Question

In a recent year, the distribution of age for senators in the United States Senate was unimodal and roughly symmetric with mean 65 years and standard deviation 10.6 years. Consider a simulation with 200 trials in which, for each trial, a random sample of 5 senators’ ages is selected and the mean age is calculated. Which of the following best describes the distribution of the 200 sample mean ages?

(A) Approximately normal with mean 65 years and standard deviation 10.6 years.

(B) Approximately normal with mean 65 years and standard deviation (10.6)/√5 years.

(C) Approximately normal with mean 65 years and standard deviation (10.6)/√200 years.

(D) Approximately uniform with mean 65 years and standard deviation (10.6)/√5 years.

(E) Approximately uniform with mean 65 years and standard deviation (10.6)/√200 years.

Answer 1

Here we have

$\mu=65,\sigma=10.6,n=5$

Since sample size is less than 5 so according to CLT sampling distribution of sample mean will be approximately uniform with mean and SD as follow:

$\mu_{\bar{x}}=\mu=65$

$\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}=\frac{10.6}{\sqrt{5}}$

Correct option:

(D) Approximately uniform with mean 65 years and standard deviation (10.6)/√5 years.