The table below gives the number of hours ten randomly selected students spent studying and their corresponding midterm exam grades. Using this data, consider the equation of the regression line, yˆ=b0+b1x, for predicting the midterm exam grade that a student will earn based on the number of hours spent studying. Keep in mind, the correlation coefficient may or may not be statistically significant for the data given. Remember, in practice, it would not be appropriate to use the regression line to make a prediction if the correlation coefficient is not statistically significant.

Hours Studying

0

0.5

1

2

2.5

3

3.5

4

4.5

5

Midterm Grades

62

75

77

79

80

81

83

88

92

98

To find the equation of the regression line \( \hat{y} = b_0 + b_1x \) for predicting the midterm exam grade based on the number of hours spent studying, we'll go through the following steps:

### Step 1: Calculate the Means
First, we calculate the mean of the number of hours studied (\( \bar{x} \)) and the mean of the midterm grades (\( \bar{y} \)).

### Step 2: Calculate the Slope \( b_1 \)
The slope \( b_1 \) of the regression line is given by:

\[
b_1 = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}
\]

Where:
- \( x_i \) are the values of hours studied
- \( y_i \) are the corresponding midterm grades

### Step 3: Calculate the Intercept \( b_0 \)
The intercept \( b_0 \) is calculated as:

\[
b_0 = \bar{y} - b_1\bar{x}
\]

### Step 4: Determine the Correlation Coefficient \( r \)
The correlation coefficient \( r \) measures the strength and direction of the linear relationship between hours studied and midterm grades.

### Step 5: Assess Statistical Significance
Finally, we determine if the correlation coefficient is statistically significant. If it is not significant, it would not be appropriate to use the regression line for predictions.

Let's perform these calculations. I'll calculate the values for you.### Regression Line Equation
The equation of the regression line for predicting the midterm exam grade (\( \hat{y} \)) based on the number of hours studied (\( x \)) is:

\[
\hat{y} = 67.42 + 5.42x
\]

### Interpretation
- **Intercept (\( b_0 = 67.42 \))**: This suggests that if a student spends 0 hours studying, their predicted midterm grade would be approximately 67.42.
- **Slope (\( b_1 = 5.42 \))**: For each additional hour spent studying, the predicted midterm grade increases by approximately 5.42 points.

### Correlation Coefficient
The correlation coefficient \( r = 0.939 \) indicates a strong positive linear relationship between the number of hours studied and the midterm grades.

### Statistical Significance
Given the strong correlation coefficient, it suggests that there is a statistically significant relationship between study hours and grades, making it appropriate to use this regression line for prediction.

Would you like further analysis, such as residuals, predictions for specific hours, or testing the significance of the correlation?

Here are some related questions:

1. Would you like to predict a student's grade based on a specific number of study hours?
2. Should we perform a residual analysis to assess the fit of the regression model?
3. Would you like to calculate the coefficient of determination \( R^2 \) to understand the variance explained by the model?
4. Do you want to visualize the regression line on a scatter plot?
5. How would you interpret the significance of the correlation coefficient?
6. Should we explore any potential outliers or influential points in the data?
7. Are you interested in conducting a hypothesis test to confirm the significance of the slope?
8. Would you like to compare this linear model with a possible non-linear model?

**Tip:** The intercept in a regression line may not always have a practical interpretation, especially when the x-value is not close to zero or if zero is outside the range of observed data.

Learn how to predict midterm exam grades based on the number of hours studied using regression analysis. Calculate the regression line equation, interpret the correlation coefficient, and assess statistical significance. Suitable for advanced high school or college students.

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