After the polls close on election day, networks compete to be the first to predi
ID: 3312222 • Letter: A
Question
After the polls close on election day, networks compete to be the first to predict which candidate will win. The predictions are based on counts in certain precincts and on exit polls. Exit polls are conducted by asking random samples of voters who have just exited from the polling booth (hence the name) for which candidate they voted. In American presidential elections, the candidate who receives the most votes in a state receives the state’s entire Electoral College vote. In practice, this means that either the Democrat or the Republican candidate will win. Suppose that the results of an exit poll in one state were recorded that out of 765 voters asked, 407 voted the Republican candidate.
(a) The poll closed at 8:00 P.M.. Can the networks conclude from their exit polls that the Republican candidate will win the state? Conduct a hypothesis test.
(b) Give a 95% confidence interval for the proportion of the votes the Republican candidate received. Give a full sentence interpretation of your confidence interval in the context of the problem.
(c) Would a 99% confidence interval be wider or narrower than the one you found in part (b)? Verify your results by computing the interval.
(d) Would a 90% confidence interval be wider or narrower than the one you found in part (b)? Verify your results by computing the interval.
I really just need help with c and d, answers to a and b can be found in example 12.5 in the link below. Thanks in advance!
http://guo.ba.ntu.edu.tw/%E6%95%99%E5%AD%B8%E8%AA%B2%E7%A8%8B/%E5%A4%A7%E5%AD%B8%E9%83%A8/%E7%B5%B1%E8%A8%88%E5%AD%B8/%E8%AC%9B%E7%BE%A9%E5%92%8C%E4%BD%9C%E6%A5%AD/Present_pdf/Chapter12.pdf
Explanation / Answer
a)
H0: p <= 0.5
H1: p > 0.5
pcap = 407/765 = 0.5320
test statistics, z = (0.5320 - 0.5)/sqrt(0.5*0.5/765) = 1.7701
p-value = 0.0384
Reject null hypothesis.
There is significant evidence to conclude that republican candidate will win.
b)
The above CI indicates, if a sample is selected from the population there are 95% chances that the proportion of population voting for republican candidate is in this CI.
c)
This CI is wider than the CI in part B.
d)
This CI is narrower than the CI in part b
n 765 p 0.532026144 z-value of 95% CI 1.9600 SE = sqrt(p*(1-p)/n) 0.01804 ME = z*SE 0.03536 Lower Limit = p - ME 0.49667 Upper Limit = p + ME 0.56738 95% CI (0.4967 , 0.5674 )