Question

Suppose that you are interested in purchasing a house in a particular area of a city and are inter- ested in the average size of the homes in that area. In a random sample of 200 homes, you find a sample mean of 2127.94 square feet and a standard deviation of 387.276 square feet. Further- more, you calculated a 99% confidence interval for the true mean size to be (2056.72, 2199.16). Why is it unnecessary to check for normality in this setting?

My attempt: It is unnecessary to check for normality, because the sample size is large enough for the central limit theorem to be used. Thus the sample distribution is approximately normal.

Answer 1

Result:

Suppose that you are interested in purchasing a house in a particular area of a city and are inter- ested in the average size of the homes in that area. In a random sample of 200 homes, you find a sample mean of 2127.94 square feet and a standard deviation of 387.276 square feet. Further- more, you calculated a 99% confidence interval for the true mean size to be (2056.72, 2199.16). Why is it unnecessary to check for normality in this setting?

My attempt: It is unnecessary to check for normality, because the sample size is large enough for the central limit theorem to be used. Thus the sample distribution is approximately normal.

Yes your attempt is correct.

The central limit theorem states that the sampling distribution of the mean of any independent, random variable will be normal or nearly normal, if the sample size is large enough, In this case the sample size of 200 is large enough to the assumption of normality.