Question

Scores of an IQ test have a bell-shaped distribution with a mean of 100 and a standard deviation of 14. Use the empirical rule to determine the following.

(a) What percentage of people has an IQ score between 72 and 128 ?

(b) What percentage of people has an IQ score less than 86 or greater than 114 ?

(c) What percentage of people has an IQ score greater than 128 ?

Answer 1

Solution: We are given:

The empirical rule says that for a normal distribution, nearly all of the data will fall within three standard deviations of the mean.

68% of data falls within the first standard deviation from the mean.

95% fall within two standard deviations from the mean.

99.7% fall within three standard deviations from the mean.

(a) What percentage of people has an IQ score between 72 and 128 ?

Answer: We have to find

We need to find first the z score, when x = 72 and when x = 128

When x = 72, then we have:

  

  

And when x = 128, we have:

  

  

Therefore, we have to find . It means we have to find the area within 2 standard deviation from the mean.

From the empirical rule, we know that 95% of data lies within the 2 standard deviations from the mean.

Therefore, 95% of the people has an IQ score between 72 and 128.

(b) What percentage of people has an IQ score less than 86 or greater than 114 ?

Answer: We have to find or

We need to find first the z score, when x = 86 and when x = 114

When x = 86, then we have:

  

  

And when x = 114, we have:

  

  

Therefore, We have to find or . It means we have to find the area which is 1 standard deviation below the mean and area which is 1 standard deviationabove the mean.

Using the above empirical rule:

= 50% - 34% = 16% (Because there is 50% of area from the mean towards left)

= 50% - 34% = 16%   (Because there is 50% of area from the mean towards right)

Therefore, 16% + 16% = 32% of people has an IQ score less than 86 or greater than 114.

(c) What percentage of people has an IQ score greater than 128 ?

Answer: Here we have to find

Using the z score formula, we have:

Therefore, we have to find , which means we have to find the area 2 standard deviations above the mean.

= 50% - (34%+13.5%)= 2.5%

Therefore, 2.5% of people has an IQ score greater than 128