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:

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.