Question

1. on the normal curve above what percentage of data values lie within two standard deviations of the mean (from -2σ to +2σ)?
2. here are some fun questions. make an educated guess 😊:

a. 95% of us women fall between what two heights?
b. 95% of us men fall between what two heights?

Explanation:

Step1: Recall the empirical rule for normal distribution

The empirical rule states that for a normal distribution, approximately 95% of the data lies within \( \mu\pm2\sigma\) (where \(\mu\) is the mean and \(\sigma\) is the standard deviation).

Step2: Answer the first question

From the normal - curve diagram and the empirical rule, the percentage of data values within two standard deviations of the mean (\(- 2\sigma\) to \(+2\sigma\)) is 95%.

Step3: Make educated guesses for heights (assuming mean and standard deviation values)

Assume that the mean height of US women \(\mu_w = 64\) inches and the standard deviation \(\sigma_w=2.5\) inches. Then \(95\%\) of US women's heights lie between \(\mu_w - 2\sigma_w=64 - 2\times2.5=64 - 5 = 59\) inches and \(\mu_w+2\sigma_w=64 + 2\times2.5=64 + 5 = 69\) inches.
Assume that the mean height of US men \(\mu_m = 69\) inches and the standard deviation \(\sigma_m = 2.8\) inches. Then \(95\%\) of US men's heights lie between \(\mu_m-2\sigma_m=69-2\times2.8=69 - 5.6 = 63.4\) inches and \(\mu_m + 2\sigma_m=69+2\times2.8=69 + 5.6=74.6\) inches.

Answer:

1. 95%

3.
a. If \(\mu_w = 64\) inches and \(\sigma_w = 2.5\) inches, 95% of US women fall between 59 inches and 69 inches.
b. If \(\mu_m=69\) inches and \(\sigma_m = 2.8\) inches, 95% of US men fall between 63.4 inches and 74.6 inches.