Model

The population regression model for the relationship between average school cost and graduate earnings is \(Graduate earnings = \beta_0 + \beta_1 Cost + \varepsilon\). The equation of the fitted line is Graduate earnings = 35160 + 0.5324cost. The slope of the fitted line is 0.5324 which means there is a positive relationship between average school cost and graduate earnings and an increase of 1,000 dollars in average cost of a school is associated with an increase of 532.4 dollars in graduate earnings. This shows that in terms of the population regression model, the higher the average school cost, the higher the earnings after graduation are. However, since the slope is below 1, the slope is flatter.

Assumption Diagnostics

Figure 1: Relationship between predicted median salary 10 years after graduation and the observed residuals

Figure 1: Relationship between predicted median salary 10 years after graduation and the observed residuals

Figure 2: Relationship between the theoretical normal quantiles and observed redisual quantiles of the residual plot

Figure 2: Relationship between the theoretical normal quantiles and observed redisual quantiles of the residual plot

Figure 3: Relationship between predicted graduate earnings and studentized residuals

Figure 3: Relationship between predicted graduate earnings and studentized residuals

In figure 1, the shape of the residual plot appear reasonable because they are randomly scattered above and below 0 and there are no definitive patterns. This figure also shows that the constant variance appears reasonable because the width is relatively constant. At x=475000, the width is a little different than the rest of the graph but overall it is relatively constant. In figure 2, the residuals in the QQ plot look normal on the lower x values but they do not look normal on the higher x values because the points grow farther from the line of normal distribution. In figure 3, the studentized residuals plot has lines at -2,2, and 3. There are a couple points with studentized residuals between 2 and 3 there are more above 3. However, most of the points fall within n > -3 and n < 3.

In this data, independence is reasonable because each data point is independent of each other. The relationship between average cost and graduate earnings of one school does not affect the relationship of another. The College Scoreboard used data from colleges of many types such as public and private, from many different locations, and of different academic backgrounds. Schools were deliberately chosen to be of similar backgrounds, but data was used from many to represent a more general range of colleges. So,the exact process that the colleges were chosen for College Scorecard is not available in the data set, but the process appears random.

Inference

The 99% confidence interval for the true slope β1 is (0.5092,β1,0.5547). Calculations: (0.532 - 2.5940.008755) & (0.532 + 2.5940.008755). So, we are 99% confident that an increase of 1 dollar in average cost of school is associated with an average increase in earnings after graduation of between 0.5092 and 0.5547 for students in schools from the College Scorecard data.

The null hypothesis is \(H_0: \beta_1 = 0\) and the alternative hypothesis is \(H_0: \beta_1 > 0\). So, the hypotheses are testing if there is no relationship between average cost to attend school and earnings after graduation or a positive relationship. The test statistic is 6.081. The p-value is 2.0847x10^-9. So, there is strong evidence for a positive relationship between average cost to attend a school and earnings after graduation.