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Answer:
Step-by-step explanation:
To find the expected age of a brother when the sister’s age is 12 years, we can use the concept of linear regression in statistics. Given that the coefficient of correlation (\( r \)) between the ages of brothers and sisters is 0.8, and we have the means and standard deviations of their ages, we can use these values to determine the expected age of the brother.
Here’s the step-by-step process to solve the problem:
1. **Identify the known values:**
- Correlation coefficient (\( r \)): 0.8
- Mean age of brothers (\( \bar{X} \)): 25 years
- Mean age of sisters (\( \bar{Y} \)): 22 years
- Standard deviation of brothers (\( \sigma_X \)): 4 years
- Standard deviation of sisters (\( \sigma_Y \)): 5 years
- Age of the sister (\( Y \)): 12 years
2. **Calculate the slope (\( b \)) of the regression line:**
The formula for the slope of the regression line of brothers' age on sisters' age is:
\[
b = r \cdot \frac{\sigma_X}{\sigma_Y}
\]
where:
- \( r = 0.8 \)
- \( \sigma_X = 4 \)
- \( \sigma_Y = 5 \)
Plugging in the values:
\[
b = 0.8 \cdot \frac{4}{5} = 0.8 \cdot 0.8 = 0.64
\]
3. **Calculate the intercept (\( a \)) of the regression line:**
The formula for the intercept of the regression line is:
\[
a = \bar{X} - b \cdot \bar{Y}
\]
where:
- \( \bar{X} = 25 \)
- \( \bar{Y} = 22 \)
- \( b = 0.64 \)
Plugging in the values:
\[
a = 25 - 0.64 \cdot 22 = 25 - 14.08 = 10.92
\]
4. **Predict the age of the brother when the sister’s age is 12 years:**
Using the regression equation:
\[
\text{Expected age of the brother} = a + b \cdot Y
\]
where:
- \( a = 10.92 \)
- \( b = 0.64 \)
- \( Y = 12 \)
Plugging in the values:
\[
\text{Expected age of the brother} = 10.92 + 0.64 \cdot 12 = 10.92 + 7.68 = 18.60
\]
Therefore, the expected age of the brother when the sister’s age is 12 years is **18.6 years**.