Question

Please help!!!!

100 juniors at a nearby private school took the ACT test.
The scores were distributed normally with a mean of 22 and a standard deviation of 3

what percentage of scores are between 22 and 25?

what percentage of scores are between 16 and 28

What percentage is less than 13

what percentage is greater than 25

approximately how many juniors scored higher than 22

and lastly how many juniors scored between 19 and 25

Answer 1

I'll be using the empirical (68-95-99.7) rule extensively. It also helps to know that the normal distribution with mean
`\mu` and standard deviation
`\sigma` is symmetric about its mean.

a.
`P(22<X<25)=P(\mu<X<\mu+\sigma)=\frac<\sigma)2\approx34\%`

b.
`P(16<X<28)=P(|X-\mu|<2\sigma)\approx95\%`

c.
`P(X<13)=P(X<\mu-3\sigma)=\fracX-\mu2\approx1.5\%`

d.
`P(X>25)=P(X>\mu+\sigma)=\fracX-\mu2\approx16\%`

e.
`P(X>22)=50\%`, so about 50 juniors scored above the average.

f.
`P(19<X<25)=P(|X-\mu|<\sigma)\approx68\%`, so about 68 juniors scored between 19 and 25.