Question

the blood platelet counts of a group of women have a bell - shaped distribution with a mean of 257.1 and a standard deviation of 63.2 (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 67.5 and 446.7?
b. what is the approximate percentage of women with platelet counts between 130.7 and 383.5?
a. approximately □% of women in this group have platelet counts within 3 standard deviations of the mean, or between 67.5 and 446.7. (type an integer or a decimal. do not round.)

Explanation:

Step1: Recall the empirical rule for normal distribution

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

Step2: Analyze part a

We are asked for the percentage of women with platelet counts within 3 standard deviations of the mean. By the empirical rule, this percentage is 99.7.

Step3: Analyze part b

First, find the number of standard deviations from the mean for the given values. Let $\mu = 257.1$ and $\sigma=63.2$. For $x_1 = 130.7$, $z_1=\frac{130.7 - 257.1}{63.2}=\frac{- 126.4}{63.2}=- 2$. For $x_2 = 383.5$, $z_2=\frac{383.5 - 257.1}{63.2}=\frac{126.4}{63.2}=2$. The empirical rule states that approximately 95% of the data lies within 2 standard deviations of the mean.

Answer:

a. 99.7
b. 95