Let's go through this step by step to calculate the required values.

(a) Calculation of $r^2$:

We need to calculate $r^2$, the coefficient of determination, which measures how well the regression line fits the data. $r^2$ is computed by squaring the correlation coefficient $r$.

Steps:

1. Organize the Data: We have two variables: the university ranking and the percentage of alumni who strongly agree their education was worth the cost.

   • Independent variable (X): Ranking
   • Dependent variable (Y): Percentage of Alumni Who Strongly Agree

2. Compute the Correlation Coefficient $r$:

   • We'll compute the correlation coefficient $r$ using the formula: [ r = \frac{n \sum (XY) - \sum X \sum Y}{\sqrt{[n \sum X^2 - (\sum X)^2][n \sum Y^2 - (\sum Y)^2]}} ] where:
   • $n$ is the number of data points
   • $X$ is the ranking
   • $Y$ is the percentage of alumni who strongly agree.

3. Square $r$ to get $r^2$.

4. Interpret $r^2$: This value tells us the proportion of variance in the percentage of alumni who strongly agree that can be explained by the university ranking.

(b) Calculation of the Standard Error $se$:

The standard error of the regression, denoted by $se$, is a measure of the average distance between the observed values and the regression line. It is calculated as:

$se = \sqrt{\frac{\sum (Y - \hat{Y})^2}{n - 2}}$ where:

• $Y$ is the observed percentage of alumni who strongly agree.
• $\hat{Y}$ is the predicted value from the regression line.
• $n$ is the number of data points.

We can now proceed to calculate both $r^2$ and $se$ using the data you provided. I'll compute these values for you.### (a) Value of $r^2$: The calculated $r^2$ is approximately 0.2827. This means that about 28.27% of the variability in the percentage of alumni who strongly agree can be explained by the linear relationship between the university ranking and the percentage of alumni who strongly agree.

(b) Value of the Standard Error $se$:

The calculated standard error $se$ is approximately 5.34265.

These values provide insight into the strength and accuracy of the regression model, with the $r^2$ showing moderate correlation and the standard error indicating the typical deviation of the data from the predicted values.

Do you have any questions or would you like further details on any step?

Here are five follow-up questions related to this problem:

1. How can we interpret a low $r^2$ value in a regression analysis?
2. What does the standard error of the regression tell us about the data's fit?
3. How would the regression model change if the correlation were stronger?
4. What could cause deviations between observed values and the regression line?
5. Can we predict future data points with confidence based on this model?

Tip: When the $r^2$ value is low, it suggests that factors other than the one measured (university ranking) may have a stronger influence on the outcome (alumni agreement).