Question

Based on the Normal model N(100,16) describing IQ scores, what percent of people's IQs would you expect to be:
a) over 80%
b) under 90%
c) between 112 and 132

Answer 1

Answer on Question #54882 – Math – Statistics and Probability

Based on the Normal model N(100,16) describing IQ scores, what percent of people's IQs would you expect to be:

a) over 80

b) under 90

c) between 112 and 132

Solution

In this example we work with notation N(mu, sigma), where mu=100, sigma=16.

a) over 80:

$$p = P(X > 80) = P\left(Z > \frac{80 - 100}{16}\right) = P(Z > -1.25) = 1 - P(Z < -1.25) = 1 - 0.1056 = 0.8944,$$

where $X$ follows N(100,16) model, $Z$ follows N(0,1) model.

Answer: 89.44%.

b) under 90:

$$p = P(X < 90) = P\left(Z < \frac{90 - 100}{16}\right) = P(Z < -0.625) = 0.2660,$$

where $X$ follows N(100,16) model, $Z$ follows N(0,1) model.

Answer: 26.60%.

c) between 112 and 132:

$$\begin{array}{l} p = P(112 < X < 132) = P\left(\frac{112 - 100}{16} < Z < \frac{132 - 100}{16}\right) = P(Z < 2) - P(Z < 0.75) \\ = 0.9772 - 0.7734 = 0.2038, \end{array}$$

where $X$ follows N(100,16) model, $Z$ follows N(0,1) model.

Answer: 20.38%.

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