Consider the following data on shoulder girth and height of a group of physically active adults. The mean shoulder girth is 107.60 cm with a standard deviation of 10.36 cm. The mean height is 171.16 cm with a standard deviation of 9.39 cm. The correlation between height and shoulder girth is 0.69.

A scatterplot has a horizontal axis labeled "Shoulder girth (cm)" with values from 85 to 135 and a vertical axis labeled "Height (cm)" with values from 145 to 200. The scatterplot has many points. A pattern goes up and right from (86, 145) to (134, 199). The points are scattered moderately from the pattern.
(a)
Write the equation of the regression line for predicting height. (Let y represent height in centimeters and let x represent shoulder girth in centimeters. Round your numerical values to three decimal places.)
ŷ = 
  
Incorrect: Your answer is incorrect.
(b)
Interpret the slope in this context. (Round your answer to three decimal places.)
For each centimeter increase in shoulder girth, we would expect height to increase on average by 
.625
 
Correct: Your answer is correct.
 centimeters.
Interpret the intercept in this context. (Round your answer to three decimal places.)
People who have a shoulder girth of 0 centimeters are expected to be on average 
 
Incorrect: Your answer is incorrect.
 centimeters tall.

To find the equation of the regression line for predicting height based on shoulder girth, we need to use the formula for a simple linear regression line:

\[
\hat{y} = b_0 + b_1x
\]

where:
- \( \hat{y} \) is the predicted height,
- \( b_0 \) is the intercept,
- \( b_1 \) is the slope,
- \( x \) is the shoulder girth.

We need the following values for the calculation:
- The correlation coefficient \( r = 0.69 \),
- The standard deviation of height \( s_y = 9.39 \),
- The standard deviation of shoulder girth \( s_x = 10.36 \),
- The mean height \( \bar{y} = 171.16 \),
- The mean shoulder girth \( \bar{x} = 107.60 \).

### Step 1: Calculate the slope (\(b_1\))

The formula for the slope is:

\[
b_1 = r \times \frac{s_y}{s_x}
\]

Substituting the values:

\[
b_1 = 0.69 \times \frac{9.39}{10.36} = 0.625
\]

So, the slope \(b_1\) is 0.625.

### Step 2: Calculate the intercept (\(b_0\))

The formula for the intercept is:

\[
b_0 = \bar{y} - b_1 \times \bar{x}
\]

Substituting the values:

\[
b_0 = 171.16 - 0.625 \times 107.60
\]

\[
b_0 = 171.16 - 67.25 = 103.91
\]

So, the intercept \(b_0\) is 103.91.

### Final regression equation:
\[
\hat{y} = 103.910 + 0.625x
\]

### Part (b): Interpretation of the intercept

The intercept represents the predicted height when the shoulder girth is 0 cm. This is not a realistic value in this context, since no one would have a shoulder girth of 0 cm, so the intercept doesn't have a meaningful interpretation in practical terms. It is simply a necessary part of the regression equation.

Would you like to go deeper into any of these steps or details?

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Here are some follow-up questions to extend your understanding:
1. How does the correlation value affect the slope of the regression line?
2. What are the assumptions of the linear regression model, and how could they be violated?
3. Why is the intercept not meaningful in this context, and what would happen if we forced it to be 0?
4. How would the regression equation change if the correlation were negative?
5. How could outliers in the data impact the regression line?

**Tip:** Always examine the residuals of your regression to check for any patterns, which can indicate if the linear model is appropriate.

Consider the following data on shoulder girth and height of a group of physically active adults. The mean shoulder girth is 107.60 cm with a standard deviation of 10.36 cm. The mean height is 171.16 cm with a standard deviation of 9.39 cm. The correlation between height and shoulder girth is 0.69.

A scatterplot has a horizontal axis labeled 'Shoulder girth (cm)' with values from 85 to 135 and a vertical axis labeled 'Height (cm)' with values from 145 to 200. The scatterplot has many points. A pattern goes up and right from (86, 145) to (134, 199). The points are scattered moderately from the pattern.
(a) Write the equation of the regression line for predicting height. (Let y represent height in centimeters and let x represent shoulder girth in centimeters. Round your numerical values to three decimal places.)
(b) Interpret the slope in this context. For each centimeter increase in shoulder girth, we would expect height to increase on average by 0.625 centimeters.

Explore how to calculate and interpret the linear regression equation for predicting height based on shoulder girth in a group of physically active adults. Learn key concepts like slope, intercept, and correlation, with practical examples. Ideal for students in advanced high school or undergraduate statistics courses.

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