Sure, let's break this down step-by-step:
### Regression Formula
The equation of the regression line is given by:
[tex]\[ \hat{Y} = 2 + 0.9X \][/tex]
where:
- [tex]\(\hat{Y}\)[/tex] is the predicted amount spent,
- [tex]\(X\)[/tex] is the browsing time in minutes.
We need to determine the predicted amount spent and compare it to the observed value for different browsing times.
### 1. Browsing Time: 25 minutes
Observed Spending: 47.5 dollars
Step 1: Calculate the predicted spending using the regression equation.
[tex]\[ \hat{Y} = 2 + 0.9 \times 25 \][/tex]
[tex]\[ \hat{Y} = 2 + 22.5 \][/tex]
[tex]\[ \hat{Y} = 24.5 \][/tex]
Step 2: Compare the observed spending (47.5 dollars) to the predicted spending (24.5 dollars).
- The observed spending of 47.5 dollars is above the predicted spending of 24.5 dollars.
### 2. Browsing Time: 32 minutes
Observed Spending: 27.89 dollars
Step 1: Calculate the predicted spending using the regression equation.
[tex]\[ \hat{Y} = 2 + 0.9 \times 32 \][/tex]
[tex]\[ \hat{Y} = 2 + 28.8 \][/tex]
[tex]\[ \hat{Y} = 30.8 \][/tex]
Step 2: Compare the observed spending (27.89 dollars) to the predicted spending (30.8 dollars).
- The observed spending of 27.89 dollars is below the predicted spending of 30.8 dollars.
### 3. Browsing Time: 4 minutes
Observed Spending: 5.6 dollars
Step 1: Calculate the predicted spending using the regression equation.
[tex]\[ \hat{Y} = 2 + 0.9 \times 4 \][/tex]
[tex]\[ \hat{Y} = 2 + 3.6 \][/tex]
[tex]\[ \hat{Y} = 5.6 \][/tex]
Step 2: Compare the observed spending (5.6 dollars) to the predicted spending (5.6 dollars).
- The observed spending of 5.6 dollars is on the predicted spending of 5.6 dollars.
### Summary
- Browsing time 25 minutes:
- Predicted amount: 24.5 dollars
- Observed value is above the regression line.
- Browsing time 32 minutes:
- Predicted amount: 30.8 dollars
- Observed value is below the regression line.
- Browsing time 4 minutes:
- Predicted amount: 5.6 dollars
- Observed value is on the regression line.
