To determine how the amount of time a customer spends browsing in the store affects the amount the customer spends, we'll use the given regression line equation [tex]$\hat{Y} = 2 + 0.9X$[/tex] where [tex]$X$[/tex] is the browsing time in minutes and [tex]$\hat{Y}$[/tex] is the predicted amount spent in dollars.
Given the browsing times and observed amounts spent, we'll calculate the predicted amounts spent and compare them with the observed values.
### For a browsing time of 21 minutes:
1. Calculate the predicted amount spent:
[tex]\[
\hat{Y} = 2 + 0.9 \times 21
\][/tex]
[tex]\[
\hat{Y} = 2 + 18.9
\][/tex]
[tex]\[
\hat{Y} = 20.9 \text{ dollars}
\][/tex]
2. Compare the observed value with the predicted value:
The observed spending for 21 minutes is 29.9 dollars.
- Difference: [tex]\( 29.9 - 20.9 = 9.0 \)[/tex]
- Since the observed value (29.9 dollars) is higher than the predicted value (20.9 dollars), it lies above the regression line.
### For a browsing time of 10 minutes:
1. Calculate the predicted amount spent:
[tex]\[
\hat{Y} = 2 + 0.9 \times 10
\][/tex]
[tex]\[
\hat{Y} = 2 + 9
\][/tex]
[tex]\[
\hat{Y} = 11 \text{ dollars}
\][/tex]
2. Compare the observed value with the predicted value:
The observed spending for 10 minutes is 7.29 dollars.
- Difference: [tex]\( 7.29 - 11 = -3.71 \)[/tex]
- Since the observed value (7.29 dollars) is lower than the predicted value (11 dollars), it lies below the regression line.
### For a browsing time of 15 minutes:
1. Calculate the predicted amount spent:
[tex]\[
\hat{Y} = 2 + 0.9 \times 15
\][/tex]
[tex]\[
\hat{Y} = 2 + 13.5
\][/tex]
[tex]\[
\hat{Y} = 15.5 \text{ dollars}
\][/tex]
2. Compare the observed value with the predicted value:
The observed spending for 15 minutes is 15.5 dollars.
- Difference: [tex]\( 15.5 - 15.5 = 0 \)[/tex]
- Since the observed value (15.5 dollars) is equal to the predicted value (15.5 dollars), it lies on the regression line.
### Summary:
- For a browsing time of 21 minutes, the predicted amount spent is 20.9 dollars, and the observed value of 29.9 dollars lies above the regression line.
- For a browsing time of 10 minutes, the predicted amount spent is 11 dollars, and the observed value of 7.29 dollars lies below the regression line.
- For a browsing time of 15 minutes, the predicted amount spent is 15.5 dollars, and the observed value of 15.5 dollars lies on the regression line.
