What is the probability that the sample proportion is between 0.2 and 0.42?

Question

Based on historical data, your manager believes that 29% of the company's orders come from first-time customers. A random sample of 166 orders will be used to estimate the proportion of first-time-customers. What is the probability that the sample proportion is between 0.2 and 0.42?

 (Enter your answer as a number accurate to 4 decimal places.)

Answer 1

population proportion = p = 0.29

n = sample size = 166

standard error = (0.29 * 0.71/166)^0.5 = 0.035

phat = sample proportion

want to find P(0.2 < phat < 0.42)

use z-statistic = (phat - p)/standard error

P(0.2 < phat < 0.42) = P( (0.2-0.29)/0.035 < z < (0.42-0.29)/0.035) = P(-2.57 < z < 3.71) =

P(z < 3.71) - P(z < -2.57) = 1 - 0.0051 = 0.9949