Can I please get some help with these two questions?

Question

You are conducting a survey and you want to keep your margin of error to 4%. How many participants should you have in your sample?

 

You conduct a survey and find that 422 students out of 813 would vote for Hillary Clinton, if they could vote. What is the margin of error for your survey?

Answer 1

For a margin of error of a 95% confidence interval, you can use the following formula:

 

n = ( p-hat*(1 - p-hat)*(1.96)^2  ) / MOE^2

 

where p-hat is the estimate of proportion success. Since you don't know what this is before taking your survey, a conservative estimate for the number in the sample is to use p-hat = 0.5. This proportion will produce the largest number of participants for your desired MOE.

 

So, 

 

n = ( (0.5)*(0.5)*(1.96)^2 ) / (0.04)^2 = 600.25 => 601 participants

 

 

For the MOE for your voting survey, the proportion voting for Hillary is 422/813 = 0.52. So p-hat = 0.52.

 

p-hat +/- (1.96)*sqrt( ( (.52)*(.48) ) / 813 ) is your 95% confidence interval for your MOE

 

0.52 +/- 0.034, so MOE = +/- 0.034

 

(0.486 , 0.554) is the confidence interval for your proportion estimate. This tells you that if you were to do many repeated surveys like this one, the true population proportion would fall in this range 95% of the time.