Question

2. Knowing the results of the 2018 SAT math exam with a mean of 531 and standard deviation of 114, answer the following questions: (10 points)

a. What percentage of test takers scored lower than 400 on the math SAT?
Your calculation:


b. What percentage scored between 600 and 700 points?
Your calculation:


c. Your score is 725. What is your percentile rank?

Answer 1

(a) 12.5%

(b) 20%

(c) 96th

Explanation:

Let X denote the results of the 2018 SAT math exam.

It is provided that the mean of exam was, μ = 531 and standard deviation of σ 114.

Assume that X follows a normal distribution.

(a)

Compute the probability of test takers who scored lower than 400 on the math SAT as follows:

`P(X<400)=P((X-\mu)/(\sigma)<(400-531)/(114))`

`=P(Z<-1.15)\\\\=0.12507\\\\\approx 0.125`

Thus, the percentage of test takers who scored lower than 400 on the math SAT is 12.5%.

(b)

Compute the probability of test takers who scored between 600 and 700 points as follows:

`P(600<X<700)=P((600-531)/(114)<(X-\mu)/(\sigma)<(700-531)/(114))`

`=P(0.61<Z<1.48)\\\\=P(Z<1.48)-P(Z<0.61)\\\\=0.93056-0.72907\\\\=0.20149\\\\\approx 0.20`

Thus, the percentage of test takers who scored between 600 and 700 points is 20%.

(c)

Compute the value of P (Z < 725) as follows:

`P(X<725)=P((X-\mu)/(\sigma)<(725-531)/(114))`

`=P(Z<1.70)\\\\=0.95543\\\\\approx 0.96`

Thus, a student who scores 725 has a 96th percentile rank.