According to a recent U.S. Bureau of Labor Statistics report, the proportion of married couples with children in which both parents work outside the home is $59\%$. You select an SRS of $50$married couples with children and let role="math" localid="1668496807604" $\hat{p}=$the sample proportion of couples in which both parents work outside the home.

a. Identify the mean of the sampling distribution of $\hat{p}$.

b. Calculate and interpret the standard deviation of the sampling distribution of$\hat{p}$. Verify that the $10\%$condition is met.

c. Describe the shape of the sampling distribution of $\hat{p}$. Justify your answer.

$$\begin{gathered} p=59\%=0.59 \\ n=50 \end{gathered}$$

The mean of the sampling distribution of the sample proportions is same to the population proportion $p.$

$$\begin{gathered} {\mu }_{\hat{p}}=p=59\% \\ =0.59 \end{gathered}$$

${\sigma }_{\hat{p}}=\sqrt{\frac{p(1-p)}{n}}$

The mean of the sampling distribution of the sample proportions is same to the population proportion $p.$

${\mu }_{\hat{p}}=p=59\%=0.59$

The standard deviation of the sampling distribution is

$$\begin{gathered} {\sigma }_{\hat{p}}=\sqrt{\frac{p(1-p)}{n}} \\ =\sqrt{\frac{0.59(1-0.59)}{50}} \\ =0.0696 \end{gathered}$$

The proportion of couples in which both parents work outside the home among $50$married couples varies on average by$0.0696$ from the mean of$0.59$.

The sampling distribution of the sample proportions $\hat{p}$is about Normal is the large counts condition is satisfied that is when $np\ge 10$and $n(1-p)\ge 10$.

$$\begin{gathered} np=50(0.59)=29.5\ge 10 \\ n(1-p)=50(1-0.59)=20.5\ge 10 \end{gathered}$$

Since the large counts condition is satisfied, the sampling distribution of the sample proportion is approximately Normal.