https://file.aiquickdraw.com/mathbot/file/1728553472223iopas54o.jpg

To solve the problem, we need to analyze the data and equation provided.

The equation of the line of best fit is:

\[
\hat{y} = -0.5x + 7.64
\]

Where:
- \(x\) is the time spent texting (in hours),
- \(\hat{y}\) is the predicted time spent exercising (in hours).

### Step-by-Step Breakdown:

1. **Predict the time spent exercising:**
   Use the equation \(\hat{y} = -0.5x + 7.64\) to find the predicted values for \(x = 4.10\) and \(x = 5.00\).

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

---

### 1. For \(x = 4.10\):
   - **Predicted time spent exercising:**
     \[
     \hat{y} = -0.5(4.10) + 7.64 = -2.05 + 7.64 = 5.59 \, \text{hours}
     \]
   - **Observed time spent exercising:** From the table, the observed value for \(x = 4.10\) is \(y = 5.59\) hours.
   - **Residual:**
     \[
     \text{Residual} = 5.59 - 5.59 = 0
     \]

---

### 2. For \(x = 5.00\):
   - **Predicted time spent exercising:**
     \[
     \hat{y} = -0.5(5.00) + 7.64 = -2.5 + 7.64 = 5.14 \, \text{hours}
     \]
   - **Observed time spent exercising:** From the table, the observed value for \(x = 5.00\) is \(y = 6.32\) hours.
   - **Residual:**
     \[
     \text{Residual} = 6.32 - 5.14 = 1.18 \, \text{hours}
     \]

---

### Summary Table:

| Time spent texting (in hours) | Observed time spent exercising (in hours) | Predicted time spent exercising (in hours) | Residual (in hours) |
|-------------------------------|-------------------------------------------|--------------------------------------------|---------------------|
| 4.10                          | 5.59                                      | 5.59                                       | 0                   |
| 5.00                          | 6.32                                      | 5.14                                       | 1.18                |

---

Let me know if you need further details or have any questions.

### 5 Related Questions:
1. How is the line of best fit calculated from a set of data points?
2. What are the uses of residuals in regression analysis?
3. How do you determine whether a line of best fit is accurate for given data?
4. What is the significance of a residual of zero?
5. How would you calculate residuals for other values of time spent texting?

### Tip:
Residuals help to show how well the predicted values match the actual data. A residual close to zero indicates a good fit between the prediction and observation.

The table and scatter plot show the time spent texting, x, and the time spent exercising, y, by each of 9 students last week. Using the equation of the line of best fit, ŷ = -0.5x + 7.64, fill in the blanks with exact answers for the predicted and residual times spent exercising for x = 4.10 and x = 5.00.

Learn how to use the line of best fit equation ŷ = -0.5x + 7.64 to calculate the predicted time spent exercising and residuals for time spent texting. This problem involves understanding linear regression, residuals, and interpreting scatter plots.

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