Answer to Question #128203 in Statistics and Probability for Abby

Question

Suppose a simple random sample of size n=50 is taken from a population of N=15,000 with a population proportion of p=0.4 having a certain characteristic. What is the probability that a simple random sample will have x > 21 having that characteristic?

(Round to four decimal places)

Answer 1

Check that the sample size is large enough:

$$n\cdot p=50\cdot0.4=20\geq10$$

$$np(1-p)=50\cdot0.4(1-0.4)=12\geq10$$

Sampling distribution for $\hat{p}$ is approximately normal:

$$\mu_{\hat{p}}=p=0.4$$

$$\sigma_{\hat{p}}=\sqrt{\dfrac{p(1-p)}{n}}=\sqrt{\dfrac{0.4(1-0.4)}{50}}\approx0.069282$$

$${21\over 50}=0.42$$

$$P(\hat{p}>0.42)=1-P(\hat{p}\leq0.42)=$$

$$=1-P(Z\leq\dfrac{0.42-0.4}{0.069282})\approx1-P(Z\leq0.288675)\approx$$

$$\approx0.386415$$

The probability that a simple random sample will have x > 21 having that characteristic is approximately $0.3864.$