Teams in the National Football League (NFL) in the US play...

Teams in the National Football League (NFL) in the US play four pre-season games each year before the regular season starts. Do teams that do well in the pre-season tend to also do well in the regular season? We are interested in whether there is a positive linear association between the number of wins in the pre-season and the number of wins in the regular season for teams in the NFL. (a) What are the null and alternative hypotheses for this test? (b) The correlation between these two variables for the 32 NFL teams over the 10 year period from 2005 to 2014 is 0.067 . Use this sample (with $n=320$ ) to calculate the appropriate test statistic and determine the p-value for the test. (c) State the conclusion in context, using a $5 \%$ significance level. (d) When an NFL team goes undefeated in the pre-season, should the fans expect lots of wins in the regular season?

Teams in the National Football League (NFL) in the US play four pre-season games each year before the regular season starts. Do teams that do well in the pre-season tend to also do well in the regular season? We are interested in whether there is a positive linear association between the number of wins in the pre-season and the number of wins in the regular season for teams in the NFL.
(a) What are the null and alternative hypotheses for this test?
(b) The correlation between these two variables for the 32 NFL teams over the 10 year period from 2005 to 2014 is 0.067 . Use this sample (with $n=320$ ) to calculate the appropriate test statistic and determine the p-value for the test.
(c) State the conclusion in context, using a $5 \%$ significance level.
(d) When an NFL team goes undefeated in the pre-season, should the fans expect lots of wins in the regular season?

Key Concepts

P-Value Interpretation

The p-value represents the probability of obtaining a test statistic at least as extreme as the one observed, assuming that the null hypothesis is true. It allows decision-makers to assess the evidence against the null hypothesis, based on whether it falls below a predetermined significance level (commonly 5%).

Statistical Significance and Contextual Conclusion

This concept involves comparing the p-value to the chosen significance level to decide whether to reject the null hypothesis. Beyond the numbers, conclusions should be stated in plain language that relates the findings back to the original research question and the practical implications of the results.

Test Statistic Calculation

For testing the significance of a Pearson correlation coefficient, the t-statistic is calculated using the formula t = r * sqrt((n - 2) / (1 - r^2)), where r is the sample correlation and n is the sample size. This statistic follows a t-distribution with n - 2 degrees of freedom under the null hypothesis.

Practical versus Statistical Significance

Even if a test yields a statistically significant result, the magnitude of the association and its real-world implications must be evaluated. This concept reminds analysts to consider whether an effect size is meaningful in practical terms, which is crucial for decision-making and interpreting research outcomes.

Pearson Correlation Coefficient

This is a measure of the linear relationship between two continuous variables. It quantifies the strength and direction of the association, ranging from -1 (perfect negative correlation) to +1 (perfect positive correlation), with 0 indicating no linear correlation.

Hypothesis Formulation

This involves stating a null hypothesis (no relationship or no effect) and an alternative hypothesis (a specific relationship or effect) about the variables of interest. In the context of testing the association between two quantitative variables, it lays the groundwork for statistical testing by clearly defining what is being evaluated.

Correlation Hypothesis Testing

This is a statistical procedure used to determine whether the observed correlation in a sample reflects a true correlation in the population. The test generally involves assessing a t-statistic derived from the correlation coefficient and comparing it to a critical value from the t-distribution.

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