Question

heights of men on a baseball team have a bell - shaped distribution with a mean of 168 cm and a standard deviation of 5 cm. using the empirical rule, what is the approximate percentage of the men between the following values? a. 158 cm and 178 cm b. 153 cm and 183 cm a. % of the men are between 158 cm and 178 cm. (round to one decimal place as needed.)

Explanation:

Step1: Recall the empirical rule

The empirical rule for a normal - distribution states that approximately 68% of the data lies within 1 standard deviation of the mean ($\mu\pm\sigma$), 95% lies within 2 standard deviations of the mean ($\mu\pm2\sigma$), and 99.7% lies within 3 standard deviations of the mean ($\mu\pm3\sigma$). The mean $\mu = 168$ cm and the standard deviation $\sigma=5$ cm.

Step2: Calculate the number of standard - deviations for part a

For the values 158 cm and 178 cm:
First, find the number of standard - deviations from the mean.
For $x = 158$ cm, $z_1=\frac{158 - 168}{5}=\frac{- 10}{5}=-2$.
For $x = 178$ cm, $z_2=\frac{178 - 168}{5}=\frac{10}{5}=2$.
Since 158 cm is 2 standard deviations below the mean and 178 cm is 2 standard deviations above the mean, by the empirical rule, the percentage of data within 2 standard deviations of the mean is 95.0%.

Step3: Calculate the number of standard - deviations for part b

For the values 153 cm and 183 cm:
For $x = 153$ cm, $z_1=\frac{153 - 168}{5}=\frac{-15}{5}=-3$.
For $x = 183$ cm, $z_2=\frac{183 - 168}{5}=\frac{15}{5}=3$.
Since 153 cm is 3 standard deviations below the mean and 183 cm is 3 standard deviations above the mean, by the empirical rule, the percentage of data within 3 standard deviations of the mean is 99.7%.

Answer:

a. 95.0%
b. 99.7%