Question

5. Based on the 2000 Census, the proportion of the California population aged 15 years old or older who are married is p = 0.524. Suppose n = 1000 persons are to be sampled from this population and the sample proportion of married persons ( ) is to be calculated. pˆa) What is the mean of the sampling distribution of ? pˆb) What is the standard deviation of the sampling distribution of ?pˆc) What is the approximate probability that less than 50% of the people in the sample are married?

Answer 1

Let $X=$ the sample proportion of items: $X\sim B(n, p)$ X\sim B(n, p).

The mean and variance for the binomial variate is

$$mean=\mu=E(X)=np, Var(X)=\sigma^2=np(1-p)$$

Given that $n=1000, p=0.524$. Then

$$mean=\mu=E(X)=np=1000(0.524)=524>10$$ $$Var(X)=\sigma^2=np(1-p)=1000(0.524)(1-0.524)=249.424$$

$$n(1-p)=1000(1-0.524)=476>10$$

We can use the normal model

$$N\big(p, \sqrt{{p(1-p) \over n}}\big)=N\big(0.524, \sqrt{{0.524(1-0.524) \over 1000}}\big)$$

to approximate the sampling distribution of $\hat{p}$

a) What is the mean of the sampling distribution of $\hat{p}$

$$mean=0.524$$

b) What is the standard deviation of the sampling distribution of $\hat{p}$

$$\sigma_{\hat{p}}=\sqrt{{0.524(1-0.524) \over 1000}}\approx0.0158$$

c) What is the approximate probability that less than 50% of the people in the sample are married?

$$P(\hat{p}<0.5)=P(z<{0.5-0.524 \over 0.0158})\approx P(Z<-1.5190)\approx0.0644$$