Question

Suppose that scores on a test are normally distributed with a mean of 80 and a standard deviation of 8. Answer the questions below. (a) What is the 70th percentile? (round to the tenths place) (b) What percentage of students score less than 70? (round to the tenths place, give the percent)

Answer 1

(a) 84.2

(b) 10.6

Explanation:

To solve this questions we can use the standardization formula, where we know that if
`X\sim N(\mu,\sigma^2)` then
`Z=(X-\mu)/(\sigma) \sim N(0,1)`

So for

(a) we know that the z score for the 70th percentile is 0.524, so using the normalization equation we have

`(X-\mu)/(\sigma)=0.524`

`X=0.524*8+80=84.192`

(b) We can procede as above and get

`P(X<70)=P((X-80)/(8)<(70-80)/(8))=P(Z<-1.25)=0.1056`