Question

A simple random sample of size n=53 is obtained from a population with μ=53 and σ=7. Does the population need to be normally distributed for the sampling distribution of x to be approximately normally​ distributed? Why? What is the sampling distribution of x​?

Answer 1

Answer:

No, because the Central Limit Theorem posits that regardless of the shape of the underlying​ population, the sampling distribution of bar x becomes approximately normal as the sample​ size (n) increases.

The sampling distribution of x is;

$\mu_{\overline x} = \mu = 53$

Step-by-step explanation:

Given that:

The sample size n = 53

The population mean μ = 53

The standard deviation σ = 7

The sampling distribution of x is;

$\mu_{\overline x} = \mu = 53$

Sampling distribution of the standard deviation is:

$\sigma _x =\dfrac{\sigma}{\sqrt{n}}$

$\sigma _x =\dfrac{7}{\sqrt{53}}$

$\sigma _x =\dfrac{7}{7.28}$

$\sigma _x =0.96$