Question

7
the manager of a gym selected a sample of 139 members at random to estimate the percentage of the gyms members that would continue to pay for a membership if the price increased. from the survey, the manager estimates that 82% of the gyms members would continue to pay for a membership if the price increased, with an associated margin of error of 6.39%. if the survey is repeated with a random sample of 278 members and the results are calculated in the same way, which of the following will be the most likely effect of using the larger random sample compared to the smaller random sample ?
a the margin of error will be lower.
b the margin of error will be higher.
c the estimate of the percentage of the gyms members that would continue to pay for a membership if the price increased will be lower.
d the estimate of the percentage of the gyms members that would continue to pay for a membership if the price increased will be higher.

Explanation:

Margin of error in surveys is inversely related to sample size—larger samples reduce sampling variability, leading to a smaller margin of error. The sample proportion estimate (82% here) is not systematically affected by sample size; it depends on the population, not sample size. Thus, the most likely effect is a lower margin of error.

Answer:

A. The margin of error will be lower.