1. A random sample of n = 100 measurements is obtained from a population with 𝜇 = 55 and 𝜎 = 20. Describe the sampling distribution for the sample means by computing the 𝜇𝑥̅ and 𝜎𝑥̅.

Since the sample size n=100 is statistically considered to be large (n>30), we assume that the mean of the sampling distribution of x̄ is equal to the population mean, μ.

Therefore;

 𝜇𝑥̅  = μ = 55

Also, we estimate the sample standard deviation (𝜎𝑥̅.) using the formula:

𝜎𝑥̅. = $\frac{\sigma}{\sqrt{n}}$

where $\sigma$ = population standard deviation

Given n=100 and $\sigma$ = 20

Then;

𝜎𝑥̅ = $\frac{\sigma}{\sqrt{n}}$ = $\frac{20}{\sqrt{100}}$ = 2

Answer: The sampling distribution of the sample means has 𝜇𝑥̅ = 55 and 𝜎𝑥̅ =2