Answer to Question #344789 in Statistics and Probability for Jcanne

Question

1. The height of the students in Bataan National High School- SHS has a mean of 151 cm with the standard deviation of 6. Assume that the variable is normally distributed.

a. What is the probability that their mean height is less than 150 cm?

b. If 50 of them are taken as samples, what is the probability that their mean height is less than 150cm?

Answer 1

a. We have a normal distribution, $\mu=151, \sigma=6.$

Let's convert it to the standard normal distribution,

$z=\cfrac{x-\mu}{\sigma}=\cfrac{150-151}{6}=-0.17;$

$P(X<150)=P(Z<-0.17)=0.4325 \\ \text{​(from z-table).}$

b. $\mu=151, \sigma=6, n=50.$

$z=\cfrac{\bar{x}-\mu}{\sigma/\sqrt{n}}=\cfrac{150-151}{6/\sqrt{50}}=-1.18,\\ P(\bar{X}<150)=P(\bar{Z}<-1.18)=0.1190.\\ \text{ (from z-table).}$