Given a population in which the probability of success is p =0.20, if a sample of 500 items is taken, then calculate the probability the proportion of successes in the sample will be between 0.18 and 0.23.
A) 0.7812
B) 0.8221
C) 0.9212
D) 0.6812

Question

Quizplus · Accepted Answer

To calculate the probability that the proportion of successes in the sample will be between 0.18 and 0.23, we use the normal approximation to the binomial distribution because the sample size is large (n = 500). The mean of the sampling distribution of the proportion is $ \mu_p = p = 0.20 $, and the standard deviation is $ \sigma_p = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.20(0.80)}{500}} $. We then convert the proportions 0.18 and 0.23 to z-scores and use the standard normal distribution to find the probabilities. The calculations yield a probability of approximately 0.8221, making option B the correct choice.

Given a Population in Which the Probability of Success Is