167
## 138 4 5 3 3 5 4.167
## 139 4 5 3 3 5 4.167
## 140 9 8 6 8 8 7.833
## 141 9 8 6 8 8 7.833
## 142 4 5 2 4 4 3.833
## 143 4 5 2 4 4 3.833
## 144 4 5 2 4 4 3.833
## 145 4 5 2 4 4 3.833
## 146 4 5 2 4 4 3.833
## 147 5 7 5 4 4 4.833
## 148 5 7 5 4 4 4.833
## 149 5 7 5 4 4 4.833
## 150 5 7 5 4 4 4.833
## 151 5 7 5 4 4 4.833
## 152 5 7 5 4 4 4.833
## 153 5 7 5 4 4 4.833
## 154 3 6 2 1 5 3.000
## 155 3 6 2 1 5 3.000
## 156 3 6 2 1 5 3.000
## 157 3 6 2 1 5 3.000
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## 463 not formal colorplot.default(evals$age, evals$bty_avg, main="Scatterplot Age Vs BTY_AVG", xlab="Age ", ylab="Beauty Avg ")
Simple linear regression
The fundamental phenomenon suggested by the study is that better looking teachers are evaluated more favorably. Let’s create a scatterplot to see if this appears to be the case:
plot(evals$score ~ evals$bty_avg)Before we draw conclusions about the trend, compare the number of observations in the data frame with the approximate number of points on the scatterplot. Is anything awry?
- Replot the scatterplot, but this time use the function
jitter()on the \(y\)- or the \(x\)-coordinate. (Use?jitterto learn more.) What was misleading about the initial scatterplot?
Answer :- The earlier plot was hidding a lot of observations, whereas using the jitter() function has led
?jitter## starting httpd help server ... doneplot.default(evals$score ~ jitter(evals$bty_avg,1))
- Let’s see if the apparent trend in the plot is something more than natural variation. Fit a linear model called
m_btyto predict average professor score by average beauty rating and add the line to your plot usingabline(m_bty). Write out the equation for the linear model and interpret the slope. Is average beauty score a statistically significant predictor? Does it appear to be a practically significant predictor?
Answer:- The slope(0.06664) is nothing to write about and we can safely say that average beauty score is not a good predictor .The Slope means that of each point of beauty average, the score will grow only 0.06664.
\[ \hat{y} = 3.88034 + 0.06664 * bty_avg \]
m_bty = lm(formula = score ~ bty_avg, data = evals)
summary(m_bty)##
## Call:
## lm(formula = score ~ bty_avg, data = evals)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.9246 -0.3690 0.1420 0.3977 0.9309
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.88034 0.07614 50.96 < 2e-16 ***
## bty_avg 0.06664 0.01629 4.09 5.08e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5348 on 461 degrees of freedom
## Multiple R-squared: 0.03502, Adjusted R-squared: 0.03293
## F-statistic: 16.73 on 1 and 461 DF, p-value: 5.083e-05plot.default(evals$score ~ evals$bty_avg)
abline(m_bty)
- Use residual plots to evaluate whether the conditions of least squares regression are reasonable. Provide plots and comments for each one (see the Simple Regression Lab for a reminder of how to make these).
Answer :- Due to environment setting issues, somehow my plot methods are getting hidden . As discussed with professor Bryer leaving it aside for now.
#plot_ss(x = evals$bty_avg, y = evals$score, data= evals, showSquares = TRUE)
# Below is the error getting while ussing plot_SS
#Error in plot_ss(x = evals$bty_avg, y = evals$score, data = evals, showSquares = TRUE) : unused argument (data = evals)Multiple linear regression
The data set contains several variables on the beauty score of the professor: individual ratings from each of the six students who were asked to score the physical appearance of the professors and the average of these six scores. Let’s take a look at the relationship between one of these scores and the average beauty score.
plot(evals$bty_avg ~ evals$bty_f1lower)
cor(evals$bty_avg, evals$bty_f1lower)As expected the relationship is quite strong - after all, the average score is calculated using the individual scores. We can actually take a look at the relationships between all beauty variables (columns 13 through 19) using the following command:
plot(evals[,13:19])These variables are collinear (correlated), and adding more than one of these variables to the model would not add much value to the model. In this application and with these highly-correlated predictors, it is reasonable to use the average beauty score as the single representative of these variables.
In order to see if beauty is still a significant predictor of professor score after we’ve accounted for the gender of the professor, we can add the gender term into the model.
m_bty_gen <- lm(score ~ bty_avg + gender, data = evals)
summary(m_bty_gen)- P-values and parameter estimates should only be trusted if the conditions for the regression are reasonable. Verify that the conditions for this model are reasonable using diagnostic plots.
Answer :- There are qutliers at both end , with more outliers at the upper end.
# checking Residuals are nearly normal
m_bty_gen <- lm(score ~ bty_avg + gender, data = evals)
summary(m_bty_gen)##
## Call:
## lm(formula = score ~ bty_avg + gender, data = evals)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.8305 -0.3625 0.1055 0.4213 0.9314
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.74734 0.08466 44.266 < 2e-16 ***
## bty_avg 0.07416 0.01625 4.563 6.48e-06 ***
## gendermale 0.17239 0.05022 3.433 0.000652 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5287 on 460 degrees of freedom
## Multiple R-squared: 0.05912, Adjusted R-squared: 0.05503
## F-statistic: 14.45 on 2 and 460 DF, p-value: 8.177e-07qqnorm(m_bty_gen$residuals)
qqline(m_bty_gen$residuals)
Looking at the below plot , we can safely assume most of the absolute resuduals values are close to the fitted values.
# check variability of the residuals is normal, by plotting absolute values of residuals against fitted values.
plot.default(abs(m_bty_gen$residuals) ~ m_bty_gen$fitted.values)
The plot shows that residulas were gathered independenlty.
#Checking resdiuals are independent
plot.default(m_bty_gen$residuals ~ c(1:nrow(evals)))
The boxplot shows that the median score of both males and females are similar in terms of evaluation scores,hence there is a linear relationship between gender and evaluation score. This concludes that there is a linear relationship between beauty average and teaching evaluation score.
#checking each variable is lineraly related to outcome.
boxplot(evals$score ~ evals$gender)
8. Is bty_avg still a significant predictor of score? Has the addition of gender to the model changed the parameter estimate for bty_avg?
Answer :- It is not a significant predictor of score, the addition of gender has improved the prediction estimates for the model.
Note that the estimate for gender is now called gendermale. You’ll see this name change whenever you introduce a categorical variable. The reason is that R recodes gender from having the values of female and male to being an indicator variable called gendermale that takes a value of \(0\) for females and a value of \(1\) for males. (Such variables are often referred to as “dummy” variables.)
As a result, for females, the parameter estimate is multiplied by zero, leaving the intercept and slope form familiar from simple regression.
\[ \begin{aligned} \widehat{score} &= \hat{\beta}_0 + \hat{\beta}_1 \times bty\_avg + \hat{\beta}_2 \times (0) \\ &= \hat{\beta}_0 + \hat{\beta}_1 \times bty\_avg\end{aligned} \]
We can plot this line and the line corresponding to males with the following custom function.
Having the problem of not able to find the method due to environmntal issues .
#multiLines(m_bty_gen)- What is the equation of the line corresponding to males? (Hint: For males, the parameter estimate is multiplied by 1.) For two professors who received the same beauty rating, which gender tends to have the higher course evaluation score?
Answer :- \[ \hat{y} = 3.74734 + 0.07416 * bty_avg + 0.17 \] Males will be the ones to higher course evaluation score.
The decision to call the indicator variable gendermale instead ofgenderfemale has no deeper meaning. R simply codes the category that comes first alphabetically as a \(0\). (You can change the reference level of a categorical variable, which is the level that is coded as a 0, using therelevel function. Use ?relevel to learn more.)
- Create a new model called
m_bty_rankwithgenderremoved andrankadded in. How does R appear to handle categorical variables that have more than two levels? Note that the rank variable has three levels:teaching,tenure track,tenured.
Answer:-
m_bty_rank <- lm(score ~ bty_avg + rank, data = evals)
summary(m_bty_rank)##
## Call:
## lm(formula = score ~ bty_avg + rank, data = evals)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.8713 -0.3642 0.1489 0.4103 0.9525
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.98155 0.09078 43.860 < 2e-16 ***
## bty_avg 0.06783 0.01655 4.098 4.92e-05 ***
## ranktenure track -0.16070 0.07395 -2.173 0.0303 *
## ranktenured -0.12623 0.06266 -2.014 0.0445 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5328 on 459 degrees of freedom
## Multiple R-squared: 0.04652, Adjusted R-squared: 0.04029
## F-statistic: 7.465 on 3 and 459 DF, p-value: 6.88e-05The interpretation of the coefficients in multiple regression is slightly different from that of simple regression. The estimate for bty_avg reflects how much higher a group of professors is expected to score if they have a beauty rating that is one point higher while holding all other variables constant. In this case, that translates into considering only professors of the same rank with bty_avg scores that are one point apart.
The search for the best model
We will start with a full model that predicts professor score based on rank, ethnicity, gender, language of the university where they got their degree, age, proportion of students that filled out evaluations, class size, course level, number of professors, number of credits, average beauty rating, outfit, and picture color.
- Which variable would you expect to have the highest p-value in this model? Why? Hint: Think about which variable would you expect to not have any association with the professor score.
Let’s run the model…
Answer :- Outfit to have no association with the professor score.
m_full <- lm(score ~ rank + ethnicity + gender + language + age + cls_perc_eval
+ cls_students + cls_level + cls_profs + cls_credits + bty_avg
+ pic_outfit + pic_color, data = evals)
summary(m_full)- Check your suspicions from the previous exercise. Include the model output in your response.
Answer :- For variable Outfit not formal estimate is -0.1126817 Std error: 0.0738800 , the t value of 0.12792 shows that the variable is more important but is not the crucial one.
- Interpret the coefficient associated with the ethnicity variable.
Answer :- 0.1234929 that means if the sample is not minoruty then it would expect to add .123 to the score.
- Drop the variable with the highest p-value and re-fit the model. Did the coefficients and significance of the other explanatory variables change? (One of the things that makes multiple regression interesting is that coefficient estimates depend on the other variables that are included in the model.) If not, what does this say about whether or not the dropped variable was collinear with the other explanatory variables?
Answer :- Removing the cls_profssingle with highest p-value i.e. 0.77806
m_full_cls_profs <- lm(score ~ rank + ethnicity + gender + language + age + cls_perc_eval
+ cls_students + cls_level + cls_credits + bty_avg
+ pic_outfit + pic_color, data = evals)
summary(m_full_cls_profs)##
## Call:
## lm(formula = score ~ rank + ethnicity + gender + language + age +
## cls_perc_eval + cls_students + cls_level + cls_credits +
## bty_avg + pic_outfit + pic_color, data = evals)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.7836 -0.3257 0.0859 0.3513 0.9551
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.0872523 0.2888562 14.150 < 2e-16 ***
## ranktenure track -0.1476746 0.0819824 -1.801 0.072327 .
## ranktenured -0.0973829 0.0662614 -1.470 0.142349
## ethnicitynot minority 0.1274458 0.0772887 1.649 0.099856 .
## gendermale 0.2101231 0.0516873 4.065 5.66e-05 ***
## languagenon-english -0.2282894 0.1111305 -2.054 0.040530 *
## age -0.0089992 0.0031326 -2.873 0.004262 **
## cls_perc_eval 0.0052888 0.0015317 3.453 0.000607 ***
## cls_students 0.0004687 0.0003737 1.254 0.210384
## cls_levelupper 0.0606374 0.0575010 1.055 0.292200
## cls_creditsone credit 0.5061196 0.1149163 4.404 1.33e-05 ***
## bty_avg 0.0398629 0.0174780 2.281 0.023032 *
## pic_outfitnot formal -0.1083227 0.0721711 -1.501 0.134080
## pic_colorcolor -0.2190527 0.0711469 -3.079 0.002205 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4974 on 449 degrees of freedom
## Multiple R-squared: 0.187, Adjusted R-squared: 0.1634
## F-statistic: 7.943 on 13 and 449 DF, p-value: 2.336e-14- Using backward-selection and p-value as the selection criterion, determine the best model. You do not need to show all steps in your answer, just the output for the final model. Also, write out the linear model for predicting score based on the final model you settle on.
m_final <- lm(score ~ gender + language + age + cls_perc_eval
+ cls_credits + bty_avg + pic_color, data = evals)
summary(m_final)##
## Call:
## lm(formula = score ~ gender + language + age + cls_perc_eval +
## cls_credits + bty_avg + pic_color, data = evals)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.81919 -0.32035 0.09272 0.38526 0.88213
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.967255 0.215824 18.382 < 2e-16 ***
## gendermale 0.221457 0.049937 4.435 1.16e-05 ***
## languagenon-english -0.281933 0.098341 -2.867 0.00434 **
## age -0.005877 0.002622 -2.241 0.02551 *
## cls_perc_eval 0.004295 0.001432 2.999 0.00286 **
## cls_creditsone credit 0.444392 0.100910 4.404 1.33e-05 ***
## bty_avg 0.048679 0.016974 2.868 0.00432 **
## pic_colorcolor -0.216556 0.066625 -3.250 0.00124 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5014 on 455 degrees of freedom
## Multiple R-squared: 0.1631, Adjusted R-squared: 0.1502
## F-statistic: 12.67 on 7 and 455 DF, p-value: 6.996e-15- Verify that the conditions for this model are reasonable using diagnostic plots.
par(mfrow=c(1,1))
qqnorm(m_final$residuals)
qqline(m_final$residuals)
b)
# check variability of the residuals is normal, by plotting absolute values of residuals against fitted values.
plot.default(abs(m_final$residuals) ~ m_final$fitted.values)
plot.default(m_final$residuals ~ c(1:nrow(evals)))
boxplot(evals$score ~ evals$ethnicity)
boxplot(evals$score ~ evals$gender)
boxplot(evals$score ~ evals$cls_credits)
The original paper describes how these data were gathered by taking a sample of professors from the University of Texas at Austin and including all courses that they have taught. Considering that each row represents a course, could this new information have an impact on any of the conditions of linear regression? Answer :- No, because the data gathered for every course is taken independently.
Based on your final model, describe the characteristics of a professor and course at University of Texas at Austin that would be associated with a high evaluation score. Answer :- Professor from not minority group speaking english .
Would you be comfortable generalizing your conclusions to apply to professors generally (at any university)? Why or why not?
Answer :- This study may be true for Texan university, but the study might have other features which are not used here hence cannot be generalised.
This is a product of OpenIntro that is released under a Creative Commons Attribution-ShareAlike 3.0 Unported. This lab was written by Mine Çetinkaya-Rundel and Andrew Bray.