Question

1. Polling is a common part of elections. Assume that an opinion poll of 400 probable voters finds that Candidate A has an advantage of 52% to 48%.
   1. Develop a 95% confidence interval for the proportion of the population favoring Candidate A.
   2. Do you think it is reasonable to claim that Candidate A is guaranteed to win the election? Why or why not?
   3. Repeat the above analysis based on a sample of 1000 voters. How did your interval/conclusion change (if it did)?

Answer 1

(a)

n = 400    

p = 0.52    

% = 95    

Standard Error, SE = √{p(1 - p)/n} =    √(0.52(1 - 0.52))/400 = 0.024979992

z- score = 1.959963985    

Width of the confidence interval = z * SE =     1.95996398454005 * 0.0249799919935936 = 0.04895988

Lower Limit of the confidence interval = P - width =     0.52 - 0.0489598846415424 = 0.47104012

Upper Limit of the confidence interval = P + width =     0.52 + 0.0489598846415424 = 0.56895988

The confidence interval is [0.471, 0.569], that is [47.1%, 56.9%]

(b) Since a part of the above confidence interval lies below 50%, we can't say that candidate A is sure to win the election.

(c)

n = 1000    

p = 0.52    

% = 95    

Standard Error, SE = √{p(1 - p)/n} =    √(0.52(1 - 0.52))/1000 = 0.015798734

z- score = 1.959963985    

Width of the confidence interval = z * SE =     1.95996398454005 * 0.0157987341265052 = 0.03096495

Lower Limit of the confidence interval = P - width =     0.52 - 0.0309649498892741 = 0.48903505

Upper Limit of the confidence interval = P + width =     0.52 + 0.0309649498892741 = 0.55096495

The confidence interval is [0.489, 0.551], that is [48.9%, 55.1%]

Since a part of the above confidence interval lies below 50%, we still can't say that candidate A is sure to win the election.