Question

According to the College Board, the mean score in 2017 for the Scholastic Aptitude Test (SAT) was 1060 points with a standard deviation of 195 points https://collegereadiness.collegeboard.org/pdf/sat‐percentile‐ranks‐gender‐race‐ ethnicity.pdf). Assume that SAT scores are normally distributed.

A. Approximately what percent of SAT‐takers score between 800 and 1200?
B. Approximately what percent of SAT‐takers score 1340 and above?
C. Approximately what is the probability of a random sample of 9 SAT‐ takers getting a group average score of 1340 and above? Is this probability higher or lower than the probability of one random student getting a score of 1340 or higher?
Show your work and intuitively explain.

Answer 1

Solution:

Given in the question
Mean = 1060points
Standard deviation = 195 points
Solution(a)
P(800<Xbar<1200) = P(Xbar<1200) - P(Xbar<800)
Z = (1200-1060)/195 = 0.7179
Z = (800-1060)/195 = -1.333
From Z table we found that
P(800<Xbar<1200) = 0.7642-0.0918 = 0.6724 or 67.24%

Solution(b)
P(Xbar>=1340) =1-P(Xbar<1340)
Z = (1340-1060)/195 = 1.4358
From z table we found p-value
P(Xbar>=1340) = 1-0.9251 = 0.0749 or 7.49%

Solution(c)
Given n= 9
P(Xbar>=1340)=1-P(Xbar<1340)
Z = (1340-1060)/195/sqrt(9)
Z = 4.3076
So from Z table we found p-value
P(Xbar>=1340) = 1-0.9999966 = 0.0000033
As we can see that probability is lower than the probability of one eandom student getting a score of 1340 or higher.