Question

The blood platelet counts of a group of women have a bell-shaped distribution with a mean of 262.3 and a standard deviation of 66.6 (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 62.5 and 462.1? b. What is the approximate percentage of women with platelet counts between 195.7 and 328.9?

Answer 1

Approximately 99% of women have platelet counts within 3 standard deviations of the mean. Approximately 68% of women have platelet counts between 195.7 and 328.9.

Step-by-step explanation:

A. According to the empirical rule, approximately 99% of the data is within 3 standard deviations of the mean. Given a mean of 262.3 and a standard deviation of 66.6, the lower limit is the mean minus 3 times the standard deviation: 262.3 - (3 * 66.6) = 62.5. The upper limit is the mean plus 3 times the standard deviation: 262.3 + (3 * 66.6) = 462.1. Therefore, the approximate percentage of women with platelet counts within 3 standard deviations of the mean is 99%. B. To find the approximate percentage of women with platelet counts between 195.7 and 328.9, you need to calculate the z-scores for these values. The z-score formula is z = (x - mean) / standard deviation. For 195.7: z = (195.7 - 262.3) / 66.6 = -1.0015. For 328.9: z = (328.9 - 262.3) / 66.6 = 1.0015. Using a standard normal distribution table or a calculator, you can find that the approximate percentage between these z-scores is 68%. Therefore, the approximate percentage of women with platelet counts between 195.7 and 328.9 is also 68%.