An alternative hypothesis (H1) is the proposition considered when evidence leads you to reject the null hypothesis (H0). The null starts by treating an observed difference as chance; the alternative treats the difference as meaningful rather than essentially zero. section 20.2 section 5.5
Use H1 to turn a practical question into a testable claim. For a two-group t-test, H0 says the group means are equal, while H1 says their true difference in means is not zero. A p-value at or below a predefined 0.05 threshold is used here to reject H0 and accept H1; a larger p-value means fail to reject H0. section 7.4.1 section 20.2
A practical hypothesis-testing workflow is:
- State H0 first: assume any observed variance is due to chance. section 20.2
- State H1 as the opposing claim that the variance is consequential or the means differ. section 12.7.2
- Choose a test suited to the data; a t-test compares continuous data from no more than two groups. section 7.4.1
- Compare the returned p-value with the selected significance threshold, then reject or fail to reject H0. section 20.2
Home versus road foul calls
For NBA 2018–19 regular-season games, a Welch two-sample t-test compares personal fouls for road and home teams. The output reports a p-value of 0.02727 and H1 that the true mean difference is not zero. Under the 0.05 rule, that result rejects equality of means and supports the alternative hypothesis. section 7.4.1
options(scipen = 999) t.test(fouls1819reg$pfR, fouls1819reg$pfH) ## ## Welch Two Sample t-test ## ## data: fouls1819reg$pfR and fouls1819reg$pfH ## t = 2.2089, df = 2457.8, p-value = 0.02727 ## alternative hypothesis: true difference in means is not equal to 0 ## 95 percent confidence interval: ## 0.04290002 0.72132763 ## sample estimates: ## mean of x mean of y ## 21.09512 20.71301
Inspect the R t.test output to connect the alternative hypothesis, p-value, confidence interval, and group means in one result. section 7.4.1
Salary distribution and championships
A second t-test compares Gini coefficients for championship-winning teams and all other teams from 1991–2018. The p-value of 0.006245 falls below 0.05, so the equal-means null is rejected in favor of an alternative that the salary-distribution difference is meaningful. section 12.7.2
