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Sagot :
Sure, let's solve this step-by-step.
1. Predicted Amount for 25 Minutes:
- Given browsing time, [tex]\( X = 25 \)[/tex] minutes.
- Regression equation: [tex]\(\hat{Y} = 2 + 0.9X\)[/tex].
Substitute [tex]\( X = 25 \)[/tex] into the equation:
[tex]\[ \hat{Y} = 2 + 0.9 \times 25 = 2 + 22.5 = 24.5 \text{ dollars} \][/tex]
- Observed amount spent, [tex]\( s = 44.5 \)[/tex] dollars.
Compare the observed amount to the predicted amount:
[tex]\[ 44.5 \text{ (observed)} > 24.5 \text{ (predicted)} \][/tex]
This means the observed value is above the regression line.
2. Predicted Amount for 30 Minutes:
- Given browsing time, [tex]\( X = 30 \)[/tex] minutes.
- Regression equation: [tex]\(\hat{Y} = 2 + 0.9X\)[/tex].
Substitute [tex]\( X = 30 \)[/tex] into the equation:
[tex]\[ \hat{Y} = 2 + 0.9 \times 30 = 2 + 27 = 29 \text{ dollars} \][/tex]
- Observed amount spent, [tex]\( s = 27.29 \)[/tex] dollars.
Compare the observed amount to the predicted amount:
[tex]\[ 27.29 \text{ (observed)} < 29 \text{ (predicted)} \][/tex]
This means the observed value is below the regression line.
3. Predicted Amount for 28 Minutes:
- Given browsing time, [tex]\( X = 28 \)[/tex] minutes.
- Regression equation: [tex]\(\hat{Y} = 2 + 0.9X\)[/tex].
Substitute [tex]\( X = 28 \)[/tex] into the equation:
[tex]\[ \hat{Y} = 2 + 0.9 \times 28 = 2 + 25.2 = 27.2 \text{ dollars} \][/tex]
- Observed amount spent, [tex]\( s = 27.2 \)[/tex] dollars.
Compare the observed amount to the predicted amount:
[tex]\[ 27.2 \text{ (observed)} = 27.2 \text{ (predicted)} \][/tex]
This means the observed value is on the regression line.
### Summary of Results:
1. Browsing time of 25 minutes:
- Predicted amount spent: [tex]\( 24.5 \)[/tex] dollars.
- Observed value in relation to the regression line: Above.
2. Browsing time of 30 minutes:
- Predicted amount spent: [tex]\( 29 \)[/tex] dollars.
- Observed value in relation to the regression line: Below.
3. Browsing time of 28 minutes:
- Predicted amount spent: [tex]\( 27.2 \)[/tex] dollars.
- Observed value in relation to the regression line: On the regression line.
1. Predicted Amount for 25 Minutes:
- Given browsing time, [tex]\( X = 25 \)[/tex] minutes.
- Regression equation: [tex]\(\hat{Y} = 2 + 0.9X\)[/tex].
Substitute [tex]\( X = 25 \)[/tex] into the equation:
[tex]\[ \hat{Y} = 2 + 0.9 \times 25 = 2 + 22.5 = 24.5 \text{ dollars} \][/tex]
- Observed amount spent, [tex]\( s = 44.5 \)[/tex] dollars.
Compare the observed amount to the predicted amount:
[tex]\[ 44.5 \text{ (observed)} > 24.5 \text{ (predicted)} \][/tex]
This means the observed value is above the regression line.
2. Predicted Amount for 30 Minutes:
- Given browsing time, [tex]\( X = 30 \)[/tex] minutes.
- Regression equation: [tex]\(\hat{Y} = 2 + 0.9X\)[/tex].
Substitute [tex]\( X = 30 \)[/tex] into the equation:
[tex]\[ \hat{Y} = 2 + 0.9 \times 30 = 2 + 27 = 29 \text{ dollars} \][/tex]
- Observed amount spent, [tex]\( s = 27.29 \)[/tex] dollars.
Compare the observed amount to the predicted amount:
[tex]\[ 27.29 \text{ (observed)} < 29 \text{ (predicted)} \][/tex]
This means the observed value is below the regression line.
3. Predicted Amount for 28 Minutes:
- Given browsing time, [tex]\( X = 28 \)[/tex] minutes.
- Regression equation: [tex]\(\hat{Y} = 2 + 0.9X\)[/tex].
Substitute [tex]\( X = 28 \)[/tex] into the equation:
[tex]\[ \hat{Y} = 2 + 0.9 \times 28 = 2 + 25.2 = 27.2 \text{ dollars} \][/tex]
- Observed amount spent, [tex]\( s = 27.2 \)[/tex] dollars.
Compare the observed amount to the predicted amount:
[tex]\[ 27.2 \text{ (observed)} = 27.2 \text{ (predicted)} \][/tex]
This means the observed value is on the regression line.
### Summary of Results:
1. Browsing time of 25 minutes:
- Predicted amount spent: [tex]\( 24.5 \)[/tex] dollars.
- Observed value in relation to the regression line: Above.
2. Browsing time of 30 minutes:
- Predicted amount spent: [tex]\( 29 \)[/tex] dollars.
- Observed value in relation to the regression line: Below.
3. Browsing time of 28 minutes:
- Predicted amount spent: [tex]\( 27.2 \)[/tex] dollars.
- Observed value in relation to the regression line: On the regression line.
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