A sociologist is interested in the proportion of people that would attend a fund raiser if the had to pay to enter the event. How large a sample must be taken to be 98% confident that the estimate is in error by at most 3% if:
a. The sample size is based on a pilot study that suggests 72% of the people involved would pay.
b. No reliable data exists.
a. To determine the sample size required based on a pilot study suggesting 72% of the people would pay, we can use the following formula:
n = [Z^2 * p * (1-p)] / E^2
Where:
n = required sample size
Z = desired level of confidence (in this case, 98% confidence corresponds to a Z-value of 2.33)
p = proportion estimated from the pilot study (0.72)
E = maximum desired margin of error (0.03)
Plugging in the values, we get:
n = [2.33^2 * 0.72 * (1-0.72)] / 0.03^2
n ≈ 815.8
Therefore, a sample size of approximately 816 should be taken to be 98% confident that the estimate is in error by at most 3% based on the pilot study.
b. In the absence of reliable data, a common approach is to use a conservative estimate of p as 0.5 (maximum variability). Using the same formula as above, we have:
n = [2.33^2 * 0.5 * (1-0.5)] / 0.03^2
n ≈ 1067.4
Therefore, a sample size of approximately 1068 should be taken to be 98% confident that the estimate is in error by at most 3% when no reliable data exists.
To determine the sample size needed for the sociologist's study, we can use the formula for sample size calculation for estimating a proportion:
n = (Z^2 * p * (1-p)) / E^2
Where:
n = sample size
Z = Z-value corresponding to the desired level of confidence
p = estimated proportion from the pilot study or assumed proportion if no reliable data exists
E = maximum allowable error (margin of error)
a. When a sample size is based on a pilot study:
In this case, the pilot study suggests that 72% of people would pay. We can use this estimate as the assumed proportion (p).
To calculate the sample size, we need the Z-value for the desired level of confidence. Let's assume a 98% confidence level, which corresponds to a Z-value of 2.33.
Given values:
p = 0.72 (estimated proportion)
E = 0.03 (maximum allowable error)
Z = 2.33 (Z-value for 98% confidence level)
Substituting these values into the formula:
n = (2.33^2 * 0.72 * (1-0.72)) / 0.03^2
n = (5.4289 * 0.72 * 0.28) / 0.0009
n = 1.0918 / 0.0009
n ≈ 1213.11
Therefore, a sample size of approximately 1214 (rounded up) would be needed to be 98% confident that the estimate is in error by at most 3% based on the pilot study results.
b. When no reliable data exists:
When reliable data does not exist, we usually assume a conservative estimate of 0.5 (50%) for the proportion (p), which maximizes the required sample size.
Using the same formula as above with the assumed proportion (p = 0.5):
n = (Z^2 * p * (1-p)) / E^2
n = (2.33^2 * 0.5 * (1-0.5)) / 0.03^2
n = (5.4289 * 0.5 * 0.5) / 0.0009
n = 1.3572 / 0.0009
n ≈ 1507.96
Therefore, a sample size of approximately 1508 (rounded up) would be needed to be 98% confident that the estimate is in error by at most 3% when no reliable data exists.