Question

the blood platelet counts of a group of women have a bell - shaped distribution with a mean of 261.6 and a standard deviation of 68.8 (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 2 standard deviations of the mean, or between 114.0 and 389.2?
b. what is the approximate percentage of women with platelet counts between 45.2 and 458.0?
a. approximately % of women in this group have platelet counts within 2 standard deviations of the mean, or between 114.0 and 389.2.
(type an integer or a decimal. do not round.)
b. approximately % of women in this group have platelet counts between 45.2 and 458.0.
(type an integer or a decimal. do not round.)

Explanation:

Step1: Recall the empirical rule

The empirical rule states that for a bell - shaped distribution:

• Approximately \(68\%\) of the data lies within \(1\) standard deviation of the mean (\(\mu\pm\sigma\)).
• Approximately \(95\%\) of the data lies within \(2\) standard deviations of the mean (\(\mu\pm2\sigma\)).
• Approximately \(99.7\%\) of the data lies within \(3\) standard deviations of the mean (\(\mu\pm3\sigma\)).

Step2: Calculate the range for part (a)

Given \(\mu = 251.6\) and \(\sigma=68.8\).
For \(2\) standard deviations: \(\mu - 2\sigma=251.6-2\times68.8 = 251.6 - 137.6=114.0\) and \(\mu + 2\sigma=251.6 + 2\times68.8=251.6 + 137.6 = 389.2\).
By the empirical rule, approximately \(95\%\) of women have platelet counts within \(2\) standard deviations of the mean.

Step3: Calculate the range for part (b)

For \(3\) standard deviations: \(\mu - 3\sigma=251.6-3\times68.8=251.6 - 206.4 = 45.2\) and \(\mu+3\sigma=251.6+3\times68.8=251.6+206.4 = 458.0\).
By the empirical rule, approximately \(99.7\%\) of women have platelet counts within \(3\) standard deviations of the mean.

Answer:

a. \(95\)
b. \(99.7\)