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 have to be a whole number.

Short answer

The true statement is option (a) Tom’s confidence interval is wider.

Step-by-step solution

Concept introduction

The normal distribution, also known as the Gaussian distribution, is a symmetric probability distribution centered on the mean, indicating that data around the mean occur more frequently than data far from it.

Explanation

Janelle, on the other hand, employs more degrees of freedom than Tom.

A lower $t*$-value is associated with a higher degree of freedom.

A lower $t*$-value means a smaller margin of error and, as a result, a smaller confidence interval.

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

Thus the answer (a) is correct.