The empirical rule is a statistical rule (also called the three-sigma rule or the 68-95-99.7 rule) that states that, for normally distributed data, almost all of the data will fall within three standard deviations on either side of the mean.
This rule states that:
68% of data within 1 standard deviation
95% of data within 2 standard deviations
99.7% of data within 3 standard deviations
The question gives that:
[tex]\begin{gathered} mean=170\text{ cm} \\ sd=7\text{ cm} \end{gathered}[/tex]
The question asks to get the percentage of values between 149 cm and 191 cm. If the empirical rule holds, it means that:
[tex]\begin{gathered} 170-7n=149 \\ and \\ 170+7n=191 \end{gathered}[/tex]
We can solve for n using any of the equations:
[tex]\begin{gathered} 7n=191-170 \\ 7n=21 \\ n=\frac{21}{7} \\ n=3 \\ or \\ 7n=170-149=21 \\ n=3 \end{gathered}[/tex]
This means that the values lie between 3 standard deviations of the mean.
Therefore, by the Empirical Rule, 99.7% of the men are between 156 cm and 184 cm.
