 Find the prediction equation and interpret the​ y-intercept and slope.

Part 5

The prediction equation is 

ModifyingAbove y with caretyequals=​(enter your response here​)plus+​(enter your response here​)x.

​(Round to three decimal places as​ needed.)
Student	Study Time	GPA	Skipped

1	14	2.6	9

2	25	3.6	1

3	15	3.3	3

4	5	2.9	5

5	10	3.0	3

6	12	3.2	3

7	5	2.9	12

8	21	3.9	0

To find the prediction equation, which is a linear regression equation of the form \( \hat{y} = a + bx \), we need to determine the slope \( b \) and the y-intercept \( a \). Given that the relationship we are looking to predict is between "Study Time" (independent variable \( x \)) and "GPA" (dependent variable \( y \)), we'll calculate \( a \) and \( b \) using the following formulas:

\[
b = \frac{n\sum(xy) - \sum x \sum y}{n\sum x^2 - (\sum x)^2}
\]
\[
a = \frac{\sum y - b \sum x}{n}
\]

### Step 1: Organize and calculate necessary sums

Given dataTo find the prediction equation, which is a linear regression equation of the form \( \hat{y} = a + bx \), we need to determine the slope \( b \) and the y-intercept \( a \). Given that the relationship we are looking to predict is between "Study Time" (independent variable \( x \)) and "GPA" (dependent variable \( y \)), we'll calculate \( a \) and \( b \) using the following formulas:

\[
b = \frac{n\sum(xy) - \sum x \sum y}{n\sum x^2 - (\sum x)^2}
\]
\[
a = \frac{\sum y - b \sum x}{n}
\]

### Step 1: Organize and calculate necessary sums

Given 

\[
\begin{array}{|c|c|c|}
\hline
\text{Study Time (x)} & \text{GPA (y)} & \text{xy} & x^2 \\
\hline
14 & 2.6 & 36.4 & 196 \\
25 & 3.6 & 90.0 & 625 \\
15 & 3.3 & 49.5 & 225 \\
5  & 2.9 & 14.5 & 25  \\
10 & 3.0 & 30.0 & 100 \\
12 & 3.2 & 38.4 & 144 \\
5  & 2.9 & 14.5 & 25  \\
21 & 3.9 & 81.9 & 441 \\
\hline
\end{array}
\]

Now calculate:

\[
\sum x = 14 + 25 + 15 + 5 + 10 + 12 + 5 + 21 = 107
\]
\[
\sum y = 2.6 + 3.6 + 3.3 + 2.9 + 3.0 + 3.2 + 2.9 + 3.9 = 25.4
\]
\[
\sum xy = 36.4 + 90.0 + 49.5 + 14.5 + 30.0 + 38.4 + 14.5 + 81.9 = 355.2
\]
\[
\sum x^2 = 196 + 625 + 225 + 25 + 100 + 144 + 25 + 441 = 1781
\]

\[
n = 8
\]

### Step 2: Calculate slope \( b \)

\[
b = \frac{n\sum(xy) - \sum x \sum y}{n\sum x^2 - (\sum x)^2}
\]
\[
b = \frac{8 \times 355.2 - 107 \times 25.4}{8 \times 1781 - 107^2}
\]

Let's calculate \( b \):

\[
b = \frac{2841.6 - 2717.8}{14248 - 11449} = \frac{123.8}{2799} \approx 0.044
\]

### Step 3: Calculate y-intercept \( a \)

\[
a = \frac{\sum y - b \sum x}{n}
\]
\[
a = \frac{25.4 - 0.044 \times 107}{8} = \frac{25.4 - 4.708}{8} = \frac{20.692}{8} \approx 2.586
\]

### Prediction Equation

\[
\hat{y} = 2.586 + 0.044x
\]

### Interpretation

- **Slope (b = 0.044):** For each additional hour of study time, the GPA is predicted to increase by 0.044 points.
- **Y-intercept (a = 2.586):** If a student has 0 hours of study time, the predicted GPA would be 2.586.

---

Do you want more details, or do you have any questions?

**Here are 8 related questions:**
1. How do you calculate the coefficient of determination (R-squared) for this regression?
2. Can you determine if the slope is statistically significant?
3. How would the prediction equation change if a new data point is added?
4. How do you calculate the residuals for each data point?
5. Can you perform a hypothesis test on the slope coefficient?
6. What does a positive slope imply about the relationship between study time and GPA?
7. How would the equation change if we included skipped classes as another independent variable?
8. How do you determine the standard error of the slope coefficient?

**Tip:** When interpreting the y-intercept in regression analysis, consider whether it's reasonable for the independent variable to be zero in your context.

Understand how to derive and interpret the prediction equation in linear regression, focusing on the y-intercept and slope. This problem involves calculating the regression line using study time and GPA data, demonstrating key statistical formulas.

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