1. ## normal distributions/probability

The Stanford-Binet IQ scores of a group of people are normally distributed with the mean of 100 and a standard deviation of 16. According to this test, what is the probability tha a randomly selected member of the group will have the following IQ?

a) an IQ less than 100
b) an IQ greater than 116
c) an IQ less than 116
d) an IQ between 84 and 116

100 -16 = 84 and 100 + 16 = 116
84<=x<=116 would be the first standard deviation
how do I calculate the probabilities ?

2. You don't really need to find the range of the first standard deviation, you know...

Let X represent the IQ scores.

$\displaystyle X \sim N(100, 16^2)$

Then,
Standardize to the z value using:

$\displaystyle z = \dfrac{x - \mu}{\sigma}$

a) $\displaystyle P(X < 100) = P(z < \dfrac{100-100}{16}) = P(z < 0)$

Then look in your table for this probability.

Same for the others.

b) $\displaystyle P(X > 116) = P(z > \dfrac{116 - 100}{16}) = P(z > 1)$

c) $\displaystyle P(X < 116) = P(z < \dfrac{116-100}{16}) = P(z < 1)$

d) $\displaystyle P(84 < X < 116) = P( \dfrac{84-100}{16} < z < \dfrac{116-100}{16}) = P(-1 <z < 1)$

Well, since you did the range for the first standard deviation, the answer for the last one should be easier. What do you know about the probability of the central part of a normal curve bounded by one standard deviation on each side from the mean?