91Ó°ÊÓ

Critical Values. Use software or Table C to find a. the critical value for a one-sided test with level \(\alpha=0.01\) based on the \(t_{1}\) distribution. b. the critical value for a \(90 \%\) confidence interval based on the \(t_{30}\) distribution. How does this compare with the critical value \(z^{\star}\) for a \(90 \%\) confidence interval based on the standard Normal distribution?

We prefer the \(t\) procedures to the z procedures for inference about a population mean because a. \(z\) requires that you know the observations are from a Normal population, while \(t\) does not. b. \(z\) requires that you know the population standard deviation \(\sigma\), while \(t\) does not. c. \(z\) requires that you can regard your data as an SRS from the population, while \(t\) does not.

Kicking a Helium-Filled Foot ball. Does a football filled with helium travel farther than one filled with ordinary air? To test this, the Columbus Dispatch conducted a study. Two identical footballs, one filled with helium and one filled with ordinary air, were used. A casual observer was unable to detect a difference in the two footballs. A novice kicker was used to punt the footballs. A trial consisted of kicking both footballs in a random order. The kicker did not know which football (the helium-filled or the air-filled football) he was kicking. The distance of each punt was recorded. Then another trial was conducted. A total of 39 trials were run. Here are the data for the 39 trials, in yards that the footballs traveled. The difference (helium minus air) is the response variable. a. Examine the data. Is it reasonable to use the \(t\) procedures? b. If your conclusion in part (a) is Yes, do the data give convincing evidence that the heliumfilled football travels farther than the air-filled football?