Question

the blood platelet counts of a group of women have a bell - shaped distribution with a mean of 258.3 and a standard deviation of 64.4. (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 129.5 and 387.1?
b. what is the approximate percentage of women with platelet counts between 193.9 and 322.7?
a. approximately □% of women in this group have platelet counts within 2 standard deviations of the mean, or between 129.5 and 387.1. (type an integer or a decimal. do not round.)

Explanation:

Step1: Recall the empirical rule

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

Step2: Calculate the range for part (a)

The mean \(\mu = 258.3\) and the standard deviation \(\sigma=64.4\).
The lower bound for 2 standard deviations below the mean is \(\mu - 2\sigma=258.3-2\times64.4 = 258.3 - 128.8=129.5\)
The upper bound for 2 standard deviations above the mean is \(\mu + 2\sigma=258.3 + 2\times64.4=258.3+128.8 = 387.1\)
So, for part (a), the percentage of women with platelet counts within 2 standard deviations of the mean is approximately \(95\%\)

Step3: Calculate the z - scores for part (b)

The z - score formula is \(z=\frac{x-\mu}{\sigma}\)
For \(x = 193.9\), \(z_1=\frac{193.9 - 258.3}{64.4}=\frac{-64.4}{64.4}=- 1\)
For \(x = 322.7\), \(z_2=\frac{322.7-258.3}{64.4}=\frac{64.4}{64.4}=1\)
According to the empirical rule, approximately 68% of the data lies within 1 standard deviation (\(z=-1\) to \(z = 1\)) of the mean.

Answer:

a. \(95\)
b. \(68\)