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 line is [tex]$\hat{Y} = 2 + 0.9X$[/tex].

1. A browsing time of 38 minutes is found to result in an amount spent of 61.2 dollars.
- What is the predicted amount spent?
- Where is the observed value in relation to the regression line?

2. A browsing time of 21 minutes is found to result in an amount spent of 19.81 dollars.
- What is the predicted amount spent?
- Where is the observed value in relation to the regression line?

3. A browsing time of 16 minutes is found to result in an amount spent of 16.4 dollars.
- What is the predicted amount spent?
- Where is the observed value in relation to the regression line?

Question

18-06-2024
Mathematics

Answered

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 line is [tex]$\hat{Y} = 2 + 0.9X$[/tex].

1. A browsing time of 38 minutes is found to result in an amount spent of 61.2 dollars.
- What is the predicted amount spent?
- Where is the observed value in relation to the regression line?

2. A browsing time of 21 minutes is found to result in an amount spent of 19.81 dollars.
- What is the predicted amount spent?
- Where is the observed value in relation to the regression line?

3. A browsing time of 16 minutes is found to result in an amount spent of 16.4 dollars.
- What is the predicted amount spent?
- Where is the observed value in relation to the regression line?

Answer :

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GinnyAnswer GinnyAnswer · Answer 1 · 2024-06-18T02:03:51-04:00

Let's solve each part of the question step-by-step using the given regression equation [tex]$\hat{Y} = 2 + 0.9X$[/tex].

### Part 1: Browsing time of 38 minutes

1. Calculate the predicted amount spent:
[tex]\[ \hat{Y} = 2 + 0.9 \times 38 \][/tex]
[tex]\[ \hat{Y} = 2 + 34.2 \][/tex]
[tex]\[ \hat{Y} = 36.2 \text{ dollars} \][/tex]

2. Compare the observed value (61.2 dollars) with the predicted amount (36.2 dollars):
[tex]\[ \text{Observed amount} - \text{Predicted amount} = 61.2 - 36.2 = 25 \text{ dollars} \][/tex]

The observed value (61.2 dollars) is 25 dollars above the regression line.

### Part 2: 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 (19.81 dollars) with the predicted amount (20.9 dollars):
[tex]\[ \text{Observed amount} - \text{Predicted amount} = 19.81 - 20.9 \approx -1.09 \text{ dollars} \][/tex]

The observed value (19.81 dollars) is approximately 1.09 dollars below the regression line.

### Part 3: Browsing time of 16 minutes

1. Calculate the predicted amount spent:
[tex]\[ \hat{Y} = 2 + 0.9 \times 16 \][/tex]
[tex]\[ \hat{Y} = 2 + 14.4 \][/tex]
[tex]\[ \hat{Y} = 16.4 \text{ dollars} \][/tex]

2. Compare the observed value (16.4 dollars) with the predicted amount (16.4 dollars):
[tex]\[ \text{Observed amount} - \text{Predicted amount} = 16.4 - 16.4 = 0 \text{ dollars} \][/tex]

The observed value (16.4 dollars) matches exactly with the regression line.

### Summary

1. For a browsing time of 38 minutes, the predicted amount spent is 36.2 dollars. The observed value (61.2 dollars) is 25 dollars above the regression line.

2. For a browsing time of 21 minutes, the predicted amount spent is 20.9 dollars. The observed value (19.81 dollars) is approximately 1.09 dollars below the regression line.

3. For a browsing time of 16 minutes, the predicted amount spent is 16.4 dollars. The observed value (16.4 dollars) is exactly on the regression line.