Question

The Empirical Rule Based on Data Set 1 “Body Data” in Appendix B, blood platelet counts of women have a bell-shaped distribution with a mean of 255.1 and a standard deviation of 65.4. (All units are 1000 cells/L.) Using the empirical rule, what is the approximate percentage of women with platelet counts

a. within 2 standard deviations of the mean, or between 124.3 and 385.9?

b. between 189.7 and 320.5?

Short answer

a. 95% of women have a platelet count within two standard deviations of the mean.

b. 68% of women have a platelet count between 189.7 and 320.5.

Step-by-step solution

Given information

Female blood platelet counts are listed for a group of females.

The mean platelet count is equal to 255.1.

The standard deviation of the platelet count is equal to 65.4.

Empirical rule

For a dataset that seems to follow a bell-shaped distribution, the following rules can be applied:

• Percentage of values between $\left(\mu -\sigma ,\mu +\sigma \right)$equal to 68%
• Percentage of values between $\left(\mu -2\sigma ,\mu +2\sigma \right)$equal to 95%
• Percentage of values between $\left(\mu -3\sigma ,\mu +3\sigma \right)$equal to 99.7%

a.

The limits are given to be above and below two standard deviations of the mean.

The values are (124.3,385.9).

According to the empirical rule stated above, the percentage of women that fall between the given platelet counts (124.3,385.9) is equal to 95%.

b.

The values of the limits are given as (189.7,320.5).

The following calculations are performed.

$$\begin{gathered} \mu -\sigma =255.1-65.4 \\ =189.7 \\ \mu +\sigma =255.1+65.4 \\ =320.5 \end{gathered}$$

It can be observed that the given values are above and below one standard deviation of the mean. Therefore, according to the rule stated above, 68% of females have a platelet count between (189.7,320.5).