Question

1) Past research indicates that 64% of U.S. voters oppose capital punishment. A pollster wishes to estimate the current proportion of U.S. voters who oppose capital punishment to see if the percentage has changed.

a)How many voters should be surveyed in order to be 95% confident that the true proportion is estimated to within 4%?b)Suppose they ignore the sample size found in part (a) and decide to select a random sample of 500 voters and find that 335 of them oppose capital punishment. Find and interpret a 95% confidence interval for the true percentage of U.S. voters who oppose capital punishment. Be sure to check the conditions and write them down!

c) Should the pollsters conclude that the percentage of current voters who oppose capital punishment has changed? Explain

Answer 1

We have given

P=0.64

a)How many voters should be surveyed in order to be 95% confident that the true proportion is estimated to within 4%?

We have to find n such that

p(|P-Pest|<0.04) =0.95

Using fact that

p(|P-Pest|/sqrt(P*(1-P)/n)<1.96) =0.95

0.04/sqrt(P*(1-P)/n)=1.96

n= 1.96^2*0.64*(1-0.64)/0.04^2

=553.19

= 554

554 voters should be surveyed in order to be 95% confident that the true proportion is estimated to within 4%

b)Suppose they ignore the sample size found in part (a) and decide to select a random sample of 500 voters and find that 335 of them oppose capital punishment.

95% confidence interval for the true percentage of U.S.

= ( p +/- Z0.025*sqrt(p*(1-p)/n))

p= 335/500 = 0.67

n= 500

Z0.025=1.96

Hence CI = (0.67 +/- 1.96*sqrt(0.67*0.33/500))

=(0.6288 , 0.7112)

=(62.88% , 71.12%)

95% confidence interval for the true percentage of U.S. =(62.88% , 71.12%)

There 95% chance that true percentage of U.S will fall in this interval.

c) Should the pollsters conclude that the percentage of current voters who oppose capital punishment has changed? Explain

No, he percentage of current voters who oppose capital punishment has noy changed because this interval contains past percentage 64%