Question

Heights of women (in inches) are approximately N(64.5,2.5) distributed. Compute the probability that the average height of 25 randomly selected women will be bigger than 66 inches.

Answer 1

Answer:

the probability that the average height of 25 randomly selected women will be bigger than 66 inches is 0.0013

Step-by-step explanation:

From the summary of the given statistical dataset

The mean and standard deviation for the sampling distribution of sample mean of 25 randomly selected women can be calculated as follows:

$\mu_{\overline x} = \mu _x$ = 64.5

$\sigma_{\overline x }= \dfrac{\sigma}{\sqrt n}$

$\sigma_{\overline x }= \dfrac{2.5}{\sqrt {25}}$

$\sigma_{\overline x }= \dfrac{2.5}{5}$

$\sigma_{\overline x }$ = 0.5

Thus X $\sim$ N (64.5,0.5)

Therefore, the probability that the average height of 25 randomly selected women will be bigger than 66 inches is:

$P(\overline X > 66) = P ( \dfrac{\overline X - \mu_\overline x}{\sigma \overline x }>\dfrac{66 - 64.5}{0.5} })$

$P(\overline X > 66) = P ( Z>\dfrac{66 - 64.5}{0.5} })$

$P(\overline X > 66) = P ( Z>\dfrac{1.5}{0.5} })$

$P(\overline X > 66) = P ( Z>3 })$

$P(\overline X > 66) = 1- P ( Z<3 })$

$P(\overline X > 66) = 1- 0.9987$

$P(\overline X > 66) =0.0013$

the probability that the average height of 25 randomly selected women will be bigger than 66 inches is 0.0013