Question

A population of unknown shape has a mean of 75. You select a sample of 20. The standard deviation of the sample is 5. Compute the probability the sample mean is: a. Between 76 and 77. (0.5 Points) b. Greater than 77. (0.5 Points)

Answer 1

Answer: a. 0.1500  b. 0.0367

Explanation:

Let X is a random variable with a distribution that may be known or unknown:

$\mu_x$ = the mean of X

$\sigma_x$ = the standard deviation of X

If we draw random samples of size n, then the random samples contains sample means $\overline{X}$, tends to be normally distributed

$\overline{X}\sim N(\mu_x,\dfrac{\sigma_x}{\sqrt{n}})$

Here, $\mu_x$ = 75

$\sigma_x$ =5

n=20

a.

$P(76$

b.

$P(\overline{X}>77)=P(\dfrac{\overline{X}-\mu}{\dfrac{\sigma}{\sqrt{n}}}>\dfrac{77-75}{\dfrac{5}{\sqrt{20}}})\\\\=P(Z>1.79) \ \ [z=\dfrac{\overline{X}-\mu}{\dfrac{\sigma}{\sqrt{n}}}]\\\\=1-P(z$