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Suppose we want to examine the relationship between scores on a popular standardized test and performance in college. We have chosen a random sample of fifteen students just finishing their first year of college, and for each student we've recorded her score on the standardized test (from 400 to 1600) and her grade point average (from 0 to 4) for her first year in college. The data are shown below, with x denoting the score on the standardized test and y denoting the first-year college grade point average. A scatter plot of the data is shown in Figure 1. The least-squares regression line for these data has a slope of approximately 0.0016. (a) What is the value of the y-intercept of the least-squares regression line for these data? Round your answer to at least four decimal places. (b)	What is the value of the sample correlation coefficient for these data? Round your answer to at least three decimal places.

### Given Data:
- Slope of the least-squares regression line (\(m\)) = 0.0016
- The data includes the standardized test scores (\(x\)) and the corresponding first-year college GPA (\(y\)) for 15 students.

### (a) Finding the y-Intercept of the Least-Squares Regression Line

The general form of the least-squares regression line is:
\[
y = mx + b
\]
where:
- \(m\) is the slope of the line.
- \(b\) is the y-intercept.

To find the y-intercept \(b\), we need to use the formula:
\[
b = \bar{y} - m\bar{x}
\]
where:
- \(\bar{y}\) is the mean of the \(y\) values (GPA).
- \(\bar{x}\) is the mean of the \(x\) values (standardized test scores).

#### Step 1: Calculate \(\bar{x}\) (mean of \(x\)) and \(\bar{y}\) (mean of \(y\)):
Let's first compute the mean values.

\[
\bar{x} = \frac{\text{sum of all } x \text{ values}}{n} \quad \text{and} \quad \bar{y} = \frac{\text{sum of all } y \text{ values}}{n}
\]

\[
\text{Sum of } x = 1290 + 820 + 1190 + 1490 + 960 + 1510 + 1430 + 850 + 650 + 580 + 1210 + 1110 + 1070 + 1370 + 1500
\]
\[
\text{Sum of } y = 3.07 + 2.22 + 2.89 + 3.11 + 2.33 + 2.91 + 2.83 + 2.34 + 2.61 + 2.15 + 3.10 + 2.75 + 3.09 + 3.03 + 3.12
\]

Using these, we can find the averages \(\bar{x}\) and \(\bar{y}\).

#### Step 2: Plug in the values to find \(b\):
Once we have \(\bar{x}\) and \(\bar{y}\), we can plug them into the equation \(b = \bar{y} - m\bar{x}\) to find the y-intercept \(b\).

### (b) Finding the Sample Correlation Coefficient \(r\)

The sample correlation coefficient \(r\) is calculated using the following 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.
- \(\sum{xy}\) is the sum of the product of paired scores.
- \(\sum{x}\), \(\sum{y}\), \(\sum{x^2}\), and \(\sum{y^2}\) are the respective sums of the \(x\) values, \(y\) values, squares of \(x\) values, and squares of \(y\) values.

### Let's calculate these values:

I'll start by computing \(\bar{x}\), \(\bar{y}\), and the y-intercept. Then, I will move to calculate the correlation coefficient.### (a) Value of the y-Intercept:
The y-intercept \(b\) of the least-squares regression line is approximately **0.9535** (rounded to four decimal places).

### (b) Value of the Sample Correlation Coefficient:
The sample correlation coefficient \(r\) for these data is approximately **0.809** (rounded to three decimal places).

### Would you like more details or have any questions?

### Related Questions:
1. What does the correlation coefficient tell us about the relationship between test scores and GPA?
2. How would the slope change if the correlation were stronger?
3. Can you explain how the least-squares method finds the line of best fit?
4. What might cause outliers in this data set, and how would they affect the correlation coefficient?
5. What is the significance of the y-intercept in this context?
6. How would you interpret the slope of the regression line in real-world terms?
7. What steps would you take to calculate the regression line manually?
8. How could this model be used to predict GPA based on test scores for future students?

### Tip:
The closer the correlation coefficient \(r\) is to 1 or -1, the stronger the linear relationship between the variables.

Learn how to calculate the y-intercept and sample correlation coefficient in regression analysis using a given set of standardized test scores and college GPA data. Understand the concepts of least-squares regression, correlation analysis, and their applications in statistics.

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