Hypothesis Testing: Large Samples

Hypothesis testing with large samples (\(n \geq 30\)) benefits from the Central Limit Theorem, which ensures that sample means are approximately normally distributed. This allows the use of \(t\)-tests even when the underlying population distribution is unknown. A well-defined 8-step framework aids in systematically evaluating hypotheses, balancing statistical rigor with practical decision-making.

Motivating Case: Public Health Crisis Response

Consider a public health application involving NYC’s historical temperature data. Using the airquality dataset (n = 153), we test whether the mean temperature exceeds 77°F. The one-sided \(t\)-test returns a test statistic of \(t = 1.1531\) and a p-value of 0.1253. Since this p-value exceeds the typical \(\alpha = 0.05\) threshold, we fail to reject the null hypothesis. Despite the observed mean being \(77.88^\circ\)F, the evidence is insufficient to conclude that temperatures are significantly above 77°F.

Poultry Farming (ChickWeight)

In contrast, consider the ChickWeight dataset where we examine whether chicks on Diet 1 have a mean weight different from 100g. With n = 220, a two-sided \(t\)-test yields \(t = 0.6926\) and a p-value of 0.4893. The observed mean weight is 102.65g, but again the large p-value indicates no significant difference from the hypothesized mean. The 95% confidence interval (95.12g, 110.17g) includes 100g, reinforcing that any deviation could easily be due to sampling variability.

These examples highlight that even with large samples, statistical significance depends on both effect size and variability—not merely sample size. Proper interpretation ensures sound, data-driven decisions.