[DATA] Invest in Education Go to www.pearsonhighered.com/sullivanstats to obtain the data file 12_3_17. The variable “Cost” represents the four-year cost including tuition, supplies, room and board, the variable “Annual ROI” represents the return on investment for graduates of the school—essentially how much you would earn on the investment of attending the school. The variable “Grad Rate” represents the graduation rate of the school.
a. In Problem 49 from Section 4.1, a scatter diagram between “Cost” and “Grad Rate” treating “Cost” as the explanatory variable suggested a positive association between the two variables. Treating “Cost” as the explanatory variable, x, test whether a negative association exists between the cost and annual ROI for graduates of four-year schools at the alpha = 0.01 level of significance. Normal probability plots suggest the residuals are normally distributed.

Verified step by step guidance

1. Step 1: Define the hypotheses for the test. Since we want to test if there is a negative association between Cost (explanatory variable, x) and Annual ROI (response variable, y), set up the hypotheses as follows: the null hypothesis \(H_0\): the slope \(\beta_1 \geq 0\) (no negative association), and the alternative hypothesis \(H_a\): the slope \(\beta_1 < 0\) (negative association).
2. Step 2: Fit the simple linear regression model \(y = \beta_0 + \beta_1 x + \epsilon\), where \(y\) is Annual ROI and \(x\) is Cost. Use the data to calculate the estimated slope \(\hat{\beta}_1\) and its standard error \(SE_{\hat{\beta}_1}\).
3. Step 3: Calculate the test statistic for the slope using the formula: \(\displaystyle t = \frac{\hat{\beta}_1 - 0}{SE_{\hat{\beta}_1}}\) This t-statistic follows a t-distribution with \(n - 2\) degrees of freedom, where \(n\) is the number of data points.
4. Step 4: Determine the critical value for the t-test at the significance level \(\alpha = 0.01\) for a one-tailed test (left tail) with \(n - 2\) degrees of freedom. This critical value will be negative since we are testing for a negative slope.
5. Step 5: Compare the calculated t-statistic to the critical value. If the t-statistic is less than the critical value, reject the null hypothesis and conclude there is sufficient evidence of a negative association between Cost and Annual ROI. Otherwise, do not reject the null hypothesis.

Key Concepts

Hypothesis Testing for Correlation

Hypothesis testing for correlation involves assessing whether there is a statistically significant relationship between two variables. Here, the null hypothesis typically states no association or a non-negative association, while the alternative suggests a negative association. The test uses sample data to determine if observed correlations differ from zero beyond random chance, based on a chosen significance level (alpha).

Linear Regression and Residuals

Linear regression models the relationship between an explanatory variable (Cost) and a response variable (Annual ROI) by fitting a line that minimizes residuals, the differences between observed and predicted values. Residual analysis, including normal probability plots, checks if residuals are approximately normally distributed, a key assumption for valid inference in regression hypothesis tests.

Significance Level and p-Value

The significance level (alpha) is the threshold for deciding whether to reject the null hypothesis, commonly set at 0.01 for strict tests. The p-value measures the probability of observing data as extreme as the sample if the null hypothesis is true. If the p-value is less than alpha, the null is rejected, indicating sufficient evidence for the alternative hypothesis—in this case, a negative association.