Question

t2.7 if the heights of a population of men are approxi- mately normally distributed, and the middle 99.7% have heights between 50\ and 70\, what is the stan- dard deviation of the heights in this population? (a) 1\ (b) 3\ (c) 4\ (d) 6\ (e) 12\

Explanation:

Step1: Recall the empirical rule

For a normal - distribution, approximately 99.7% of the data lies within 3 standard deviations of the mean, i.e., \(\mu - 3\sigma\) and \(\mu+3\sigma\).

Step2: Calculate the range

The range of the middle 99.7% of the data is \(R=(7\times12)-(5\times12)=24\) inches.

Step3: Relate range to standard - deviation

The range \(R = (\mu + 3\sigma)-(\mu - 3\sigma)=6\sigma\).

Step4: Solve for standard - deviation

If \(R = 6\sigma\) and \(R = 24\) inches, then \(\sigma=\frac{R}{6}\). Substituting \(R = 24\) into the formula, we get \(\sigma=\frac{24}{6}=4\) inches.

Answer:

(c) \(4''\)