Use the Normal model ​N(​100,16​) describing IQ scores to answer the following.

Question

​a) What percent of​ people's IQs are expected to be over 80 ​?

​b) What percent of​ people's IQs are expected to be under 85​?

​c) What percent of​ people's IQs are expected to be between 116 and 128​?

​(Round to one decimal place as​ needed.)

Answer 1

X ~ Normal(100, 16)

μ = 100

σ = 16

z = (X - μ) / σ

We need to use the z-score formula to find the probability for each one.

1.) z = (80 - 100) / 16 = -20 / 16 = -1.25

P(z > -1.25) = 1 - P(z ≤ -1.25) = 1 - 0.1056 = 0.8944

About 89.44% of people's IQs are expected to be over 80.

2.) z = (85 - 100) / 16 = -15 / 16 = -0.94

P(z < -0.94) = 0.1736

About 17.36% of people's IQs are expected to be under 85.

3.) z₁ = (116 - 100) / 16 = 16 / 16 = 1.00

z₂ = (128 - 100) / 16 = 28 / 16 = 1.75

P(z₁ < 1.00) = 0.8413 P(z₂ < 1.75) = 0.9599

P(1.00 < z < 1.75) = 0.9599 - 0.8413 = 0.1186

About 11.86% of people's IQs are expected to be between 116 and 128.