t'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. a) How many of this younger age group must we survey in order to estimate the proportion of non-grads to within 4% with 90% confidence?

Given : 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.

So the most near proportion value of never graduated from high school among the 25 to 30 age group is p=0.22.

The formula to find the sample size is given by :-

[tex]n=p(1-p)\times(\dfrac{z_{\alpha/2}}{E})^2[/tex]

Given : Confidence level : 0.90

Significance level : [tex]\alpha: 1-0.90=0.1[/tex]

Critical value : [tex]z_{\alpha/2}=1.645[/tex]

Margin of error : [tex]E=0.04[/tex]

Then , the required sample size will be :-

[tex]n=(0.22)(1-0.22)\times(\dfrac{1.645}{0.04})^2\\\\\Rightarrow\ n=290.22118125\approx290[/tex]

Hence, the minimum required sample size = 290