Find the minimum sample size when we want to construct a 90% 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 her. The desired margin of error is 0.08.
A) n = 44
B) n = 72
C) n = 73
D) n = 103

Question

Quizplus · Accepted Answer

We can use the formula for the margin of error, E, of a confidence interval for a population proportion:

E = z*sqrt(p̂(1-p̂)/n)

where p̂ is the sample proportion, n is the sample size, and z is the z-score corresponding to the desired confidence level.

Since we want a 90% confidence interval, we use a z-score of 1.645 (from a standard normal distribution table). We also know that p̂ = 22/100 = 0.22 (the proportion of support for candidate A in the sample).

Plugging in these values and solving for n, we get:

0.08 = 1.645*sqrt(0.22(1-0.22)/n)

Squaring both sides and multiplying by n gives:

n = (1.645^2)*0.22(1-0.22)/(0.08^2)

n ≈ 72.8708

Rounding up to the next integer, we get a minimum sample size of 73, which corresponds to choice C.

Find the Minimum Sample Size When We Want to Construct