Question

The blood platelet counts of a group of women have a​ bell-shaped distribution with a mean of 250.8 and a standard deviation of 69.3. ​(All units are 1000 ​cells/muμ​L.) 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 181.5 and 320.1?b. What is the approximate percentage of women with platelet counts between 112.2 and 389.4?

Answer 1

a) 68%; b) 99.5%

Explanation:

The empirical rule states that 68% of data falls within 1 standard deviation of the mean; 95% of data falls within 2 standard deviations of the mean; and 99.7% of data falls within 3 standard deviations of the mean.

For part a,

We are asked the approximate percentage of women whose platelet counts are within 1 standard deviation of the mean. According to the empirical rule, this is 68%.

For part b,

We are given the endpoints of the interval. The lower endpoint is 181.5; this is 250.8-181.5 = 138.6 away from the mean. Dividing by the standard deviation, 69.3, we have

138.6/69.3 = 2

This is 2 standard deviations away from the mean.

The higher endpoint is 320.1; this is 320.1-181.5 = 138.6 away from the mean. Dividing by the standard deviation, 69.3, we have

138.6/69.3 = 2

This is standard deviations away from the mean.

This means this interval includes about 95% of women.