Answer to Question #322893 in Statistics and Probability for Nate

Question

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

7. 𝑛 = 9

𝜇 = 5.3

𝜎 = 3

Mean:

Variance:

Standard Deviation:

8. 𝑛 = 11

𝜇 = 4.23

𝜎 = 5

Mean:

Variance:

Standard Deviation:

Answer 1

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

7. Mean:

$\mu_{\bar x} =\mu=5.3.$

Variance:

$\sigma^2_{\bar x}=\cfrac{\sigma^2}{n}=\cfrac{3^2}{9}=1.$

Standard deviation:

$\sigma_{\bar x}=\cfrac{\sigma}{\sqrt n}=\cfrac{3}{\sqrt 9}=1.$

8. Mean:

$\mu_{\bar x} =\mu=4.23.$

Variance:

$\sigma^2_{\bar x}=\cfrac{\sigma^2}{n}=\cfrac{5^2}{11}=2.27.$

Standard deviation:

$\sigma_{\bar x}=\cfrac{\sigma}{\sqrt n}=\cfrac{5}{\sqrt {11}}=1.51.$