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Run a regression analysis on the following data set, where [tex]\( y \)[/tex] is the final grade in a math class and [tex]\( x \)[/tex] is the average number of hours the student spent working on math each week.

[tex]\[
\begin{array}{|c|c|}
\hline
\text{Hours/Week } (x) & \text{Grade } (y) \\
\hline
5 & 50 \\
10 & 71 \\
11 & 83.4 \\
13 & 79.2 \\
14 & 91.6 \\
15 & 85 \\
15 & 93 \\
16 & 96.4 \\
20 & 100 \\
20 & 100 \\
\hline
\end{array}
\][/tex]

1. **State the regression equation**: [tex]\( y = m \cdot x + b \)[/tex], with constants accurate to two decimal places.

[tex]\(\boxed{\phantom{x}}\)[/tex]

2. **What is the predicted value for the final grade when a student spends an average of 8 hours each week on math?**

Grade = [tex]\(\boxed{\phantom{x}}\)[/tex] (Round to 2 decimal places.)

Answer :

To solve this problem, we need to perform a regression analysis to find the relationship between the average number of hours a student spends on math each week and their final grade. We are provided with the following data:

```
\begin{tabular}{|r|r|}
\hline
hours/week (x) & Grade (y) \\
\hline
5 & 50 \\
10 & 71 \\
11 & 83.4 \\
13 & 79.2 \\
14 & 91.6 \\
15 & 85 \\
15 & 93 \\
16 & 96.4 \\
20 & 100 \\
20 & 100 \\
\hline
\end{tabular}
```

### Step 1: Perform Linear Regression

The goal of linear regression is to find the best-fitting line through the data, which can be represented by the equation:

[tex]\[ y = m \cdot x + b \][/tex]

where:
- [tex]\( y \)[/tex] is the predicted final grade,
- [tex]\( x \)[/tex] is the average number of hours a student spends on math each week,
- [tex]\( m \)[/tex] is the slope of the line,
- [tex]\( b \)[/tex] is the y-intercept.

Using the provided data, we perform linear regression analysis to determine the slope [tex]\( m \)[/tex] and the intercept [tex]\( b \)[/tex].

From the analysis, we find the regression equation as:

[tex]\[ y = 3.2 \cdot x + 40.43 \][/tex]

### Step 2: Predict the Final Grade for 8 Hours of Study

Now that we have the equation, we can predict the final grade for a student who spends an average of 8 hours each week on math. Substitute [tex]\( x = 8 \)[/tex] into the regression equation:

[tex]\[ y = 3.2 \cdot 8 + 40.43 \][/tex]

Calculate:

[tex]\[ y = 25.6 + 40.43 = 66.03 \][/tex]

Therefore, the predicted final grade for a student spending 8 hours per week on math is 66.03.

To summarize, the regression equation is [tex]\( y = 3.2 \cdot x + 40.43 \)[/tex], and for a student who studies for 8 hours each week, the predicted final grade is 66.03.

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