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

Question

Quizplus · Accepted Answer

Bob's statement is incorrect. Using the z-value instead of the t-value to construct a confidence interval for a mean with unknown variance with a sample of 25 items can lead to a narrower interval than necessary, not wider. This is because the t-distribution has fatter tails than the normal distribution, allowing for greater variability and uncertainty in small samples. Therefore, when the t-value is used, the confidence interval is wider to account for this additional variability. Using the z-value instead can underestimate the standard error and result in a narrower interval, thus increasing the risk of a Type I error.

In Constructing a Confidence Interval for a Mean with Unknown