Final answer:
The true statement is that the 99% confidence interval is wider than the 95% confidence interval, regardless of sample size, due to a higher confidence level requiring a greater range to ensure the interval likely contains the true population mean.
Step-by-step explanation:
The true statement regarding the comparison between a 95% confidence interval and a 99% confidence interval for a population mean is: The 99% confidence interval will be wider than the 95% confidence interval regardless of the size of Joe's sample.
This occurs because a higher confidence level implies a greater certainty that the interval contains the true population mean, which is achieved by increasing the range of the interval. As the confidence level rises from 95% to 99%, the z* value (or t* value for small samples) used to calculate the margin of error increases. Consequently, the margin of error increases, resulting in a wider confidence interval.
For instance, if the sample mean is x and the error bound for a population mean (EBM) is the margin of error, the confidence interval estimate will have the form: (x − EBM, x + EBM). Thus, an increase in the confidence level leads to an increase in the margin of error and therefore a wider interval.