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A boutique wants to determine how the amount of time a customer spends browsing in the store affects the amount the customer spends.

The equation of the regression is [tex]$\hat{Y} = 2 + 0.9X$[/tex].

1. For a browsing time of 25 minutes resulting in an amount of [tex]$44.5, what is the predicted amount spent?

Where is the observed value in relation to the regression line?

2. For a browsing time of 30 minutes resulting in an amount spent of $[/tex]27.29, what is the predicted amount spent?

Where is the observed value in relation to the regression line?

3. For a browsing time of 28 minutes resulting in an amount spent of $27.2, what is the predicted amount spent?

Where is the observed value in relation to the regression line?

Sagot :

GinnyAnswer
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.
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