The accompanying data are a subset of data from the report "Great Jobs, Great Lives."† The values are approximate values read from a scatterplot. Students at a number of universities were asked if they agreed that their education was worth the cost. One variable in the table is the percentage of students at the university who responded that they strongly agree. The other variable in the table is the U.S. News & World Report ranking of the university.
Ranking	Percentage of Alumni
Who Strongly Agree
28	53
29	59
30	62
37	56
45	54
47	63
52	55
54	61
57	70
60	59
65	66
66	55
72	65
75	57
82	67
88	59
98	75
A button hyperlink to the SALT program that reads: Use SALT.
(a)
What is the value of
r2
for this data set? (Round your answer to four decimal places.)

Incorrect: Your answer is incorrect.
Write a sentence interpreting this value in context. (Hint: See Example 4.13. Round your answer to two decimal places.)
Approximately

Incorrect: Your answer is incorrect.
% of the variability in the percentage of alumni who strongly agree can be explained by the linear relationship between the percentage of alumni who strongly agree and ranking.
(b)
What is the value of
se
for this data set? (Round your answer to five decimal places.)

Incorrect: Your answer is incorrect.

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).

The accompanying data are a subset of data from the report 'Great Jobs, Great Lives.' Students at a number of universities were asked if they agreed that their education was worth the cost. One variable in the table is the percentage of students at the university who responded that they strongly agree. The other variable in the table is the U.S. News & World Report ranking of the university. (a) What is the value of r^2 for this data set? (b) What is the value of se for this data set?

Learn how to calculate the coefficient of determination (r^2) and the standard error (se) for a data set involving university rankings and alumni agreement. This problem covers key statistical concepts including correlation coefficients and linear regression, suitable for students in Grades 11-12 and undergraduate levels.

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