The table and scatter plot show the time spent watching TV, x, and the time spent doing homework, y, by cuch of 11 students last week.

The equation of the line of best lit is -0.5x-22.63.

Time spent watching TV,x (in hours)

Time spent doing homework, y (in hours)

4.60

22.48

6.00

19.60

6.80

16.58

8.20

21.14

10.50

15.49

10.50

18.80

11.90

16.68

13.40

13.40

15.10

16.53

16.50

16.70

18.00

17.50

Time spent doing homework (in hours)

Time spent watching TV (in hours)

Use 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 watching TV (in hours)

11.90

18.00

Observed tam spent doing namework (in hours)

Predicted time spent doing homework (m hours)

X

Resident (in hours)



To solve this problem, we'll use the equation of the line of best fit provided: 

\[ y = -0.5x + 22.63 \]

Where:
- \( y \) is the time spent doing homework (in hours),
- \( x \) is the time spent watching TV (in hours).

We are asked to fill in the blanks for the observed and predicted times spent doing homework and calculate the residuals for the given times spent watching TV.

### Given Data:

- **Time spent watching TV**: \( x = 11.90 \) and \( x = 18.00 \)
- **Observed time spent doing homework**: (You need to use the data table)

We'll calculate the predicted time spent doing homework using the line of best fit for both values of \( x \) and then calculate the residuals (the difference between the observed and predicted values).

### Step 1: Calculate Predicted Time Spent Doing Homework

#### For \( x = 11.90 \):

\[
y_{\text{pred}} = -0.5(11.90) + 22.63
\]

\[
y_{\text{pred}} = -5.95 + 22.63 = 16.68 \text{ hours}
\]

#### For \( x = 18.00 \):

\[
y_{\text{pred}} = -0.5(18.00) + 22.63
\]

\[
y_{\text{pred}} = -9.00 + 22.63 = 13.63 \text{ hours}
\]

### Step 2: Calculate Residuals

The residual is calculated as:

\[
\text{Residual} = y_{\text{obs}} - y_{\text{pred}}
\]

Where:
- \( y_{\text{obs}} \) is the observed time spent doing homework.
- \( y_{\text{pred}} \) is the predicted time spent doing homework.

#### For \( x = 11.90 \):

- **Observed**: From the table, when \( x = 11.90 \), \( y_{\text{obs}} = 16.68 \) hours.
- **Predicted**: \( y_{\text{pred}} = 16.68 \) hours.

\[
\text{Residual} = 16.68 - 16.68 = 0.00 \text{ hours}
\]

#### For \( x = 18.00 \):

- **Observed**: From the table, when \( x = 18.00 \), \( y_{\text{obs}} = 17.50 \) hours.
- **Predicted**: \( y_{\text{pred}} = 13.63 \) hours.

\[
\text{Residual} = 17.50 - 13.63 = 3.87 \text{ hours}
\]

### Summary:

| Time spent watching TV (in hours) | Observed time spent doing homework (in hours) | Predicted time spent doing homework (in hours) | Residual (in hours) |
|----------------------------------|----------------------------------------------|----------------------------------------------|---------------------|
| 11.90                            | 16.68                                        | 16.68                                        | 0.00                |
| 18.00                            | 17.50                                        | 13.63                                        | 3.87                |

These are the values you should fill in for the blanks provided.

Would you like further explanations, or do you have any other questions?

---

### Follow-Up Questions:

1. How does the residual indicate the accuracy of the prediction?
2. What does a residual of zero indicate about the observed and predicted values?
3. How would the residual change if the actual observed time was higher than predicted?
4. Can the equation of the line of best fit be used to make reliable predictions for any value of \( x \)?
5. What would happen to the residual if the observed time spent doing homework was less than the predicted time?

**Tip:** Residuals help identify how well a model fits the data. A small residual means the prediction is close to the actual value, while a large residual indicates a larger error in prediction.

Explore how linear regression is used to predict homework time based on TV watching time using the equation of a line. Learn to calculate residuals and understand their significance in evaluating prediction accuracy. Suitable for high school students.

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