Answer to Question #322886 in Statistics and Probability for Nate

Question

A. Find the mean (𝝁𝑿), variance (𝝈 𝑿

𝟐) and standard deviation (𝝈𝑿) of the sampling distribution of the means given the sample size, and the means and variances of the population. Sample size n were randomly selected from the population.

1. 𝑛 = 3

𝜇 = 6

𝜎² = 2.4

Mean:

Variance:

Standard Deviation:

2. 𝑛 = 25

𝜇 = 20

𝜎² = 5.5

Mean:

Variance:

Standard Deviation:

Answer 1

We'll use the properties of sampling distributions of sample means.

1. Mean:

$\mu_{\bar x} =\mu=6.$

Variance:

$\sigma^2_{\bar x}=\cfrac{\sigma^2}{n}=\cfrac{2.4}{3}=0.8.$

Standard deviation:

$\sigma_{\bar x}=\cfrac{\sigma}{\sqrt n}=\sqrt{\sigma^2_{\bar x}}=\sqrt{0.8}=0.89.$

2. Mean:

$\mu_{\bar x} =\mu=20.$

Variance:

$\sigma^2_{\bar x}=\cfrac{\sigma^2}{n}=\cfrac{5.5}{25}=0.22.$

Standard deviation:

$\sigma_{\bar x}=\cfrac{\sigma}{\sqrt n}=\sqrt{\sigma^2_{\bar x}}=\sqrt{0.22}=0.47.$