Question

In recent years, SAT scores are approximately normal with a mean of 1100 and a SD of 130, while ACT scores are also approximately normal with a mean of 22 and a SD of 6.5. Without any studying, you took both tests and scored 1250 on the SAT and 24 on the ACT.

a) Did you do better on the SAT or the ACT? Make sure to find the percentile (what percent of the test takers scored lower than you?) of each score and compare.

b) You know you can do better, so you plan to spend the entire summer preparing to retake the SAT, and you’re aiming to score in the 97th percentile. What score will put you in the 97th percentile? For full credit, show all your work.

c) What proportion of the students would score between 1250 to 1400 on the SAT?

Answer 1

a)

For SAT scores

μ=1100

σ=130

x=1250

z=(x-μ)/σ

=(1250-1100)/130

=1.15

P(x<1250) = P(z<1.15) = 0.8749 =87.49 percentile

For ACT scores

μ=22

σ=6.5

x=24

z=(x-μ)/σ

=(24-22)/6.5

=0.31

P(x<24) = P(z<0.31)

               =0.6217 = 62.17 percentile

So,Score on SAT was better because we get 87.49 percentile as compared to 62.17 percentile which was obtained on ACT

b)

Here P=97% =0.97

For P=0.97, Z score is 1.88

x=z*σ + μ

=1.88*130 + 1100

=1344.4

c)

μ=1100

σ=130

For x=1250

z=(x-μ)/σ

=(1250-1100)/130

=1.15

For x=1400

z=(x-μ)/σ

=(1400-1100)/130

=2.31

P(1250<x<1400) = P(1.15<z<2.31)

                              =P(z<2.31) - P(z<1.15)

                              =0.9896 - 0.8749 { using normal distribution table)

                              =0.1147