Consider a train that runs from Seattle to Los Angeles. The mean travel time from one stop to the next is 128 minutes, with a standard deviation of 112 minutes. The mean distance traveled from one stop to the next is 107 miles with a standard deviation of 100 miles. The correlation between travel time and distance is 0.631.

A scatterplot has a horizontal axis labeled "Distance (miles)" with values from 0 to 350 and a vertical axis labeled "Travel Time (minutes)" with values from 0 to 380. The scatterplot has 16 points. A pattern goes up and right from (10, 5) to (345, 360). The points are scattered widely from the pattern.
(a)
Write the equation of the regression line for predicting travel time. (Let y represent travel time in minutes and let x represent distance traveled in miles.)
ŷ =    
Incorrect: Your answer is incorrect.
(b)
Interpret the slope in this context.
For each mile increase in distance traveled, we would expect travel time to increase on average by  
Incorrect: Your answer is incorrect.
 minutes.
Interpret the intercept in this context.
When the distance traveled is 0 miles, the travel time is expected to be on average   
Incorrect: Your answer is incorrect.
 minutes.

To solve the problem of finding the regression equation, interpreting the slope, and intercept in this context, we follow these steps:

### Part (a): Writing the regression equation

The general form of the linear regression equation is:
\[
\hat{y} = b_0 + b_1 x
\]
Where:
- \(\hat{y}\) is the predicted value of the response variable (in this case, travel time in minutes),
- \(x\) is the explanatory variable (distance in miles),
- \(b_1\) is the slope of the regression line (change in travel time for each additional mile traveled),
- \(b_0\) is the intercept (predicted travel time when distance = 0).

We can calculate the slope \(b_1\) using the following formula:
\[
b_1 = r \cdot \frac{s_y}{s_x}
\]
Where:
- \(r = 0.631\) is the correlation between travel time and distance,
- \(s_y = 112\) is the standard deviation of travel time,
- \(s_x = 100\) is the standard deviation of distance.

Now, calculate the slope \(b_1\):
\[
b_1 = 0.631 \cdot \frac{112}{100} = 0.631 \cdot 1.12 = 0.70752
\]
So, the slope \(b_1 = 0.70752\).

To calculate the intercept \(b_0\), use the formula:
\[
b_0 = \bar{y} - b_1 \bar{x}
\]
Where:
- \(\bar{y} = 128\) is the mean travel time,
- \(\bar{x} = 107\) is the mean distance traveled.

Now, calculate the intercept \(b_0\):
\[
b_0 = 128 - (0.70752 \cdot 107) = 128 - 75.70464 = 52.29536
\]
So, the intercept \(b_0 = 52.29536\).

Therefore, the equation of the regression line is:
\[
\hat{y} = 52.30 + 0.71x
\]
### Part (b): Interpreting the slope

The slope \(b_1 = 0.71\) means that for each additional mile traveled, we expect the travel time to increase on average by **0.71 minutes**.

### Part (c): Interpreting the intercept

The intercept \(b_0 = 52.30\) means that when the distance traveled is 0 miles, the expected travel time is **52.30 minutes**. This could represent the time spent on things like boarding, waiting, or initial procedures that do not depend on the distance traveled.

### Summary:
(a) The regression equation is:
\[
\hat{y} = 52.30 + 0.71x
\]

(b) For each additional mile traveled, the expected increase in travel time is 0.71 minutes.

(c) When no distance is traveled, the expected travel time is 52.30 minutes.

---

Would you like further details or clarifications?  
Here are 5 questions to explore further:
1. How do changes in correlation affect the slope in a regression model?
2. Why is it important to check the scatter of data points around the regression line?
3. Can the intercept always be interpreted meaningfully in real-world contexts?
4. How would a perfect correlation (r = 1) impact the slope and intercept?
5. What role does the standard deviation of the variables play in the regression equation?

**Tip:** In regression analysis, outliers or widely scattered points can heavily influence the results, so it's important to check the residuals for any deviations from normal patterns.

Consider a train that runs from Seattle to Los Angeles. The mean travel time from one stop to the next is 128 minutes, with a standard deviation of 112 minutes. The mean distance traveled from one stop to the next is 107 miles with a standard deviation of 100 miles. The correlation between travel time and distance is 0.631.

A scatterplot has a horizontal axis labeled 'Distance (miles)' with values from 0 to 350 and a vertical axis labeled 'Travel Time (minutes)' with values from 0 to 380. The scatterplot has 16 points. A pattern goes up and right from (10, 5) to (345, 360). The points are scattered widely from the pattern.
(a)
Write the equation of the regression line for predicting travel time. (Let y represent travel time in minutes and let x represent distance traveled in miles.)
(b)
Interpret the slope in this context.
(c)
Interpret the intercept in this context.

Learn how to calculate the regression line for predicting travel time based on distance for a train route. This problem uses correlation and standard deviations to find the slope and intercept. The example covers the mean travel time (128 minutes) and mean distance (107 miles). Suitable for students in Grades 11-12.

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