if no preliminary sample is taken to estimate p, how large a sample is necessary to be 87% sure that the point estimate will be within a distance of 0.04 from p?
To determine the sample size needed to be 87% confident that the point estimate will be within a distance of 0.04 from the population parameter, we need to use the formula for sample size calculation:
n = (Z^2 * p * (1-p)) / E^2
Where:
n = sample size
Z = Z-score (corresponding to the desired confidence level)
p = estimated probability (since no preliminary sample is taken, we can use 0.5 as a conservative estimate)
E = margin of error/ distance from the point estimate to the population parameter
To find the Z-score corresponding to an 87% confidence level, we need to look up the value from a standard normal distribution table or use statistical software. The Z-score for an 87% confidence level is approximately 1.16.
Plugging in the values into the formula, we get:
n = (1.16^2 * 0.5 * (1 - 0.5)) / 0.04^2
n = (1.3456 * 0.25) / 0.0016
n = 0.3364 / 0.0016
n ≈ 210.25
Therefore, you would need a sample size of at least 211 to be 87% confident that the point estimate will be within a distance of 0.04 from the population parameter.