Question

The distribution of heights of adult men in the U.S. is approximately normal with mean 69 inches and standard deviation 2.5 inches. Use what you know about a normal distribution and the 68-95-99.7 rule to answer the following.

NOTE: If your answer is a percent, such as 25 percent, enter: "25 PERCENT" (without the quotes). If your answer is in inches, such as 10 inches, enter: "10 INCHES" (without the quotes and with a space between the number and the INCHES). If your answer is an interval, such as 14 to 15 inches, then enter: "14 TO 15 INCHES" (without the quotes). Do not use extra zeros and do not include a decimal point unless your answer is not a whole number. Your answer must be entered in the correct format.

(a) About what percent of men are between 64 and 66.5 inches?
Answer:
(b) About what percent of men are shorter than 66.5 inches?
Answer:
(c) Between what approximate heights do the middle 95 percent of men fall?
Answer:

Answer 1

(a) To find the percent of men between 64 and 66.5 inches, you can use the 68-95-99.7 rule for a normal distribution. According to this rule:

Approximately 68% of the data falls within one standard deviation of the mean.

Approximately 95% falls within two standard deviations.

Approximately 99.7% falls within three standard deviations.

First, calculate the standard deviation (σ):

Standard Deviation (σ) = 2.5 inches

Now, find the distance between the mean (69 inches) and each of the two heights:

For 64 inches:

Distance from the mean = 69 - 64 = 5 inches

For 66.5 inches:

Distance from the mean = 69 - 66.5 = 2.5 inches

Now, calculate how many standard deviations each of these distances represents:

For 64 inches:

Number of standard deviations = Distance / σ = 5 / 2.5 = 2

For 66.5 inches:

Number of standard deviations = Distance / σ = 2.5 / 2.5 = 1

Since approximately 68% of the data falls within one standard deviation of the mean, you can say that about 68% of men are between 64 and 66.5 inches.

Answer: 68 PERCENT

(b) To find the percent of men shorter than 66.5 inches, you can use the same information. Since 66.5 inches is one standard deviation below the mean, you know that approximately 68% of men are taller than 66.5 inches. Therefore, the percent of men shorter than 66.5 inches is the complement of this:

100% - 68% = 32%

Answer: 32 PERCENT

(c) To find the approximate heights between which the middle 95 percent of men fall, you can use the 68-95-99.7 rule again.

Since 95% of the data falls within two standard deviations of the mean, you need to find the heights that are two standard deviations above and below the mean:

For the lower bound:

Mean - 2σ = 69 - (2 * 2.5) = 69 - 5 = 64 inches

For the upper bound:

Mean + 2σ = 69 + (2 * 2.5) = 69 + 5 = 74 inches

So, the middle 95% of men fall between approximately 64 inches and 74 inches.

Answer: 64 TO 74 INCHES

Answer 2

Answer:13.5 PERCENT

16 PERCENT

64 TO 74 INCHES