Answer:
The sample size needed to guarantee a margin of error of at most 0.08 in a 95 percent confidence interval for an unknown population proportion p can be found using the formula:
n = (p(1-p)) / E^2
Where n is the sample size, p is the sample proportion, E is the margin of error, and Z is the Z-score for the desired level of confidence. Since we are using a 95 percent confidence level, the Z-score is approximately 1.96.
To find the smallest sample size that will guarantee a margin of error of at most 0.08, we need to plug in the values for E and Z into the formula and solve for n.
n = (p(1-p)) / (0.08)^2
Since we don't know the population proportion p, we can use a rough estimate of 0.5 to get a rough estimate of the sample size. This is known as the conservative sample size, because it is based on the worst-case scenario (i.e. the proportion is exactly 0.5).
n = (0.5 * (1 - 0.5)) / (0.08)^2
n = (0.25) / (0.0064)
n = 3906.25
So, the smallest sample size that will guarantee a margin of error of at most 0.08 is 3906.25.
It should be rounded up to 3907.
Explanation: