Question
At the final exam in a statistics class, the professor asks each student to indicate how many hours he or she studied for the exam. After grading the exam, the professor computes the least-squares regression line for predicting the final exam score from the number of hours studied. The equation of the line is .y=50+5x
Antoine studied for 6 hours, what do you predict his exam score will be
Antoine studied for 6 hours, what do you predict his exam score will be
Answers
GPT-4o mini
To predict Antoine's exam score based on the number of hours he studied, you can use the provided least-squares regression line equation, which is:
\[ y = 50 + 5x \]
In this equation, \( y \) represents the predicted exam score, and \( x \) represents the number of hours studied.
Since Antoine studied for 6 hours, you can substitute \( x = 6 \) into the equation:
\[ y = 50 + 5(6) \]
\[ y = 50 + 30 \]
\[ y = 80 \]
Thus, you predict that Antoine's exam score will be **80**.
\[ y = 50 + 5x \]
In this equation, \( y \) represents the predicted exam score, and \( x \) represents the number of hours studied.
Since Antoine studied for 6 hours, you can substitute \( x = 6 \) into the equation:
\[ y = 50 + 5(6) \]
\[ y = 50 + 30 \]
\[ y = 80 \]
Thus, you predict that Antoine's exam score will be **80**.
(b) Emma studied for 2 hours longer than Jeremy did. How much higher do you predict Emma’s score to be?
GPT-4o mini
Let's denote the number of hours Jeremy studied as \( x \). Thus, Emma studied for \( x + 2 \) hours.
To find the predicted scores for both Jeremy and Emma using the regression line \( y = 50 + 5x \):
1. **Jeremy's Predicted Score**:
\[
y_{\text{Jeremy}} = 50 + 5x
\]
2. **Emma's Predicted Score** (studied 2 hours more than Jeremy):
\[
y_{\text{Emma}} = 50 + 5(x + 2) = 50 + 5x + 10 = 60 + 5x
\]
Now, we can find the difference in predicted scores between Emma and Jeremy:
\[
\text{Difference} = y_{\text{Emma}} - y_{\text{Jeremy}} = (60 + 5x) - (50 + 5x)
\]
\[
\text{Difference} = 60 + 5x - 50 - 5x = 10
\]
Therefore, Emma's predicted score is **10 points** higher than Jeremy's predicted score, regardless of how many hours Jeremy studied.
To find the predicted scores for both Jeremy and Emma using the regression line \( y = 50 + 5x \):
1. **Jeremy's Predicted Score**:
\[
y_{\text{Jeremy}} = 50 + 5x
\]
2. **Emma's Predicted Score** (studied 2 hours more than Jeremy):
\[
y_{\text{Emma}} = 50 + 5(x + 2) = 50 + 5x + 10 = 60 + 5x
\]
Now, we can find the difference in predicted scores between Emma and Jeremy:
\[
\text{Difference} = y_{\text{Emma}} - y_{\text{Jeremy}} = (60 + 5x) - (50 + 5x)
\]
\[
\text{Difference} = 60 + 5x - 50 - 5x = 10
\]
Therefore, Emma's predicted score is **10 points** higher than Jeremy's predicted score, regardless of how many hours Jeremy studied.