Question

You are curious if there is a difference between men and woman for the approval percentage of a politician. Out of 400 men 307 approve and out of 500 women 340 approve.

(A) Find a 95% confidence interval for diffrence in proportion of men and women of the political candidate

(B) Does Your interval overlap 0? What does this signify?

(C) At the 5% significance level, test the claim that the proportion of men vs woman who approve of the politician is the same. Does this indicate the support is same among men and women ?

Answer 1

• Here X is the random variable denoting the number of approval for men regarding politician and let Y be random variable denoting the the number of approval for women.
• Here we have to find the 95% confidence interval for the difference in the proportion
• Let =X/n1=307/400=0.7675 and =Y/n2=340/500=0.68
• The comparison of population proportions corresponds to comparing two
  binomial probabilities on the basis of single observations X and Y from  binomial (n1,p1) and binomial
  (n2,p2), respectively.
• Considering only the case where n1 and n2 are large, so that the normal
  approximation can be used, the pivotal quantity used in practice is:
• 
• where and are the MLEs of X/n1 and Y/n2 respectively
• The 95% confidence interval is given by
• 0.95=P(-1.96<N(0,1)<1.96)
• 0.95=P(-1.96<<1.96)
• substituting the estimated proportions the confidence interval obtained is given by.
• (0.029<(p1-p2)<0.1456)
• our interval does not contains the 0, it means there exist a difference in the approval rate of men and women regarding politician.
• to test the hypothesis claim that the proportion of men vs woman who approve of the politician is the same.
• let the null hypothesis be that
• H0:p1=p2 vs H1:p1p2
• The test statistics is given by
• 
• where
• there by substituting the appropriate estimated values the test statistics is given by
• 2.09.
• The value of test statistics obatined is way greater than 5% critical value of 1.649.
• Therefore we have enough evidence to reject the null hypothesis.
• Therefore we conclude there exist a difference in the approval rate of men and women regarding politician