Question

A researcher wishes to estimate the proportion of adults living in rural areas who own a gun. He wishes to achieve a margin of error of 2.1 percentage points. What is the minimum sample size needed?

Answer 1

To estimate the minimum sample size needed, we use the formula n = (Z^2 * p * (1-p)) / E^2, where Z is the z-score corresponding to the desired confidence level, p is the estimated proportion, and E is the margin of error. Plugging in the values, the minimum sample size needed is 2228.

Step-by-step explanation:

To determine the minimum sample size needed, we use the formula:

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

Where:
- n is the minimum sample size
- Z is the z-score corresponding to the desired confidence level (in this case, 90%)
- p is the estimated proportion of adults living in rural areas who own a gun
- E is the margin of error (2.1 percentage points)

Since we don't have an estimate of the proportion (p), we can use 0.5 as a conservative estimate. Plugging in the values, we have:

Z = 1.645 (corresponding to a 90% confidence level)
p = 0.5
E = 0.021 (2.1 percentage points)

Substituting these values into the formula:

n = (1.645^2 * 0.5 * (1-0.5)) / 0.021^2

Simplifying the equation, we get:

n = 2227.022

Therefore, the minimum sample size needed is 2228.