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.