Catherine Chao, Director of Marketing Research, needs a sample of Kansas City households to participate in the testing of a new toothpaste package.If 40% of the households in Kansas City prefer the new package, the probability that Catherine's random sample of 300 households will have a sample proportion between 0.35 and 0.45 is closest to ___________.
A)0.9232
B)0.0768
C)0.4616
D)0.0384
E)0.8976

Question

Quizplus · Accepted Answer

Using the Central Limit Theorem, we can approximate the sampling distribution of the sample proportion with a normal distribution because the sample size is large (n=300). The mean of the sampling distribution is equal to the population proportion (p = 0.40), and the standard deviation (standard error) of the sampling distribution is sqrt[p(1-p)/n] = sqrt[0.40(0.60)/300] = 0.02887. We can use the normal distribution to find the probability that the sample proportion is between 0.35 and 0.45. Using a Z-score calculation for both 0.35 and 0.45 gives us Z-scores of (0.35-0.40)/0.02887 = -1.73 and (0.45-0.40)/0.02887 = 1.73, respectively. Looking up these Z-scores in a standard normal distribution table or using a calculator, we find that the area between these Z-scores is approximately 0.9232, indicating the probability.

Catherine Chao, Director of Marketing Research, Needs a Sample of Kansas