Question

Given a population mean of 27.1 with a standard deviation of 8.4 and a sample size of 10, answer these questions.

Answer 1

Answer:

a) 2.66

b) 9

Step-by-step explanation:

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean $\mu$ and standard deviation $\sigma$, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$.

part A: what is the standard deviation of the sampling distribution of x hat? show your work.

We have that $\sigma = 8.4, n = 10$. So

$s = \frac{\sigma}{\sqrt{n}} = \frac{8.4}{\sqrt{10}} = 2.66$

PartB: what does the sample size need to be if you want the standard deviation of the sampling distribution of x hat to be 2.8?

This is n when $s = 2.8$. So

$s = \frac{\sigma}{\sqrt{n}}$

$2.8 = \frac{8.4}{\sqrt{n}}$

$2.8\sqrt{n} = 8.4$

$\sqrt{n} = \frac{8.4}{2.8}$

$\sqrt{n} = 3$

$(\sqrt{n})^2 = 3^2$

$n = 9$