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.
