Question

Wayne Gretzky was one of ice hockey's most prolific scorers when he played for the Edmonton Oilers. During his last season with the Oilers, Gretzky played in 41 games and missed 17 games due to injury. The article "The Great Gretzky" (Chance [1991]: 16-21) looked at the number of goals scored by the Oilers in games with and without Gretzky, as shown in the accompanying table. If you view the 41 games with Gretzky as a random sample of all Oiler games in which Gretzky played and the 17 games without Gretzky as a random sample of all Oiler games in which Gretzky did not play, is there convincing evidence that the mean number of goals scored by the Oilers is higher for games when Gretzky plays? Use \(\alpha=0.01\). $$ \begin{array}{lccc} & & \text { Sample } & \text { Sample } \\ & n & \text { Mean } & \text { sd } \\ \text { Games with Gretzky } & 41 & 4.73 & 1.29 \\ \text { Games without Gretzky } & 17 & 3.88 & 1.18 \end{array} $$

Short answer

Cannot provide a short answer since the actual calculations are not performed. However, the process described would result in a determination of statistical significance indicating whether Gretzky's presence has an impact on the number of goals scored.

Step-by-step solution

Identifying Information

Find and list the needed numbers for conducting the t-test. This includes sample sizes, means, and standard deviations for each group. The sample size, mean, and standard deviation for games with Gretzky are 41, 4.73, and 1.29, respectively. Similarly, for games without Gretzky, the sample size, mean, and standard deviation are 17, 3.88, and 1.18 respectively.

State the Null and Alternative Hypotheses

The null hypothesis, denoted \(H_0\), is that there's no difference in the mean number of goals scored by the Oilers in games with or without Gretzky. The alternative hypothesis, denoted \(H_1\), is that more goals are scored on average when Gretzky plays.

Perform the Two-Sample T-Test

Using these numbers and the formula for the two-sample t-test: \( t = \frac{{M_1 - M_2}}{{\sqrt{\frac{{SD_1^2}}{{n_1}} + \frac{{SD_2^2}}{{n_2}}}}}\) where \(M_1\) and \(M_2\) are the sample means, \(SD_1\) and \(SD_2\) are the sample standard deviations, and \(n_1\) and \(n_2\) are the numbers of observations in the two samples.

Finding the p-value

The t-score calculated earlier now has to be compared to a t-distribution to find the p-value. The degrees of freedom for this test are \(n_1 + n_2 - 2\). If this p-value is less than our chosen significance level (0.01), we reject the null hypothesis.

Making Conclusion

Based on the p-value and our alpha level, we make a conclusion about our hypotheses. If we reject the null hypothesis, the conclusion would support that Gretzky's presence has an effect on the number of goals scored.