Question

Assume that SAT scores are normally distributed with a mean of 1000 and a standard deviation of 150. Use this information to answer the following questions. Round final answers to the nearest whole number.

What is the lowest SAT score that can be in the top 10% of testers?

What is the highest SAT score that can be in the bottom 5% of testers?

Between which two SAT scores do the middle 50% of testers lie?

Answer 1

Given that, mean (μ) = 1000 and standard deviation = 150

a) We want to find, the value of x such that, P(X > x) = 0.10

Therefore, the lowest SAT score that can be in the top 10% of testers is 1192

b) We want to find, the value of x such that, P(X < x) = 0.05

Therefore, the highest SAT score that can be in the bottom 5% of testers is 753

c) We want to find, the values of x1 and x2 such that, P(x1 < X < x2) = 0.50

First we find the z-score such that, P(-z < Z < z) = 0.50

=> 2 * P(Z < z) - 1 = 0.50

=> 2 * P(Z < z) = 1.50

=> P(Z < z) = 0.75

Using standard normal z-table we get z-score corresponding probability of 0.75 is, 0.67

=> P(-0.67 < Z < 0.67) = 0.50

For z = -0.67

x1 = (-0.67 * 150) + 1000 = -100.5 + 1000 = 899.5 ≈ 900

For z = 0.67

x2 = (0.67 * 150) + 1000 = 100.5 + 1000 = 1100.5 ≈ 1101

Therefore, between 900 and 1101 SAT scores do the middle 50% of testers lie.