What is the probability that the sample proportion will be at least 2 percent more than the population proportion?

Question

In a recent year, the Better Business Bureau settled 75% of complaints they received. You have been hired by the Bureau to investigate complaints this year involving computer stores. You plan to select a random sample of complaints to estimate the proportion of complaints the Bureau is able to settle. Suppose your sample size is 206. What is the probability that the sample proportion will be at least 2 percent more than the population proportion?

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

Answer 1

phat = sample proportion

n = 206 (sample size)

Want to find P(phat > 0.77) (0.77 is 2% more or 0.02 more than 0.75)

mean proportion (p) = population proportion = 0.75

std error mean proportion = sqrt(p * (1-p)/n) = (sqrt(0.75 * 0.25/206) = 0.03

z = (phat - p)/std error.

P(phat > 0.77) = P(z > (0.77-0.75)/0.03) = P(z > 0.67) = P(z < -0.67) = 0.2514