Question

Estimating BMI The body mass index (BMI) of all American young

women is believed to follow a Normal distribution with a standard deviation of about $7.5$. How large a sample would be needed to estimate the mean BMI μ in this population to within $\pm 1$ with $99\%$ confidence?

Short answer

Sample size is$373$.

Step-by-step solution

Given Information

It is given that $\left(\sigma \right)=7.5$

Confidence level $=99\%$

$E=1$

Concept Used

Formula to be used is
$n={\left(\frac{z_{\frac{a}{2}}\times \sigma }{E}\right)}^2$

Corresponding to$99\%$confidence interval,$Z_{\frac{a}{2}}=2.575$

Calculation

Hence, $n={\left(\frac{z_{\frac{a}{2}\times \sigma }^m}{m}\right)}^2$

$={\left(\frac{2.575\times 7.5}{1}\right)}^2$

$372.97$

$\approx 373$

Sample size is$373$