Question

the blood platelet counts of a group of women have a bell - shaped distribution with a mean of 262.3 and a standard deviation of 62.2 (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 3 standard deviations of the mean, or between 75.7 and 448.9?
b. what is the approximate percentage of women with platelet counts between 137.9 and 386.7?

a. approximately % of women in this group have platelet counts within 3 standard deviations of the mean, or between 75.7 and 448.9.
(type an integer or a decimal. do not round.)
b. approximately % of women in this group have platelet counts between 137.9 and 386.7
(type an integer or a decimal. do not round.)

Explanation:

Step1: Recall the empirical rule

The empirical rule for a normal (bell - shaped) distribution states that approximately 99.7% of the data lies within 3 standard deviations of the mean, 95% lies within 2 standard deviations of the mean, and 68% lies within 1 standard deviation of the mean.

Step2: Calculate the number of standard deviations for part b

First, find the number of standard deviations from the mean for the values 137.9 and 306.7. Let $\mu = 202.3$ and $\sigma=62.2$.
For $x = 137.9$, the number of standard deviations $z_1=\frac{137.9 - 202.3}{62.2}=\frac{- 64.4}{62.2}\approx - 1$.
For $x = 306.7$, the number of standard deviations $z_2=\frac{306.7 - 202.3}{62.2}=\frac{104.4}{62.2}\approx1.68$. But since we are using the empirical - rule approximations, we note that 137.9 and 306.7 are approximately 1 standard deviation below and above the mean respectively. Approximately 68% of the data lies within 1 standard deviation of the mean.

Answer:

a. 99.7
b. 68