question 10 0.1 pts\nuse the given data to find the minimum sample size required to estimate the population…

Explanation:

Step1: Find the z - score

For a 99% confidence level, the significance level $\alpha=1 - 0.99 = 0.01$. Then $\alpha/2=0.005$. The z - score $z_{\alpha/2}=z_{0.005}$. Looking up in the standard normal table, $z_{0.005} = 2.576$.

Step2: Use the formula for sample size

The formula for the sample size $n$ when estimating a population proportion is $n=\frac{z_{\alpha/2}^{2}\hat{p}(1 - \hat{p})}{E^{2}}$, where $\hat{p}$ is the estimated proportion from a prior study and $E$ is the margin of error. Given $\hat{p}=0.12$, $1-\hat{p}=1 - 0.12=0.88$, $E = 0.04$ and $z_{\alpha/2}=2.576$. Substitute the values into the formula: [ \begin{align*} n&=\frac{(2.576)^{2}\times0.12\times0.88}{(0.04)^{2}}\ &=\frac{6.635776\times0.1056}{0.0016}\ &=\frac{0.700738}{0.0016}\ & = 437.96125 \end{align*} ] Since the sample size $n$ must be an integer, we round up to the next whole number.

Answer:

438