Find the minimum sample size when we want to construct a 95% confidence interval on the population proportion for the support of candidate A in the following mayoral election. Candidate A is facing two opposing candidates. In a preselected poll of 100 residents, 22 supported candidate B and 14 supported candidate C. The desired margin of error is 0.06.
A) n = 173
B) n = 174
C) n = 245
D) n = 246

Question

Quizplus · Accepted Answer

We need to find the minimum sample size, which means we want to be as efficient as possible without sacrificing the desired level of confidence.

First, we need to find the proportion of people who support candidate A in the preselected poll:

Proportion of support for A = 1 - Proportion of support for B or C 
= 1 - (22/100 + 14/100) 
= 1 - 0.36 
= 0.64

Next, we need to find the critical value for a 95% confidence interval. Using a standard normal distribution table or calculator, we find the critical value to be 1.96.

Then, we use the formula for the margin of error:

Margin of Error = Z* * sqrt(pq/n)

where Z* is the critical value, p is the proportion of support for candidate A, q is 1-p, and n is the sample size.

Re-arranging the formula gives us:

n = (Z*/ ME)^2 * pq

Plugging in the values, we get:

n = (1.96/0.06)^2 * 0.64 * (1-0.64) 
n = 246.13

Rounding up to the nearest whole number gives us a minimum sample size of 246, which is answer choice D.

Find the Minimum Sample Size When We Want to Construct