Question

Say the correlation coefficient between the heights of husbands and wives is \(0.70\) and the mean male height is 5 feet 10 inches with standard deviation 2 inches, and the mean female height is 5 feet 4 inches with standard deviation \(1 \frac{1}{2}\) inches. Assuming a bivariate normal distribution, what is the best guess of the height of a woman whose husband's height is 6 feet? Find a \(95 \%\) prediction interval for her height.

Short answer

The predicted height of the woman is 5 feet 5 inches. Without the sample size, a 95% prediction interval cannot be directly computed.

Step-by-step solution

Convert heights to inches

First, convert the mean male height and the height of the specific husband given to the same units (inches) for consistency. So, 5 feet 10 inches becomes 70 inches, and 6 feet becomes 72 inches.

Calculate the slope of the regression line

Next, calculate the slope (beta) of the regression line using the formula: \(\beta = r \left( \frac{\sigma_Y}{\sigma_X} \right)\), where \(r\) is the correlation coefficient (0.70), \(\sigma_X\) is the standard deviation of male height (2 inches), and \(\sigma_Y\) is the standard deviation of female height (\(1.5\) inches). The slope, \(\beta\), is \(0.70 \times (1.5/2) = 0.525\).

Calculate the intercept of the regression line

Now calculate the intercept (alpha) of the regression line using the formula: \(\alpha = \bar{Y} - \beta \bar{X} \), where \(\bar{X}\) is the mean male height (70 inches), and \(\bar{Y}\) is the mean female height (64 inches). The intercept, \(\alpha\), is \(64 - 0.525 \times 70 = 27.25\) inches.

Predict the woman's height

Given the husband's height (72 inches), predict the woman's height using the formula: \(Y = \alpha + \beta X\). So, the predicted height, \(Y\), is \(27.25 + 0.525 \times 72 = 65\) inches, or 5 feet 5 inches.

Calculate the prediction interval

To calculate the 95% prediction interval for her height, apply the formula: \(Y \pm t \sigma_Y \sqrt{1 + 1/n + (X - \bar{X})^2/(\sigma_X^2(n - 1))}\), where \(n\) is the sample size. Without the sample size, we can't directly compute the prediction interval. In real-life situations, the sample size would usually be given or estimated based on available data.