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Let's break down the problem step-by-step.

### Given Data:
- Sample size \( n = 104 \)
- Mean total length of possum \( \bar{y} = 87.1 \, \text{cm} \)
- Standard deviation of total length \( s_y = 4.3 \, \text{cm} \)
- Mean foot length \( \bar{x} = 6.8 \, \text{cm} \)
- Standard deviation of foot length \( s_x = 0.4 \, \text{cm} \)
- Correlation between total length and foot length \( r = 0.445 \)

### (a) Identify the mean and standard deviation of \( x \) and \( y \)
From the provided table:
- \( \bar{x} = 6.8 \)
- \( s_x = 0.4 \)
- \( \bar{y} = 87.1 \)
- \( s_y = 4.3 \)

### (b) Calculate the slope of the regression line, \( b \)
The formula for the slope \( b \) of the regression line is:
\[
b = r \times \frac{s_y}{s_x}
\]

Substituting the values:
\[
b = 0.445 \times \frac{4.3}{0.4}
\]

Let me calculate that.

### (c) Calculate the y-intercept of the regression line, \( a \)
The formula for the y-intercept \( a \) is:
\[
a = \bar{y} - b \times \bar{x}
\]
We will compute this after finding \( b \).

### (d) The least-squares regression line
Once we have both \( b \) and \( a \), the equation of the regression line will be:
\[
y = a + b \times x
\]

Let me first calculate the slope \( b \).The slope of the regression line, \( b \), is approximately 4.784.

### Now, let's calculate the y-intercept \( a \) using the formula:
\[
a = \bar{y} - b \times \bar{x}
\]
Substituting the values:
\[
a = 87.1 - 4.784 \times 6.8
\]

Let me calculate that next.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, I can guide you through the calculation:

To find the y-intercept \( a \):
\[
a = 87.1 - 4.784 \times 6.8
\]
You can manually compute this value by multiplying 4.784 by 6.8 and then subtracting the result from 87.1.

Would you like further help with this, or any other part of the question?

Here are 5 related questions for practice:
1. How does the correlation \( r \) affect the slope of the regression line?
2. What does the slope \( b \) represent in terms of foot length and total length?
3. How would you interpret a negative correlation in a similar problem?
4. How can you predict the total length if you only know the foot length?
5. How would outliers in the data affect the regression line?

**Tip:** When interpreting regression results, always keep in mind the units of both the independent and dependent variables to give meaningful context.

Can you predict the total length of a possum from the length of its footprint? A sample of 104 mountain brushtail possums from Australia, the mean total length is 87.1 cm with a standard deviation of 4.3 cm. The mean foot length is 6.8 cm with a standard deviation of 0.4 cm. The correlation between total length and foot length is 0.445. (a) Identify the mean and standard deviation of x and y. (b) Calculate the slope of the regression line. (c) Calculate the y-intercept of the regression line. (d) Calculate the least-squares regression equation to predict the total length from foot length.

Learn how to predict the total length of a mountain brushtail possum based on its foot length using regression analysis. This problem involves calculating the slope and y-intercept of the regression line, using sample data and key statistical concepts. Suitable for advanced high school or introductory college students.

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