Question

Scores of an IQ test have a bell-shaped distribution with a mean of 100 and a standard deviation of 10. Use the empirical rule to determine the following

 (a) What percentage of people has an IQ score between 70 and 1307

 (b) What percentage of people has an IQ score less than 90 or greater than 110? 

 (c) What percentage of people has an IQ score greater than 120?

Answer 1

The Empirical Rule
The Empirical Rule, which is also known as the three-sigma rule or the 68-95-99.7 rule, represents a high-level guide that can be used to estimate the proportion of a normal distribution that can be found within 1, 2, or 3 standard deviations of the mean. According to this rule, if the population of a given data set follows a normal, bell-shaped distribution in terms of the population mean (M) and standard deviation (SD), then the following is true of the data:

An estimated 68% of the data within the set is positioned within one standard deviation of the mean; i.e., 68% lies within the range [M - SD, M + SD].
An estimated 95% of the data within the set is positioned within two standard deviations of the mean; i.e., 95% lies within the range [M - 2SD, M + 2SD].
An estimated 97.7% of the data within the set is positioned within three standard deviations of the mean; i.e., 99.7% lies within the range [M - 3SD, M + 3SD].

Results
Approx. 99.7% of the data lies between ± 3 SD, or between 70 and 130
Approx. 95% of the data lies between ± 2 SD, or between 80 and 120
Approx. 68% of the data lies between ± 1 SD, or between 90 and 110
Therefore the answer is 99.7% approximately.