91Ó°ÊÓ

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 \(\alpha=0.01\). $$ \begin{array}{lccc} & & \text { Sample } & \text { Sample } \\ & n & \text { Mean } & \text { sd } \\ \text { Games with Gretzky } & 41 & 4.73 & 1.29 \\ \text { Games without Gretzky } & 17 & 3.88 & 1.18 \end{array} $$

Step by step solution

01

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.

02

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.

03

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.

04

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.

05

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.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It starts by establishing two opposing hypotheses:

• The null hypothesis (\(H_0\)): assumes no significant difference between specified populations, meaning any observed differences are due to random chance.
• The alternative hypothesis (\(H_1\)): suggests that there is a statistically significant difference between the populations.

In the context of Wayne Gretzky's impact on hockey games, \(H_0\) posits that the mean number of goals scored by the Oilers does not differ whether he plays or not. \(H_1\) suggests that they score more when he is in the game. The goal is to use sample data to evaluate whether \(H_0\) can be rejected in favor of \(H_1\). This forms the foundation for assessing variables and effects in various fields through hypothesis testing.

P-Value

The p-value is a critical component in hypothesis testing, serving as a tool to help determine the strength of the evidence against the null hypothesis. It represents the probability of observing data at least as extreme as the sample data, assuming that the null hypothesis is true. In lay terms, it answers the question: "If our assumption (\(H_0\)) is correct, how "surprising" is our data?"

After performing a Two-Sample T-Test, we obtain a t-score, which helps in finding the p-value by comparing it to a standard t-distribution. A smaller p-value indicates stronger evidence against \(H_0\). If this p-value is less than the chosen significance level (usually 0.05, or 0.01 in our Gretzky example), we reject \(H_0\). This would suggest that Gretzky's participation indeed affects the mean goals scored.

Statistical Significance

Statistical significance is the determination of whether the observed difference or relationship in your data is not merely a product of random chance. It relies on the concept of a significance level, often denoted as alpha (\(\alpha\)), which sets a threshold for decision-making.

In the exercise provided, \(\alpha\) is set at 0.01, indicating a stringent threshold for rejecting the null hypothesis. If the p-value obtained from statistical testing is less than \(\alpha\), it indicates a statistically significant result. In simpler terms, you have enough evidence to claim that the presence of Gretzky indeed influences the number of goals scored, beyond random variation.

This concept helps researchers make well-grounded decisions and uphold the reliability of conclusions drawn from data analyses.

Degrees of Freedom

Degrees of freedom are a statistical concept that refers to the number of values in a calculation that are free to vary. When performing a two-sample t-test, degrees of freedom play an important role in determining the exact shape of the t-distribution curve, which affects the p-value calculation.

The formula used for this particular test is: \(n_1 + n_2 - 2\), where \(n_1\) and \(n_2\) represent the sizes of the two samples. In our Gretzky example, the games with and without him have different sample sizes, so the degrees of freedom are 41 + 17 - 2 = 56. Higher degrees of freedom generally lead to more accurate representations of the population from your sample, assisting in precise hypothesis testing.

Understanding degrees of freedom allows researchers to better assess how well their sample data can represent the broader population phenomenon they are investigating.