Question

A set of 200 test scores are normally distributed with a mean of 70 and a standard deviation of 5. Approximately how many students scored between a 60 and an 80

Answer 1

Answer: About 191 students scored between a 60 and an 80.

Explanation:

Given : A set of 200 test scores are normally distributed with a mean of 70 and a standard deviation of 5.

i.e.
`\mu=70` and
`\sigma=5`

let x be the random variable that denotes the test scores.

Then, the probability that the students scored between a 60 and an 80 :

`P(60<x<80)=P((60-70)/(5)<(x-\mu)/(\sigma)<(80-70)/(5))\\\\=P(-2<z<2)\ \ [\because z=(x-\mu)/(\sigma)]\\\\=P(z<2)-P(z<-2)\ \ [\because\ P(z_1<Z<z_2)=P(Z<z_2)-P(Z<z_1)]\\\\=P(z<2)-(1-P(z<2))\ \ [\because\ P(Z<-z)=1-P(Z<z)]\\\\=2P(z<2)-1\\\\=2(0.9772)- 1 \ \ [\text{By z-table}]\\\\=0.9544`

The number of students scored between a 60 and an 80 = 0.9544 x 200

= 190.88 ≈ 191

Hence , about 191 students scored between a 60 and an 80.