AP® Statistics · Unit 4 · Topic 4.5
Concluding a Mean Test
Concluding a mean test means comparing your p-value to alpha and stating, in context, whether you have convincing evidence for the alternative hypothesis — without claiming you proved anything.
Every t-test for means ends the same way: compare the p-value to alpha. If p-value < alpha, reject H0 and say you have convincing evidence for Ha, stated in the actual context of means being compared. If p-value ≥ alpha, fail to reject H0, and say you do not have convincing evidence for Ha. The words matter. You never say you proved H0 true or accepted it; you only say the data didn't give enough evidence against it. A well-written conclusion names the p-value, the alpha level, and the groups being compared, in a sentence a non-statistician could follow.
Significance and importance are different questions. With a huge sample, the standard error shrinks, so even a tiny, boring gap between two averages can produce a small p-value. That tells you the gap is probably real, not that it matters in the real world. A one-point difference in average test scores across 50,000 students might be statistically significant and still useless for making a decision — always ask about effect size, not just the p-value.
This conclusion step is also where mean tests connect to confidence intervals. A two-sided t-test at alpha 0.05 rejects H0: mu1 = mu2 exactly when a 95 percent confidence interval for mu1 - mu2 excludes the hypothesized value, usually zero. The interval adds what the test alone can't: a range of plausible sizes for the difference, which is how you judge practical importance after you've settled statistical significance.
Key terms
- p-value
- The probability of getting a test statistic as extreme as observed, assuming H0 is true.
- statistical significance
- A result unlikely to occur by chance alone, determined by comparing the p-value to alpha.
- practical significance
- Whether the size of an effect is large enough to matter in a real-world sense, separate from statistical significance.
- fail to reject H0
- The conclusion reached when evidence is insufficient to support the alternative hypothesis, not proof H0 is true.
- confidence interval
- A range of plausible values for a parameter, which can be used to test a hypothesis by checking if it excludes the null value.
Watch it explained
- How it worksHow do you write a t-test conclusion that names the p-value and the context?
- Common mistakeWhy isn't statistical significance the same as a difference that actually matters?
- Common mistakeWhy isn't failing to reject the null in a mean test the same as proving the two means are equal?
- Ties togetherWhy does rejecting a mean test at alpha 0.05 always match a 95 percent t-interval that misses the null value?
Check yourself
End of topic.