Question

Among first year students at a certain university, scores on the Verbal SAT follow the normal curve; the average is around 550 and the SD is about 100.

a) what percentage of these students have scores in the range 400 to 700?
b) There were about 1000 students with scores in the range 450-650 on the Verbal SAT. About _______ of them had scores in the range 500 to 600. Fill in blanks. Explain briefly

Answer 1

Part (a):


`\mathbb P(400<X<700)=\mathbb P\left((400-550)/(100)<(X-550)/(100)<(700-550)/(100)\right)`

`=\mathbb P(-1.5<Z<1.5)`

`=\mathbb P(Z<1.5)-\mathbb P(Z<-1.5)`

`\approx0.9332-0.0668=0.8664=86.64\%`

Part (b):


`\mathbb P(450<X<650)=\mathbb P(-1<Z<1)\approx0.68`

So if 1000 students make up 68% of the total student body, then there are
`0.68x=1000\implies x\approx1471` total students. You then have


`\mathbb P(500<X<600)=\mathbb P(-0.5<Z<0.5)\approx0.3829`

as the proportion of students scoring between 500 and 600. Of the 1471 or so students, this amounts to about 563 students.