Question

the blood platelet counts of a group of women have a bell - shaped distribution with a mean of 201.1 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 2 standard deviations of the mean, or between 136.7 and 385.5?
b. what is the approximate percentage of women with platelet counts between 74.5 and 447.7?
a. approximately % of women in this group have platelet counts within 2 standard deviations of the mean, or between 136.7 and 385.5. (type an integer or a decimal. do not round.)

Explanation:

Step1: Recall empirical rule

The empirical rule for a normal - distribution states that approximately 95% of the data lies within 2 standard deviations of the mean, and approximately 99.7% of the data lies within 3 standard deviations of the mean.

Step2: Analyze part a

We are asked to find the percentage of women with platelet counts within 2 standard deviations of the mean.
By the empirical rule, the percentage of data within 2 standard deviations of the mean in a bell - shaped (normal) distribution is 95%.

Step3: Analyze part b

First, find the number of standard deviations from the mean for the values 74.5 and 447.7.
Let $\mu = 201.1$ and $\sigma=62.2$.
For $x = 74.5$, the number of standard deviations $z_1=\frac{74.5 - 201.1}{62.2}=\frac{- 126.6}{62.2}\approx - 2.035\approx - 2$
For $x = 447.7$, the number of standard deviations $z_2=\frac{447.7 - 201.1}{62.2}=\frac{246.6}{62.2}\approx3.965\approx4$
Since 74.5 is approximately 2 standard deviations below the mean and 447.7 is approximately 4 standard deviations above the mean, and we know that about 99.7% of the data lies within 3 standard deviations of the mean, and the distribution is symmetric, almost all (very close to 100%) of the data lies between 74.5 and 447.7.

Answer:

a. 95
b. 99.7 (approximate, since 447.7 is a bit more than 3 standard deviations above the mean but very close to covering almost all of the distribution)