Final answer:
Reducing the tolerance by half would cause the minimum sample size to increase. This is due to the inverse relationship between sample size and error bound to maintain a specified level of confidence.
Step-by-step explanation:
If the tolerance (or allowable error-bound) is reduced by half, then the minimum sample size needed for the population mean estimate would increase. This is because the size of the sample and the error bound are inversely related to each other when the level of confidence is kept the same. To maintain that level of confidence with a smaller error bound, the sample size must be increased to reduce the variability of the sample means, thereby tightening the confidence interval around the population mean.
To determine the exact factor by which the sample size increases, we would need to know the specific criteria for the confidence interval being used, but it generally follows that the sample size increases by a factor related to the square of the decrease in tolerance. For example, if the error bound is halved, the sample size would increase by a factor of four to maintain the same level of confidence.
When considering the standard deviation of the sampling distribution of the means (the standard error), it decreases as the sample size increases. This smaller variation among the sample means estimates the population mean more precise.