Question

Suppose the proportion of people affected by gluten sensitivity in the United States is 0.2. If we let (p^) be the proportion in a random sample of size n = 120, write the mean, standard deviation and sampling distribution of sample proportion, and find the approximate probability that the value of (p^) is

1. less than the population proportion (p) by 0.01 or more

2. within 0.01 of the p

3. not within 0.01 of the p

Answer 1

mu(pcap) = 0.2

sigma(pcap) = sqrt(0.2 * 0.8/120) = 0.0365

P(within 0.01 of p)
P(0.19 <= X <= 0.21) = P((0.21 - 0.2)/0.0365) <= z <= (0.21 - 0.2)/0.0365)
= P(-0.27 <= z <= 0.27) = P(z <= 0.27) - P(z <= -0.27)
= 0.6064 - 0.3936
= 0.2128

P(not within) = 1 - 0.2128 = 0.7872