Question

A variable of a population has a mean of μ=83 and a standard deviation of σ=12. a. Identify the sampling distribution of the sample mean for samples of size 36 . b. In answering part (a), what assumptions did you make about the distribution of the variable? c. Can you answer part (a) if the sample size is 25 instead of 36 ? Why or why not?

Answer 1

a. The sampling distribution of the sample mean for samples of size 36 follows a normal distribution. b. The assumptions are that the population distribution is approximately normal and the sample size is large enough for the Central Limit Theorem to apply. c. The Central Limit Theorem applies when the sample size is large enough, so the sampling distribution of the sample mean for samples of size 25 would also follow a normal distribution.

Step-by-step explanation:

a. The sampling distribution of the sample mean for samples of size 36 follows a normal distribution. This is known as the Central Limit Theorem.

b. The assumptions made about the distribution of the variable are that the population distribution is approximately normal and the sample size is large enough for the Central Limit Theorem to apply.

c. Yes, we can answer part (a) if the sample size is 25 instead of 36. The Central Limit Theorem applies when the sample size is large enough, typically considered to be n ≥ 30. Therefore, the sampling distribution of the sample mean for samples of size 25 would also follow a normal distribution.