Question

14.31 The online article "Death Metal in the Operating Room" (www.npr.org, December 24,2009 ) describes an experiment investigating the effect of playing music during surgery. One conclusion drawn from this experiment was that doctors listening to music that contained vocal elements took more time to complete surgery than doctors listening to music without vocal elements. Suppose that \(\mu_{1}\) denotes the mean time to complete a specific type of surgery for doctors listening to music with vocal elements and \(\mu_{2}\) denotes the mean time for doctors listening to music with no vocal elements. Further suppose that the stated conclusion was based on a \(95 \%\) confidence interval for \(\mu_{1}-\mu_{2},\) the difference in treatment means. Which of the following three statements is correct? Explain why you chose this statement. Statement 1: Both endpoints of the confidence interval were negative. Statement 2: The confidence interval included \(0 .\) Statement 3: Both endpoints of the confidence interval were positive.

Short answer

The correct statement is Statement 3: Both endpoints of the confidence interval were positive. This is because the conclusion was that doctors listening to music with vocal elements (\(\mu_{1}\)) took longer than doctors listening to music without such elements (\(\mu_{2}\)). Therefore, the difference \(\mu_{1}-\mu_{2}\) must be greater than zero, which is only possible if both endpoints of the confidence interval are positive.

Step-by-step solution

Analyzing Statements

First, we need to review each of the statements and consider what each statement would imply about the data. If a 95% confidence interval for the difference in means (\(\mu_{1}-\mu_{2}\)) included 0, then we would not be able to assert that there was a significant difference between the two groups. If both endpoints were positive, it would indicate that \(\mu_{1}\) is greater than \(\mu_{2}\). If both were negative, it would mean that \(\mu_{2}\) is greater than \(\mu_{1}\).

Selecting the Correct Statement

Given that the experimental conclusion was that doctors who listened to vocal music took longer (i.e., \(\mu_{1}\) is greater than \(\mu_{2}\)), a confidence interval that reflected this would need to have both endpoints positive. Thus, the correct statement would be Statement 3: Both endpoints of the confidence interval were positive.