Question

suppose the heights of the members of a population follow a normal distribution. if the mean height of the population is 65 inches and the standard deviation is 3 inches, 99.7% of the population will have a height within which range?

choice

a. 59 inches to 71 inches
b. 53 inches to 77 inches
c. 56 inches to 74 inches
d. 62 inches to 68 inches

Explanation:

Step1: Recall the empirical rule

For a normal distribution, the empirical rule states that approximately 99.7% of the data lies within $3$ standard deviations of the mean. The formula for the range is $\mu - 3\sigma$ to $\mu + 3\sigma$, where $\mu$ is the mean and $\sigma$ is the standard deviation.

Step2: Identify $\mu$ and $\sigma$

Given $\mu = 65$ inches and $\sigma = 3$ inches.

Step3: Calculate the lower bound

Lower bound $=\mu - 3\sigma = 65 - 3\times3 = 65 - 9 = 56$ inches.

Step4: Calculate the upper bound

Upper bound $=\mu + 3\sigma = 65 + 3\times3 = 65 + 9 = 74$ inches.

Answer:

C. 56 inches to 74 inches