Question

the blood platelet counts of a group of women have a bell - shaped distribution with a mean of 256.2 and a standard deviation of 61.8. (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 194.4 and 318.0?
b. what is the approximate percentage of women with platelet counts between 70.8 and 441.6?
a. approximately $square$% of women in this group have platelet counts within 1 standard deviation of the mean, or between 194.4 and 318.0.
(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 \(68\%\) of the data lies within \(1\) standard deviation of the mean, approximately \(95\%\) lies within \(2\) standard deviations of the mean, and approximately \(99.7\%\) lies within \(3\) standard deviations of the mean.

Step2: Calculate the number of standard deviations for part (a)

For the range \(194.4\) to \(318.0\):
The mean \(\mu = 256.2\) and the standard deviation \(\sigma=61.8\)
\(\mu-\sigma=256.2 - 61.8=194.4\)
\(\mu+\sigma=256.2 + 61.8=318.0\)
This is within \(1\) standard deviation of the mean.

Step3: Calculate the number of standard deviations for part (b)

For the range \(70.8\) to \(441.6\):
\(\mu - 3\sigma=256.2-3\times61.8=256.2 - 185.4 = 70.8\)
\(\mu+3\sigma=256.2 + 3\times61.8=256.2+185.4 = 441.6\)
This is within \(3\) standard deviations of the mean.

Answer:

a. Approximately \(68\%\) of women in this group have platelet counts within \(1\) standard deviation of the mean, or between \(194.4\) and \(318.0\).
b. Approximately \(99.7\%\) of women in this group have platelet counts between \(70.8\) and \(441.6\).