Q.55 - The Practice of Statistics for AP

Question

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 M in this population to within $\pm 1v$ with $99\%$ confidence? Show your work. 

Step-by-Step Solution

Answer

sample size$n=373$

Step 1: Given information

Given in the question that, The body mass index (BMI) of all American young women is believed to follow a Normal distribution with a standard deviation of about $7.5$. 

Step 2: Explanation

Given,

$E=1$

$\sigma =7.5$

sample size

For confidence level $1-\alpha =0.99$, determine $z_{\alpha /2}=z_{0.005}$ using table A (look up $0.005$ in the table, the $z$-score is then the found $z$-score with opposite sign):

$z_{\alpha /2}=2.575$

$$\begin{gathered} n={\left(\frac{z_{\alpha /2}\sigma }{E}\right)}^2 \\ ={\left(\frac{2.575\times 7.5}{1}\right)}^2 \\ \approx 373 \end{gathered}$$