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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