In constructing a confidence interval for a mean with unknown variance with a sample of 25 items,Beth used z instead of t."Well,at least my interval will be wider than necessary,so it was a conservative error," said she.Is Beth's statement correct?
A)Yes.
B)No.
C)It depends on μ.

Question

Quizplus · Accepted Answer

Beth's statement is incorrect. Using z instead of t leads to a narrower interval than necessary, not a wider one. This is because the z-distribution assumes knowledge of the population variance, while the t-distribution accounts for the uncertainty introduced by using the sample standard deviation in place of the population standard deviation. Since the sample standard deviation tends to underestimate the population standard deviation, the t-distribution takes this into account by producing wider intervals to compensate for the increased uncertainty. By using z instead of t, Beth missed out on this adjustment for uncertainty and would have underestimated the true interval.

In Constructing a Confidence Interval for a Mean with Unknown