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 about 68% of the data lies within 1 standard deviation of the mean ($\mu\pm\sigma$), about 95% lies within 2 standard deviations ($\mu\pm2\sigma$), and about 99.7% lies within 3 standard deviations ($\mu\pm3\sigma$). 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:
The z - score is calculated as $z=\frac{x - \mu}{\sigma}$. For $x = 158$, $z_1=\frac{158 - 168}{5}=\frac{- 10}{5}=-2$. For $x = 178$, $z_2=\frac{178 - 168}{5}=\frac{10}{5}=2$.
The percentage of data between $z=-2$ and $z = 2$ is about 95% according to the empirical rule. Since the normal distribution is symmetric about the mean, the percentage of data between the mean ($z = 0$) and $z = 2$ is $\frac{95\%}{2}=47.5\%$.

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

For the values 153 cm and 183 cm:
For $x = 153$, $z_1=\frac{153 - 168}{5}=\frac{-15}{5}=-3$. For $x = 183$, $z_2=\frac{183 - 168}{5}=\frac{15}{5}=3$.
The percentage of data between $z=-3$ and $z = 3$ is about 99.7% according to the empirical rule.

Answer:

a. 47.5
b. 99.7