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Sagot :
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.
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.
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