Question

A political researcher takes a survey of 300 randomly selected registered voters in atlanta, and each person was asked who they plan on voting for in the 2020 mayoral election. 101 said they plan on voting for candidate a, 184 said they plan on voting for candidate b, and 15 were unsure or plan to vote for another candidate. In question 8, the political researcher estimated p, the population proportion of all registered voters in atlanta who plan to vote for candidate a, with a 95% confidence interval. The researcher now wants to estimate p with a margin of error of 3%. What is the minimum sample size needed?.

Answer 1

To estimate the proportion of voters for candidate A with a margin of error of 3% at a 95% confidence level, the minimum sample size needed is approximately 1067 registered voters.

Step-by-step explanation:

To determine the minimum sample size needed for estimating the population proportion p of voters who plan to vote for candidate A with a margin of error of 3%, one can use the sample size formula for a proportion:

\(n = (Z^2 * p * (1 - p)) / E^2\)

Where:

n is the sample size,

Z is the z-score

p is the estimated population proportion (from previous surveys or pilot studies),

E is the expected margin of error.

However, when p is unknown, a conservative estimate where p = 0.5 is used as it provides the maximum variance, thus the largest sample size. The formula becomes:

\(n = (1.96^2 * 0.5 * 0.5) / 0.03^2\)

Calculating this gives us:

\(n = (3.8416 * 0.25) / 0.0009\)

\(n = (0.9604) / 0.0009\)

\(n ≈ 1067\)

Therefore, the minimum sample size needed with a margin of error of 3% at a 95% confidence level is approximately 1067 registered voters.