Question

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 𝜇_𝑥̅ and 𝜎_𝑥̅. 

Answer 1

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;

 𝜇_𝑥̅  = μ = 55

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

𝜎_𝑥̅. = $\frac{\sigma}{\sqrt{n}}$

where $\sigma$ = population standard deviation

Given n=100 and $\sigma$ = 20

Then;

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

Answer: The sampling distribution of the sample means has 𝜇_𝑥̅ = 55 and 𝜎_𝑥̅ =2