Chapter 6: Q. 51 (page 380)

51. His and her earnings A study of working couples measures the income $X$ of the husband and the income $Y$ of the wife in a large number of couples in which both partners are employed. Suppose that you knew the means $μ_{X}$ and $μ_{Y}$ and the variances $σ_{X}^{2}$ and $σ_{γ}^{2}$ of both variables in the population.
(a) Is it reasonable to take the mean of the total income $X + Y$ to be $μ_{X} + μ_{Y}$ ? Explain your answer.
(b) Is it reasonable to take the variance of the total income to be $σ_{X}^{2} + σ_{y}^{2}$ ? Explain your answer.

Short Answer

Expert verified

(a) Yes, it is reasonable to take the mean of the total income $X + Y$ to $μ_{X} + μ_{Y}$ .

(b) No, it is not reasonable because the property of the variance of $σ_{X + Y}^{2} = σ_{X}^{2} + σ_{Y}^{2}$ .

Step by step solution

Part (a) Step 1: Given information

The income $X$ of the husband and the income $Y$ of the wife in a large number of couples in which both partners are employed. The mean of the total income $X + Y$ to be $μ_{X} + μ_{Y}$ .

Part (a) Step 2: Explanation

According to the information,

Mean $= μ_{X}$ and $μ_{Y}$
Variance $= σ^{2} x$ and $σ^{2} Y$
Yes, the mean of $X + Y$ is equal to $μ_{X} + μ_{Y}$

Because it is known that,
$μ_{X + Y} = μ_{X} + μ_{Y}$

Part (b) Step 1: Given information

The variances $σ_{X}^{2}$ and $σ_{γ}^{2}$ of both variables in the population..
The variance of the total income to be $σ_{X}^{2} + σ_{y}^{2}$ .

Part (b) Step 2: Explanation

The property will hold true, if it is believed that the variable $X$ and $Y$ are independent. Otherwise, it cannot be said that $σ_{X + Y}^{2} = σ_{X}^{2} + σ_{Y}^{2}$ .