Question

The variables collected for this sample are average starting salary upon graduation ($), the percentage of applicants to the full-time program who were accepted, the average GMAT test score of students entering the program, program per-year tuition ($), and percent of students with job offers at time of graduation.

1. Build a simple linear regression model with each of the predictors and the dependent variable. Provide the R2 for each simple linear regression model. Which predictor has the highest correlation with the dependent variable?

University	Tuition per year ($)	Average GMAT score	Acceptance Rate (%)	Graduates employed at graduation (%)	Average starting salary and bonus ($)
1	61875	732	7.1	73.6	142834
2	58875	726	11.0	76.9	144750
3	62424	728	20.7	84.3	142574
4	61520	724	23.5	87.4	137615
5	63454	713	13.8	79.6	142936
6	61596	713	23.2	80.7	136357
7	63148	716	18.2	75.7	139006
8	61605	716	22.1	83.8	142489
9	60744	721	18.1	74.9	135933
10	58000	690	25.1	81.8	137154
11	58975	719	23.7	69.4	126871
12	58192	692	30.0	79.6	132316
13	51500	699	26.7	84.5	111974
14	58300	687	31.2	75.5	131865
15	48100	678	30.7	81.7	128347
16	52200	691	47.4	70.3	118938
17	46800	686	34.8	71.7	115694
18	51786	684	31.6	64.4	114129
19	47950	688	41.3	77.1	113830

Answer 1

data = read.table("clipboard", header = TRUE)

# Model 1
m1 = lm(data$AverageStartingSalaryAndBonus ~ data$TuitionPerYear)
summary(m1)

##
## Call:
## lm(formula = data$AverageStartingSalaryAndBonus ~ data$TuitionPerYear)
##
## Residuals:
##    Min     1Q Median     3Q    Max
## -9444 -2634 -1220   2914 12839
##
## Coefficients:
##                      Estimate Std. Error t value Pr(>|t|)   
## (Intercept)         31900.800 14023.174   2.275   0.0362 *
## data$TuitionPerYear     1.738      0.244   7.124 1.71e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5803 on 17 degrees of freedom
## Multiple R-squared: 0.7491, Adjusted R-squared: 0.7343
## F-statistic: 50.75 on 1 and 17 DF, p-value: 1.705e-06

R2_m1 = summary(m1)$r.squared;R2_m1

## [1] 0.7490727

corr_m1 = sqrt(R2_m1);corr_m1

## [1] 0.8654899

# Model 2
m2 = lm(data$AverageStartingSalaryAndBonus ~ data$AverageGMATScore)
summary(m2)

##
## Call:
## lm(formula = data$AverageStartingSalaryAndBonus ~ data$AverageGMATScore)
##
## Residuals:
##    Min     1Q Median     3Q    Max
## -16489 -6429   1081   6693 12735
##
## Coefficients:
##                        Estimate Std. Error t value Pr(>|t|)   
## (Intercept)           -185599.4    76932.3 -2.413 0.027424 *
## data$AverageGMATScore     449.3      109.0   4.121 0.000713 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 8194 on 17 degrees of freedom
## Multiple R-squared: 0.4998, Adjusted R-squared: 0.4703
## F-statistic: 16.98 on 1 and 17 DF, p-value: 0.0007135

R2_m2 = summary(m2)$r.squared;R2_m2

## [1] 0.4997507

corr_m2 = sqrt(R2_m2);corr_m2

## [1] 0.7069305

# Model 3
m3 = lm(data$AverageStartingSalaryAndBonus ~ data$AcceptanceRate)
summary(m3)

##
## Call:
## lm(formula = data$AverageStartingSalaryAndBonus ~ data$AcceptanceRate)
##
## Residuals:
##    Min     1Q Median     3Q    Max
## -18110 -3967   1423   5404   8329
##
## Coefficients:
##                     Estimate Std. Error t value Pr(>|t|)   
## (Intercept)         153739.4     4645.6   33.09 < 2e-16 ***
## data$AcceptanceRate   -886.0      171.7   -5.16 7.85e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 7232 on 17 degrees of freedom
## Multiple R-squared: 0.6103, Adjusted R-squared: 0.5874
## F-statistic: 26.63 on 1 and 17 DF, p-value: 7.848e-05

R2_m3 = summary(m3)$r.squared;R2_m3

## [1] 0.6103376

corr_m3 = sqrt(R2_m3);corr_m3

## [1] 0.7812411

# Model 4
m4 = lm(data$AverageStartingSalaryAndBonus ~ data$GraduatesEmployedAtGraduation)
summary(m4)

##
## Call:
## lm(formula = data$AverageStartingSalaryAndBonus ~ data$GraduatesEmployedAtGraduation)
##
## Residuals:
##    Min     1Q Median     3Q    Max
## -24790 -6525   2086   6444 14529
##
## Coefficients:
##                                    Estimate Std. Error t value Pr(>|t|)
## (Intercept)                         71185.7    32437.4   2.195   0.0424 *
## data$GraduatesEmployedAtGraduation    776.1      417.3   1.860   0.0803 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 10560 on 17 degrees of freedom
## Multiple R-squared: 0.1691, Adjusted R-squared: 0.1202
## F-statistic: 3.459 on 1 and 17 DF, p-value: 0.08029

R2_m4 = summary(m4)$r.squared;R2_m4

## [1] 0.1690816

corr_m4 = sqrt(R2_m4);corr_m4

## [1] 0.4111954

# TuitionPerYear has highest correlation value with the Response variable "AverageStartingSalaryAndBonus"