Question

​It's believed that as many as 22​% of adults over 50 never graduated from high school. We wish to see if this percentage is the same among the 25 to 30 age group. What sample size would allow us to increase our confidence level to​ 95% while reducing the margin of error to only 5​%?

Answer 1

$\text{We can calculate the sample size n}\\ \text{ using the next formula}\\ n=(\frac{Z_\frac{\alpha}{2}}{ME})^2\times\hat p(1-\hat p)\\ \text{where, ME represents the margin of error},\\ Z_\frac{\alpha}{2}=Z_\frac{0.05}{2}=Z_{0.025}=1.96\\ \hat p=0.22, ME=0.05,\\ \therefore n=(\frac{1.96}{0.05})^2\times0.22(1-0.22)\approx 264\\$