Lorraine was in a hurry when she computed a confidence interval for μ. Because σ was not known, she used a Student's t distribution. However, she accidentally used degrees of freedom n instead of n - 1. Will her confidence interval be longer or shorter than one found using the correct degrees of freedom n - 1? Explain.

Shorter. As the degrees of freedom increase, the value for tc decreases.

When computing a confidence interval for μ using a Student's t distribution, the degrees of freedom play a critical role. The degrees of freedom reflect the sample size used in the calculation and impact the critical value of the t-distribution.

In Lorraine's case, she accidentally used the degrees of freedom n instead of n - 1. This means that she used a larger value for the degrees of freedom, resulting in a smaller critical value for the t-distribution. A smaller critical value corresponds to a wider confidence interval.

Therefore, Lorraine's confidence interval will be wider (longer) than one found using the correct degrees of freedom (n - 1). This wider interval indicates less precision in estimating the true population mean μ. Using the correct degrees of freedom allows for a more accurate estimation and a narrower confidence interval.

Lorraine's confidence interval will be shorter than one found using the correct degrees of freedom (n - 1). Let me explain why.

When calculating a confidence interval for the population mean (μ) using the Student's t distribution, we use the formula:

CI = x̄ ± t * (s / √n),

where x̄ is the sample mean, t is the critical value from the t-distribution based on the desired confidence level and degrees of freedom, s is the sample standard deviation, and n is the sample size.

In Lorraine's case, she used n degrees of freedom instead of n - 1. The reason we typically use n - 1 degrees of freedom is because it gives us a slightly larger standard error, which results in a wider confidence interval. This correction accounts for the fact that we are estimating the population standard deviation (σ) using the sample standard deviation (s).

By mistakenly using n degrees of freedom, Lorraine effectively used a smaller divisor in the denominator of the formula, resulting in a larger estimated standard error compared to when using the correct degrees of freedom. This larger estimated standard error will lead to smaller margin of error and, consequently, a shorter confidence interval.

To get the correct confidence interval, Lorraine should have used n - 1 degrees of freedom when calculating the critical value from the t-distribution. Using the correct degrees of freedom ensures that the confidence interval adequately accounts for the uncertainty introduced by estimating the population standard deviation based on the sample.