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Answer :
To solve the problem of finding the regression equation for the given data and predicting the final grade when a student spends an average of 9 hours, let's go through the steps involved in a linear regression analysis.
### Step 1: Understand the Data
We have a set of data where:
- [tex]\( x \)[/tex] represents the average number of hours spent on math each week.
- [tex]\( y \)[/tex] represents the final grade in a math class.
The data set is as follows:
- Hours/week (x): 4, 5, 5, 6, 7, 9, 16, 17, 18, 18
- Grades (y): 42.6, 50, 57, 58.4, 68.8, 65.6, 92.4, 85.8, 91.2, 100
### Step 2: Calculate the Mean
First, find the average (mean) for both [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
### Step 3: Calculate the Slope (m) and Intercept (b)
To find the slope [tex]\( m \)[/tex], use the formula:
[tex]\[
m = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}
\][/tex]
And to find the intercept [tex]\( b \)[/tex], use the formula:
[tex]\[
b = \bar{y} - m \cdot \bar{x}
\][/tex]
### Step 4: Formulate the Regression Equation
Now we can write the regression equation in the form:
[tex]\[
y = m \cdot x + b
\][/tex]
Based on the analysis, the regression equation obtained is:
[tex]\[
y = 3.20x + 37.56
\][/tex]
### Step 5: Predict the Grade for 9 Hours/Week
Now, we want to predict the final grade when the average number of hours spent is 9.
Substitute [tex]\( x = 9 \)[/tex] into the regression equation:
[tex]\[
y = 3.20 \cdot 9 + 37.56
\][/tex]
Calculate this expression to find the predicted grade:
[tex]\[
y = 28.8 + 37.56 = 66.36
\][/tex]
Thus, the predicted value for the final grade when a student spends an average of 9 hours is approximately 66.38.
### Step 1: Understand the Data
We have a set of data where:
- [tex]\( x \)[/tex] represents the average number of hours spent on math each week.
- [tex]\( y \)[/tex] represents the final grade in a math class.
The data set is as follows:
- Hours/week (x): 4, 5, 5, 6, 7, 9, 16, 17, 18, 18
- Grades (y): 42.6, 50, 57, 58.4, 68.8, 65.6, 92.4, 85.8, 91.2, 100
### Step 2: Calculate the Mean
First, find the average (mean) for both [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
### Step 3: Calculate the Slope (m) and Intercept (b)
To find the slope [tex]\( m \)[/tex], use the formula:
[tex]\[
m = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}
\][/tex]
And to find the intercept [tex]\( b \)[/tex], use the formula:
[tex]\[
b = \bar{y} - m \cdot \bar{x}
\][/tex]
### Step 4: Formulate the Regression Equation
Now we can write the regression equation in the form:
[tex]\[
y = m \cdot x + b
\][/tex]
Based on the analysis, the regression equation obtained is:
[tex]\[
y = 3.20x + 37.56
\][/tex]
### Step 5: Predict the Grade for 9 Hours/Week
Now, we want to predict the final grade when the average number of hours spent is 9.
Substitute [tex]\( x = 9 \)[/tex] into the regression equation:
[tex]\[
y = 3.20 \cdot 9 + 37.56
\][/tex]
Calculate this expression to find the predicted grade:
[tex]\[
y = 28.8 + 37.56 = 66.36
\][/tex]
Thus, the predicted value for the final grade when a student spends an average of 9 hours is approximately 66.38.
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