Question

the blood platelet counts of a group of women have a bell - shaped distribution with a mean of 250.5 and a standard deviation of 68.1. (all units are 1000 cells/μl.) using the empirical rule, find each approximate percentage below.
a. what is the approximate percentage of women with platelet counts within 1 standard deviation of the mean, or between 182.4 and 318.6?
b. what is the approximate percentage of women with platelet counts between 46.2 and 454.8?
a. approximately □% of women in this group have platelet counts within 1 standard deviation of the mean, or between 182.4 and 318.6. (type an integer or a decimal. do not round.)
b. approximately □% of women in this group have platelet counts between 46.2 and 454.8. (type an integer or a decimal. do not round.)

Explanation:

Step1: Recall empirical rule for 1 - standard - deviation

The empirical rule for a normal (bell - shaped) distribution states that approximately 68% of the data lies within 1 standard deviation of the mean.
The lower limit is $\mu-\sigma=250.5 - 68.1=182.4$ and the upper limit is $\mu+\sigma=250.5 + 68.1=318.6$.

Step2: Recall empirical rule for 3 - standard - deviations

First, find the number of standard deviations from the mean for the given values.
The lower value $46.2$: $z_1=\frac{46.2 - 250.5}{68.1}=\frac{- 204.3}{68.1}=-3$
The upper value $454.8$: $z_2=\frac{454.8 - 250.5}{68.1}=\frac{204.3}{68.1}=3$
The empirical rule states that approximately 99.7% of the data lies within 3 standard deviations of the mean.

Answer:

a. 68
b. 99.7