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” by Hamermesh and Parker found that instructors who are viewed to be better looking receive higher instructional ratings.

Here, you will analyze the data from this study in order to learn what goes into a positive professor evaluation.

Getting Started

Load packages

In this lab, you will explore and visualize the data using the tidyverse suite of packages. The data can be found in the companion package for OpenIntro resources, openintro.

Let’s load the packages.

library(tidyverse)
library(openintro)
library(GGally)
#create user defined function 

multiLines  <- function(model, ...){
  if(class(model)!="lm"){
    warning("Model must be the output of the function lm()")
  }
  
  if(length(model$xlevels)!=1){
    warning("Model must contain exactly one categorical predictor")
  }
  
  if(length(model$coef)-length(model$xlevels[[1]])!=1){
    warning("Model must contain exactly one non-categorical predictor")
  }
  
  palette <- c("#E69F00", "#56B4E9", "#D55E00", "#009E73", "#CC79A7", "#F0E442", "#0072B2")
  
  baseIntercept <- model$coef[1]
  nLines <- length(model$xlevels[[1]])
  intercepts <- c(baseIntercept, rep(0, nLines-1))
  indicatorInd <- c(1, rep(0, nLines)) # used to find slope parameter by process of elimination
  
  for(i in 1:(nLines-1)){
    indicatorName <- paste(names(model$contrasts),model$xlevels[[1]][1+i], sep = "")
    intercepts[i+1] <- baseIntercept + model$coef[names(model$coef)==indicatorName]
    indicatorInd <- indicatorInd + (names(model$coef)==indicatorName)
  }
  
  slope <- model$coef[!indicatorInd]
  
  num_pred = which(names(model$model[,-1]) != names(model$xlevels)) + 1
  cat_pred = which(names(model$model[,-1]) == names(model$xlevels)) + 1
  
  model$model$COL = NA
  model$model$PCH = NA
  for(i in 1:nLines){
    model$model$COL[model$model[,cat_pred] == levels(model$model[,cat_pred])[i]] = adjustcolor(palette[i],0.40)
    model$model$PCH[model$model[,cat_pred] == levels(model$model[,cat_pred])[i]] = i+14
  }
  
  plot(model$model[,1] ~ jitter(model$model[,num_pred]), col = model$model$COL, pch = model$model$PCH,
       ylab = names(model$model)[1],
       xlab = names(model$model)[num_pred])
    
  for(j in 1:nLines){
    abline(intercepts[j], slope, col = palette[j], lwd = 2, ...)
  }

  if(slope > 0){legend_pos = "bottomright"}
  if(slope < 0){legend_pos = "topleft"}  
  
  legend(legend_pos, col = palette[1:nLines], lty = 1, legend = levels(model$model[,cat_pred]), lwd = 2)
}

This is the first time we’re using the GGally package. You will be using the ggpairs function from this package later in the lab.

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. The result is a data frame where each row contains a different course and columns represent variables about the courses and professors. It’s called evals.

glimpse(evals)
## Rows: 463
## Columns: 23
## $ course_id     <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1…
## $ prof_id       <int> 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5,…
## $ score         <dbl> 4.7, 4.1, 3.9, 4.8, 4.6, 4.3, 2.8, 4.1, 3.4, 4.5, 3.8, 4…
## $ rank          <fct> tenure track, tenure track, tenure track, tenure track, …
## $ ethnicity     <fct> minority, minority, minority, minority, not minority, no…
## $ gender        <fct> female, female, female, female, male, male, male, male, …
## $ language      <fct> english, english, english, english, english, english, en…
## $ age           <int> 36, 36, 36, 36, 59, 59, 59, 51, 51, 40, 40, 40, 40, 40, …
## $ cls_perc_eval <dbl> 55.81395, 68.80000, 60.80000, 62.60163, 85.00000, 87.500…
## $ cls_did_eval  <int> 24, 86, 76, 77, 17, 35, 39, 55, 111, 40, 24, 24, 17, 14,…
## $ cls_students  <int> 43, 125, 125, 123, 20, 40, 44, 55, 195, 46, 27, 25, 20, …
## $ cls_level     <fct> upper, upper, upper, upper, upper, upper, upper, upper, …
## $ cls_profs     <fct> single, single, single, single, multiple, multiple, mult…
## $ cls_credits   <fct> multi credit, multi credit, multi credit, multi credit, …
## $ bty_f1lower   <int> 5, 5, 5, 5, 4, 4, 4, 5, 5, 2, 2, 2, 2, 2, 2, 2, 2, 7, 7,…
## $ bty_f1upper   <int> 7, 7, 7, 7, 4, 4, 4, 2, 2, 5, 5, 5, 5, 5, 5, 5, 5, 9, 9,…
## $ bty_f2upper   <int> 6, 6, 6, 6, 2, 2, 2, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 9, 9,…
## $ bty_m1lower   <int> 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 7, 7,…
## $ bty_m1upper   <int> 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 6, 6,…
## $ bty_m2upper   <int> 6, 6, 6, 6, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 6, 6,…
## $ bty_avg       <dbl> 5.000, 5.000, 5.000, 5.000, 3.000, 3.000, 3.000, 3.333, …
## $ pic_outfit    <fct> not formal, not formal, not formal, not formal, not form…
## $ pic_color     <fct> color, color, color, color, color, color, color, color, …

We have observations on 21 different variables, some categorical and some numerical. The meaning of each variable can be found by bringing up the help file:

?evals

Exploring the data

  1. 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.

This is an observation study because data is collected during an actual student evaluation. The original question “whether beauty leads directly to the differences in course evaluations,” might not be directly answerable with the current study design. rephrasing the question could be more significant. is beauty has negative and positive correlation with course evaluation or wether gender has an impact on course evaluation.

  1. 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?
# Create a histogram to visualize the distribution of scores
ggplot(evals, aes(x = score)) +
  geom_histogram(binwidth = 0.5, fill = "red4", color = "blue") +
  labs(title = "Course Evaluation Scores", x = "Score", y = "Frequency")

  1. Excluding score, select two other variables and describe their relationship with each other using an appropriate visualization.
evals %>% ggplot(aes(x = gender, y = age, fill = gender)) +
  geom_boxplot() +
  labs(title = "Distribution of Ages by Gender",
       x = "Gender", y = "Age")

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:

ggplot(data = evals, aes(x = bty_avg, y = score)) +
  geom_point()

Before you 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?

  1. Replot the scatterplot, but this time use geom_jitter as your layer. What was misleading about the initial scatterplot?
ggplot(data = evals, aes(x = bty_avg, y = score)) +
  geom_jitter()

we can actually see overlapping values in the geom_jitter while the initial graph show value where density weren’t to high

  1. Let’s see if the apparent trend in the plot is something more than natural variation. Fit a linear model called m_bty to predict average professor score by average beauty rating. 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? the equation of the line is : score = 3.88 + .06664 * bty_avg. the R square is very low which means the average beauty score doesn’t seem like a significant predictor.

Add the line of the bet fit model to your plot using the following:

ggplot(data = evals, aes(x = bty_avg, y = score)) +
  geom_jitter() +
  geom_smooth(method = "lm")

The blue line is the model. The shaded gray area around the line tells you about the variability you might expect in your predictions. To turn that off, use se = FALSE.

ggplot(data = evals, aes(x = bty_avg, y = score)) +
  geom_jitter() +
  geom_smooth(method = "lm", se = FALSE)

  1. 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).
library(DATA606)
## 
## Welcome to CUNY DATA606 Statistics and Probability for Data Analytics 
## This package is designed to support this course. The text book used 
## is OpenIntro Statistics, 4th Edition. You can read this by typing 
## vignette('os4') or visit www.OpenIntro.org. 
##  
## The getLabs() function will return a list of the labs available. 
##  
## The demo(package='DATA606') will list the demos that are available.
m_bty <- lm(evals$score ~ evals$bty_avg)
summary(m_bty)
## 
## Call:
## lm(formula = evals$score ~ evals$bty_avg)
## 
## 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 ***
## evals$bty_avg  0.06664    0.01629    4.09 5.08e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.5348 on 461 degrees of freedom
## Multiple R-squared:  0.03502,    Adjusted R-squared:  0.03293 
## F-statistic: 16.73 on 1 and 461 DF,  p-value: 5.083e-05
#Historgream
hist(m_bty$residuals)

# how they distributed
plot(resid(m_bty))

qqnorm(m_bty$residuals)
qqline(m_bty$residuals)

The residue plot appears slightly straight on the line with no skewness.

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.

ggplot(data = evals, aes(x = bty_f1lower, y = bty_avg)) +
  geom_point()

evals %>% 
  summarise(cor(bty_avg, bty_f1lower))
## # A tibble: 1 × 1
##   `cor(bty_avg, bty_f1lower)`
##                         <dbl>
## 1                       0.844

As expected, the relationship is quite strong—after all, the average score is calculated using the individual scores. You can actually look at the relationships between all beauty variables (columns 13 through 19) using the following command:

evals %>%
  select(contains("bty")) %>%
  ggpairs()

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 you’ve accounted for the professor’s gender, you can add the gender term into the model.

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-07
  1. 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.
qqnorm(m_bty_gen$residuals)
qqline(m_bty_gen$residuals)

plot(m_bty_gen$residuals ~ evals$bty_avg)
abline(h = 0, lty = 3)

A p-value less than 0.05 is typically considered to be statistically significant. Our P value is 0.05 so the conditions for the regression is reasonable.

  1. Is bty_avg still a significant predictor of score? Has the addition of gender to the model changed the parameter estimate for bty_avg?

Yes, the addition of gender makes beauty more significant because all the parameters for the model improve.

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 male and female to being an indicator variable called gendermale that takes a value of \(0\) for female professors and a value of \(1\) for male professors. (Such variables are often referred to as “dummy” variables.)

As a result, for female professors, the parameter estimate is multiplied by zero, leaving the intercept and slope form familiar from simple regression.

\[ \begin{aligned} \widehat{score} &= \hat{\beta}_0 + \hat{\beta}_1 \times bty\_avg + \hat{\beta}_2 \times (0) \\ &= \hat{\beta}_0 + \hat{\beta}_1 \times bty\_avg\end{aligned} \]

ggplot(data = evals, aes(x = bty_avg, y = score, color = pic_color)) +
 geom_smooth(method = "lm", formula = y ~ x, se = FALSE)

  1. What is the equation of the line corresponding to those with color pictures? (Hint: For those with color pictures, the parameter estimate is multiplied by 1.) For two professors who received the same beauty rating, which color picture tends to have the higher course evaluation score?

The equation of the line is score = (3.74734 + 0.17239) + 0.07416(bty_avg)

The decision to call the indicator variable gendermale instead of genderfemale 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.)

  1. Create a new model called m_bty_rank with gender removed and rank added 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.
m_bty_rank = lm(evals$score~evals$bty_avg + evals$rank)
summary(m_bty_rank)
## 
## Call:
## lm(formula = evals$score ~ evals$bty_avg + evals$rank)
## 
## 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 ***
## evals$bty_avg           0.06783    0.01655   4.098 4.92e-05 ***
## evals$ranktenure track -0.16070    0.07395  -2.173   0.0303 *  
## evals$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-05
names(m_bty_rank)
##  [1] "coefficients"  "residuals"     "effects"       "rank"         
##  [5] "fitted.values" "assign"        "qr"            "df.residual"  
##  [9] "contrasts"     "xlevels"       "call"          "terms"        
## [13] "model"
multiLines(m_bty_rank)

The 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, gender, ethnicity, 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.

  1. 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.

The cls_profssingle has the highest p-value 0.77806

Let’s run the model…

m_full <- lm(score ~ rank + gender + ethnicity + language + age + cls_perc_eval 
             + cls_students + cls_level + cls_profs + cls_credits + bty_avg 
             + pic_outfit + pic_color, data = evals)
summary(m_full)
## 
## Call:
## lm(formula = score ~ rank + gender + ethnicity + language + age + 
##     cls_perc_eval + cls_students + cls_level + cls_profs + cls_credits + 
##     bty_avg + pic_outfit + pic_color, data = evals)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.77397 -0.32432  0.09067  0.35183  0.95036 
## 
## Coefficients:
##                         Estimate Std. Error t value Pr(>|t|)    
## (Intercept)            4.0952141  0.2905277  14.096  < 2e-16 ***
## ranktenure track      -0.1475932  0.0820671  -1.798  0.07278 .  
## ranktenured           -0.0973378  0.0663296  -1.467  0.14295    
## gendermale             0.2109481  0.0518230   4.071 5.54e-05 ***
## ethnicitynot minority  0.1234929  0.0786273   1.571  0.11698    
## languagenon-english   -0.2298112  0.1113754  -2.063  0.03965 *  
## age                   -0.0090072  0.0031359  -2.872  0.00427 ** 
## cls_perc_eval          0.0053272  0.0015393   3.461  0.00059 ***
## cls_students           0.0004546  0.0003774   1.205  0.22896    
## cls_levelupper         0.0605140  0.0575617   1.051  0.29369    
## cls_profssingle       -0.0146619  0.0519885  -0.282  0.77806    
## cls_creditsone credit  0.5020432  0.1159388   4.330 1.84e-05 ***
## bty_avg                0.0400333  0.0175064   2.287  0.02267 *  
## pic_outfitnot formal  -0.1126817  0.0738800  -1.525  0.12792    
## pic_colorcolor        -0.2172630  0.0715021  -3.039  0.00252 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.498 on 448 degrees of freedom
## Multiple R-squared:  0.1871, Adjusted R-squared:  0.1617 
## F-statistic: 7.366 on 14 and 448 DF,  p-value: 6.552e-14
  1. Check your suspicions from the previous exercise. Include the model output in your response.

The cls_profssingle has the highest p-value 0.77806

  1. Interpret the coefficient associated with the ethnicity variable.

Professors that are not minority score 0.123 higher than those who are minority.

  1. 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?

The variables cls_levelupper has the highest p values 0.292200. The value is approximately the same as previous model. the rest of the variables are slightly lower as well

# Perform backward-selection
final_model <- step(m_full, direction = "backward", criterion = "p-value")
## Start:  AIC=-630.9
## score ~ rank + gender + ethnicity + language + age + cls_perc_eval + 
##     cls_students + cls_level + cls_profs + cls_credits + bty_avg + 
##     pic_outfit + pic_color
## 
##                 Df Sum of Sq    RSS     AIC
## - cls_profs      1    0.0197 111.11 -632.82
## - cls_level      1    0.2740 111.36 -631.76
## - cls_students   1    0.3599 111.44 -631.40
## - rank           2    0.8930 111.98 -631.19
## <none>                       111.08 -630.90
## - pic_outfit     1    0.5768 111.66 -630.50
## - ethnicity      1    0.6117 111.70 -630.36
## - language       1    1.0557 112.14 -628.52
## - bty_avg        1    1.2967 112.38 -627.53
## - age            1    2.0456 113.13 -624.45
## - pic_color      1    2.2893 113.37 -623.46
## - cls_perc_eval  1    2.9698 114.06 -620.69
## - gender         1    4.1085 115.19 -616.09
## - cls_credits    1    4.6495 115.73 -613.92
## 
## Step:  AIC=-632.82
## score ~ rank + gender + ethnicity + language + age + cls_perc_eval + 
##     cls_students + cls_level + cls_credits + bty_avg + pic_outfit + 
##     pic_color
## 
##                 Df Sum of Sq    RSS     AIC
## - cls_level      1    0.2752 111.38 -633.67
## - cls_students   1    0.3893 111.49 -633.20
## - rank           2    0.8939 112.00 -633.11
## <none>                       111.11 -632.82
## - pic_outfit     1    0.5574 111.66 -632.50
## - ethnicity      1    0.6728 111.78 -632.02
## - language       1    1.0442 112.15 -630.49
## - bty_avg        1    1.2872 112.39 -629.49
## - age            1    2.0422 113.15 -626.39
## - pic_color      1    2.3457 113.45 -625.15
## - cls_perc_eval  1    2.9502 114.06 -622.69
## - gender         1    4.0895 115.19 -618.08
## - cls_credits    1    4.7999 115.90 -615.24
## 
## Step:  AIC=-633.67
## score ~ rank + gender + ethnicity + language + age + cls_perc_eval + 
##     cls_students + cls_credits + bty_avg + pic_outfit + pic_color
## 
##                 Df Sum of Sq    RSS     AIC
## - cls_students   1    0.2459 111.63 -634.65
## - rank           2    0.8140 112.19 -634.30
## <none>                       111.38 -633.67
## - pic_outfit     1    0.6618 112.04 -632.93
## - ethnicity      1    0.8698 112.25 -632.07
## - language       1    0.9015 112.28 -631.94
## - bty_avg        1    1.3694 112.75 -630.02
## - age            1    1.9342 113.31 -627.70
## - pic_color      1    2.0777 113.46 -627.12
## - cls_perc_eval  1    3.0290 114.41 -623.25
## - gender         1    3.8989 115.28 -619.74
## - cls_credits    1    4.5296 115.91 -617.22
## 
## Step:  AIC=-634.65
## score ~ rank + gender + ethnicity + language + age + cls_perc_eval + 
##     cls_credits + bty_avg + pic_outfit + pic_color
## 
##                 Df Sum of Sq    RSS     AIC
## - rank           2    0.7892 112.42 -635.39
## <none>                       111.63 -634.65
## - ethnicity      1    0.8832 112.51 -633.00
## - pic_outfit     1    0.9700 112.60 -632.65
## - language       1    1.0338 112.66 -632.38
## - bty_avg        1    1.5783 113.20 -630.15
## - pic_color      1    1.9477 113.57 -628.64
## - age            1    2.1163 113.74 -627.96
## - cls_perc_eval  1    2.7922 114.42 -625.21
## - gender         1    4.0945 115.72 -619.97
## - cls_credits    1    4.5163 116.14 -618.29
## 
## Step:  AIC=-635.39
## score ~ gender + ethnicity + language + age + cls_perc_eval + 
##     cls_credits + bty_avg + pic_outfit + pic_color
## 
##                 Df Sum of Sq    RSS     AIC
## <none>                       112.42 -635.39
## - pic_outfit     1    0.7141 113.13 -634.46
## - ethnicity      1    1.1790 113.59 -632.56
## - language       1    1.3403 113.75 -631.90
## - age            1    1.6847 114.10 -630.50
## - pic_color      1    1.7841 114.20 -630.10
## - bty_avg        1    1.8553 114.27 -629.81
## - cls_perc_eval  1    2.9147 115.33 -625.54
## - gender         1    4.0577 116.47 -620.97
## - cls_credits    1    6.1208 118.54 -612.84
summary(final_model)
## 
## Call:
## lm(formula = score ~ gender + ethnicity + language + age + cls_perc_eval + 
##     cls_credits + bty_avg + pic_outfit + pic_color, data = evals)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.8455 -0.3221  0.1013  0.3745  0.9051 
## 
## Coefficients:
##                        Estimate Std. Error t value Pr(>|t|)    
## (Intercept)            3.907030   0.244889  15.954  < 2e-16 ***
## gendermale             0.202597   0.050102   4.044 6.18e-05 ***
## ethnicitynot minority  0.163818   0.075158   2.180 0.029798 *  
## languagenon-english   -0.246683   0.106146  -2.324 0.020567 *  
## age                   -0.006925   0.002658  -2.606 0.009475 ** 
## cls_perc_eval          0.004942   0.001442   3.427 0.000666 ***
## cls_creditsone credit  0.517205   0.104141   4.966 9.68e-07 ***
## bty_avg                0.046732   0.017091   2.734 0.006497 ** 
## pic_outfitnot formal  -0.113939   0.067168  -1.696 0.090510 .  
## pic_colorcolor        -0.180870   0.067456  -2.681 0.007601 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4982 on 453 degrees of freedom
## Multiple R-squared:  0.1774, Adjusted R-squared:  0.161 
## F-statistic: 10.85 on 9 and 453 DF,  p-value: 2.441e-15
  1. 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.
model <- 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(model)
## 
## 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
  1. Verify that the conditions for this model are reasonable using diagnostic plots.

Insert your answer here

plot(final_model)

qqline(final_model$residuals)

  1. 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?

the observations need to be independent for each student so they can provider a better understanding on how teacher performs for every single class they teach.

  1. 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.

These observations cannot help describe characteristics of another school. there are still a lot information to collect from the evaluation in order to make that suggestion

  1. Would you be comfortable generalizing your conclusions to apply to professors generally (at any university)? Why or why not?

No I wouldn’t be comfortable generalizing your conclusions to apply to professors generally at university. every College has different ethnic group of student that would answer questions different.