Question

A normally distributed population has mean 57.7 and standard deviation 12.1.

a. Find the probability that a single randomly selected element X of the population is less than 45

b. Find the mean and standard deviation of 𝑥̅for samples of size 16.

c. Find the probability that the mean of a sample of size 16 drawn from this population is less than 45.

Answer 1

$\mu=57.7,\sigma=12.1$

(a) $P(X<45)=P(z<\dfrac{45-57.7}{12.1})=P(Z<-1.0409)=0.14695$

(b) Sample mean $\mu_X=\mu=57.7$ , standard deviation $s=\dfrac{\sigma}{\sqrt{n}}=\dfrac{12.1}{\sqrt{16}}=\dfrac{12.1}{4}=3.025$

(c) $P(\bar{X}<45)=P(z<\dfrac{45-57.7}{3.025})=P(z<-4.19835)= 0.00001$