https://file.aiquickdraw.com/mathbot/file/1726370049282jbbub7o6.jpg

To complete the table, we'll use the equation of the line of best fit, which is given as:

\[
\hat{y} = -0.4x + 6.95
\]

Here, \(x\) is the time spent texting, and \(\hat{y}\) is the predicted time spent exercising.

### Step 1: Predicted Time Spent Exercising
For each value of \(x\) (time spent texting), we'll substitute it into the equation to find the predicted time spent exercising, \(\hat{y}\).

1. **For \(x = 4.00\) (hours of texting):**
   \[
   \hat{y} = -0.4(4.00) + 6.95 = -1.6 + 6.95 = 5.35 \, \text{hours}
   \]

2. **For \(x = 5.00\) (hours of texting):**
   \[
   \hat{y} = -0.4(5.00) + 6.95 = -2.00 + 6.95 = 4.95 \, \text{hours}
   \]

### Step 2: Residuals
The residual is the difference between the observed time spent exercising and the predicted time spent exercising. That is:
\[
\text{Residual} = \text{Observed} - \text{Predicted}
\]

We'll use the observed values from the table in the image to calculate the residuals:

1. **For \(x = 4.00\)** (observed exercise time = 4.50 hours):
   \[
   \text{Residual} = 4.50 - 5.35 = -0.85 \, \text{hours}
   \]

2. **For \(x = 5.00\)** (observed exercise time = 4.30 hours):
   \[
   \text{Residual} = 4.30 - 4.95 = -0.65 \, \text{hours}
   \]

### Final Table

| Time spent texting (in hours) | Observed time spent exercising (in hours) | Predicted time spent exercising (in hours) | Residual (in hours) |
|-------------------------------|-------------------------------------------|--------------------------------------------|---------------------|
| 4.00                          | 4.50                                      | 5.35                                       | -0.85               |
| 5.00                          | 4.30                                      | 4.95                                       | -0.65               |

Let me know if you'd like further clarification or more details on this problem!

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Here are 5 related questions to explore:
1. What is the significance of the residual in this context?
2. How would the equation of the line of best fit change if we had more data?
3. What does the slope of -0.4 represent in real-world terms?
4. How do you determine the accuracy of this line of best fit?
5. How can residuals be used to assess the quality of a model?

### Tip:  
Residuals closer to zero indicate that the model fits the data well, while larger residuals suggest that the model may not be accurately predicting the observed values.

Use the equation of the line of best fit, and use the table to fill in the blanks. Predict time spent exercising and calculate residuals based on the given values for time spent texting.

Explore how to use the equation of the line of best fit ŷ = -0.4x + 6.95 to predict time spent exercising based on time spent texting. This problem involves linear regression concepts, calculating residuals, and understanding the relationship between two variables. Suitable for Grades 9-12 students.

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