Question: Heights for women follow a normal distribution with mean of 65 inches and standard deviation 3…

Heights for women follow a normal distribution with mean of 65 inches and standard deviation 3 inches. Answer the following using the 68-95-99.7 properties of the normal distribution.

17. What percent of women are between 59 and 71 inches tall?

18. What percent of women are over 71 inches tall?

19. What percent of women are between 62 and 65 inches tall?

20. What percent of women are less than 59 inches tall?

21. What percent of women are over 74 inches tall?

Solution

The problem involves applying the properties of the normal distribution with a mean of 65 inches and a standard deviation of 3 inches. Let’s solve each question using the 68-95-99.7 rule. 17) What percent of women are between 59 and 71 inches tall? 59 inches is two standard deviations below the mean (65 - 23 = 59). 71 inches is two standard deviations above the mean (65 + 23 = 71). According to the 68-95-99.7 rule, approximately 95% of data falls within two standard deviations of the mean. Therefore, about 95% of women are between 59 and 71 inches tall. 18) What percent of women are over 71 inches tall? 71 inches is two standard deviations above the mean. The area beyond two standard deviations in a normal distribution is approximately 2.5% (since 95% is within, leaving 5% outside, split equally between both tails). Therefore, about 2.5% of women are over 71 inches tall. 19) What percent of women are between 62 and 65 inches tall? 62 inches is one standard deviation below the mean (65 - 3 = 62). 65 inches is the mean. The area between the mean and one standard deviation below is approximately 34% (since 68% are within one standard deviation on either side, and we consider half of it). Therefore, about 34% of women are between 62 and 65 inches tall. 20) What percent of women are less than 59 inches tall? 59 inches is two standard deviations below the mean. The area below two standard deviations is approximately 2.5% (as explained earlier). Therefore, about 2.5% of women are less than 59 inches tall. 21) What percent of women are over 74 inches tall? 74 inches is three standard deviations above the mean (65 + 3*3 = 74). The area beyond three standard deviations is approximately 0.15% (since 99.7% is within, leaving 0.3% outside, split equally between both tails). Therefore, about 0.15% of women are over 74 inches tall.