Question

Scores of an IQ test have a​ bell-shaped distribution with a mean of 100 and a standard deviation of 12. Use the empirical rule to determine the following. ​(a) What percentage of people has an IQ score between 88 and 112​? ​(b) What percentage of people has an IQ score less than 76 or greater than 124​? ​(c) What percentage of people has an IQ score greater than 136​?

Answer 1

According to the empirical rule, approximately 68.27%, 95.45% and 99.73% observations lie within 1, 2 and 3 standard deviations from the mean.

a)

We know that approximately 68.27% observations lie in the interval

So, approximately 68.27% observations lie between 88 and 112.

b)

We know that approximately 95.45% observations lie in the interval

So, approximately 95.45% observations lie between 76 and 124.

This means that approximately 4.55% observations have an IQ score less than 76 or greater than 124​.

c)

We know that approximately 99.73% observations lie in the interval

So, approximately 99.73% observations lie between 64 and 136.

This means approximately 99.73/2 = 49.865% observations lie between mean 100 and 136.

Since 50% values lie to the left of mean, so percentage of values less than 136 = 50+49.865 = 99.865%

Hence percentage of values greater than 136 = 100-99.865 = 0.135%