## 15.11

Body mass index of young women. Example 14.1 (page 360) assumed that the body mass index (BMI) of all American young women follows a Normal distribution with standard deviation $\sigma=7.5.$ How large a sample would be needed to estimate the mean BMI $\mu$ in this population to within $\pm 1$ with 95% confidence?

## Solution

This is direct from the formula

$n = \bigg( \displaystyle{\frac{z^\ast \sigma}{m}} \bigg)^2,$

where $n$ is the number of samples, $\sigma$ is the standard deviation of the Normal distribution, $m$ is the margin of error, and $z^\ast$ is the critical value for the corresponding confidence.

In our case, the problem gives us $\sigma=7.5$, $m=1$, and we compute the critical value from table C: $z^\ast = 1.960$ for a confidence of 95%.  Application of the formula gives then $n = 216.09.$ Remember to round always up; therefore the answer must be: we need a sample of at least 217 young women to estimate a mean BMI within $\pm 1$ with 95% confidence.

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