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

Answer:

(a) 12.5%

(b) 20%

(c) 96th

Step-by-step 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:

$P(X<400)=P(\frac{X-\mu}{\sigma}<\frac{400-531}{114})$

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

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:

$P(600<X<700)=P(\frac{600-531}{114}<\frac{X-\mu}{\sigma}<\frac{700-531}{114})$

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

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:

$P(X<725)=P(\frac{X-\mu}{\sigma}<\frac{725-531}{114})$

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

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