Question

Chebyshev’s Theorem 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 Chebyshev’s theorem, what do we know about the percentage of women with platelet counts that are within 3 standard deviations of the mean? What are the minimum and maximum platelet counts that are within 3 standard deviations of the mean?

Short answer

The minimum percentage of women with platelet counts within three standard deviations of the mean equals 89%.

The maximum platelet count within three standard deviations of the mean is equal to 451.3, and the minimum platelet count within three standard deviations of the mean is equal to 58.9.

Step-by-step solution

Given information

Female platelet counts are provided.

The mean female platelet count is equal to 255.1, and the standard deviation of the female platelet count is equal to 65.4.

Chebyshev’s theorem

Chebyshev’s theorem is applied to any dataset that approximately follows the bell-shaped probability distribution.

It explains that a minimum of $1-\frac{1}{K^2}$observations fall within K standard deviations of the mean of the data.

Thus, to compute the percentage of women that have a platelet count within K=3 standard deviations of the mean, the following calculation is done:

$$\begin{gathered} 1-\frac{1}{K^2}=1-\frac{1}{3^2} \\ =1-\frac{1}{9} \\ =\frac{8}{9} \end{gathered}$$

Therefore, $\frac{8}{9}$or approximately 89%of women have a platelet count within three standard deviations of the mean.

The limits become as follows:

$$\begin{gathered} \mu -3\sigma =255.1-3\left(65.4\right) \\ =58.9 \\ \mu +3\sigma =255.1+3\left(65.4\right) \\ =451.3 \end{gathered}$$

Thus, the maximum value of platelet count within these limits is equal to 451.3, and the minimum value of platelet count within these limits is equal to 58.9.