Question

A quiz question gives random samples of $n=10$observations from each of two Normally distributed populations. Tom uses a table of t distribution critical values and $9$degrees of freedom to calculate a $95\%$confidence interval for the difference in the two population means. Janelle uses her calculator's two-sample t Interval with $16.87$degrees of freedom to compute the $95\%$confidence interval. Assume that both students calculate the intervals correctly. Which of the following is true?

(a) Tom's confidence interval is wider.

(b) Janelle's confidence Interval is wider.

(c) Both confidence Intervals are the same.

(d) There is insufficient information to determine which confidence interval is wider.

(e) Janelle made a mistake, degrees of freedom has to be a whole number.

Short answer

The correct answer is:

(a) Tom's confidence interval is wider.

Step-by-step solution

Given information

We are given that Janelle uses a higher degree of freedom than Tom.

Explanation

A higher degree of freedom results in a lower$t^{*}-value.$

A lower $t^{*}-value$results in a lower margin of error and thus also a narrower confidence interval.

Then we know that Janelle's confidence interval is narrower than Tom's confidence interval, or Tom's confidence interval is wider than Janelle's confidence level.

Thus answer (a) is correct.