 The table and scatter, plot show the time spent texting, x, and the time spent exercising, y, by each of 9 students last week.

The equation of the line of best fit is y=-0.4x+7.07.

Time spent texting, x (in hours)

Time spent exercising, y (in hours)

2.10

6.44

2.20

554

2.80

6.96

3.50

5.24

4.00

5.50

4.50

5.24

5.00

6.50

3.28

5.50

5.90

5.50

Time spent texting (in hours)

Vise the equation of the line of best fit, and use the table above, to fill in the blanks below.

Give exact answers, not rounded approximations.

Time spent texting (in hours)

5.00

5.90

Observed time spent exercising (in hours)

Predicted time spent exercising (in hours)

X

Residual (in hours)



To complete the table by calculating the predicted time spent exercising and the residuals for each data point, we'll follow the steps below. We'll use the equation of the line of best fit, \( y = -0.4x + 7.07 \), to predict the time spent exercising based on the time spent texting.

### Step-by-Step Calculation:

1. **Equation of the line of best fit**:
   \[
   y = -0.4x + 7.07
   \]
   This equation gives the predicted value of \( y \) (time spent exercising) for any given \( x \) (time spent texting).

2. **Residual**:
   \[
   \text{Residual} = \text{Observed } y - \text{Predicted } y
   \]

Now, let's calculate the predicted values and residuals for each pair of \( x \) and observed \( y \).

### Calculations:

#### For \( x = 5.00 \) hours:
- **Predicted \( y \)**: 
  \[
  y = -0.4(5.00) + 7.07 = -2 + 7.07 = 5.07 \text{ hours}
  \]
- **Observed \( y \)**: 6.50 hours (from the table).
- **Residual**:
  \[
  \text{Residual} = 6.50 - 5.07 = 1.43 \text{ hours}
  \]

#### For \( x = 5.90 \) hours:
- **Predicted \( y \)**:
  \[
  y = -0.4(5.90) + 7.07 = -2.36 + 7.07 = 4.71 \text{ hours}
  \]
- **Observed \( y \)**: 5.90 hours (from the table).
- **Residual**:
  \[
  \text{Residual} = 5.90 - 4.71 = 1.19 \text{ hours}
  \]

### Final Table:

| Time spent texting (in hours) | Observed time spent exercising (in hours) | Predicted time spent exercising (in hours) | Residual (in hours) |
|-------------------------------|-------------------------------------------|--------------------------------------------|---------------------|
| 5.00                           | 6.50                                      | 5.07                                       | 1.43                |
| 5.90                           | 5.90                                      | 4.71                                       | 1.19                |

Let me know if you need any more details or explanations!

Here are 5 related questions for practice:
1. What would be the predicted time spent exercising for 2.80 hours of texting?
2. Calculate the residual for a student who spent 3.50 hours texting and 5.24 hours exercising.
3. How does the residual indicate how well the model fits a specific data point?
4. If the slope of the line of best fit was -0.6 instead of -0.4, how would this change the predictions?
5. Why is it important to consider residuals when evaluating the effectiveness of a regression model?

**Tip**: Residuals close to zero indicate that the model's predictions are close to the actual observed values, meaning the model fits the data well for those points.

Learn how to calculate predicted time spent exercising from texting time using a linear regression model. This problem involves applying the equation of a line of best fit and calculating residuals. Suitable for high school students studying linear regression.

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