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 2.8%. What is the minimum sample size needed?

Formula to find sample size:

n = [(z-value)^2 * p * q]/E^2
... where n = sample size, z-value is found using a z-table for whatever confidence level is used, when no value is stated in the problem p = .50, q = 1 - p, ^2 means squared, * means to multiply, and E = .028.

Plug values into the formula and calculate n.

I hope this will help.

To calculate the minimum sample size needed to estimate the proportion of adults living in rural areas who own a gun with a margin of error of 2.8%, we need to use the formula for sample size determination in survey research:

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

Where:
n = sample size needed
Z = Z-score (corresponding to the desired level of confidence)
p = estimated proportion of the population with the characteristic of interest (gun ownership)
E = desired margin of error (2.8% = 0.028)

Since we do not have an estimated proportion of gun ownership in rural areas, we can use a conservative estimate of 0.5 (50%).

Using a standard Z-score for a 95% confidence level (corresponding to a 2-sided confidence interval), which is approximately 1.96, the formula becomes:

n = (1.96^2 * 0.5 * (1-0.5)) / 0.028^2

Calculating this expression:

n = (3.8416 * 0.25) / 0.000784

n ≈ 9.604 / 0.000784

n ≈ 12268.4

Therefore, the minimum sample size needed to estimate the proportion of adults living in rural areas who own a gun with a margin of error of 2.8% is approximately 12,268.

To determine the minimum sample size needed to estimate the proportion of adults living in rural areas who own a gun with a given margin of error, we can use the formula for sample size calculation:

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

Where:
n is the sample size
Z is the Z-score corresponding to the desired level of confidence (e.g., for a 95% confidence level, Z would be approximately 1.96)
p is the estimated proportion of adults living in rural areas who own a gun (we'll assume a conservative estimate of p = 0.5 since we don't have an exact value)
E is the desired margin of error (2.8% or 0.028 in decimal form)

Plugging in the values, we have:

n = (1.96^2 * 0.5 * (1 - 0.5)) / (0.028^2)

Calculating this equation will give us the minimum sample size needed.