Answer to Question #318135 in Statistics and Probability for Joy

Question

the average grade of senior high school students in general mathematics is 89 and the standard deviation is 2.5 assume that the variable ks normally distributed.

a. what is the probability that the grades is will be larger than 92 if a grade selected?

b. if a sample of 15 grades is selected, what is the probability that the mean of the sample will be greater than 92?

Answer 1

We have a normal distribution, $\mu=89, \sigma=2.5, n_1=1, n_2=15.$

Let's convert it to the standard normal distribution,

$z=\cfrac{\bar{x}-\mu}{\sigma/\sqrt{n}}$

a.

$z_1=\cfrac{92-89}{2.5/\sqrt{1}}=1.2,$

$P(\bar{X}>92)=1-P(\bar{X}<92)=\\ =1-P(Z<1.2)=$

$=1-0.88493=0.11507$ ​(from z-table).

b.

$z_2=\cfrac{92-89}{2.5/\sqrt{15}}=4.65,$

$P(\bar{X}>92)=1-P(\bar{X}<92)=\\ =1-P(Z<4.65)=$

$=1-1=0$ ​(from z-table).