Module 8: Inference for One Proportion
Hypothesis Test for a Population Proportion (3 of 3)
Hypothesis Test for a Population Proportion (3 of 3)
Learning outcomes
- Conduct a hypothesis test for a population proportion. State a conclusion in context.
- Interpret the P-value as a conditional probability in the context of a hypothesis test about a population proportion.
- Distinguish statistical significance from practical importance.
- From a description of a study, evaluate whether the conclusion of a hypothesis test is reasonable.
More about the P-Value
The P-value is a probability that describes the likelihood of the data if the null hypothesis is true. More specifically, the P-value is the probability that sample results are as extreme as or more extreme than the data if the null hypothesis is true. The phrase “as extreme as or more extreme than” means farther from the center of the sampling distribution in the direction of the alternative hypothesis.
More generally, we view the P-value a description of the strength of the evidence against the null hypothesis and in support of the alternative hypothesis. But the P-value is a probability about sample results, not about the null or alternative hypothesis.
One More Note about P-Values and the Significance Level
You may wonder why 5% is often selected as the significance level in hypothesis testing and why 1% is also a commonly used level. It is largely due to just convenience and tradition. When Ronald Fisher (one of the founders of modern statistics) published one of his tables, he used a mathematically convenient scale that included 5% and 1%. Later, these same 5% and 1% levels were used by other people, in part just because Fisher was so highly esteemed. But mostly, these are arbitrary levels.
The idea of selecting some sort of relatively small cutoff was historically important in the development of statistics. But it’s important to remember that there is really a continuous range of increasing confidence toward the alternative hypothesis, not a single all-or-nothing value. There isn’t much meaningful difference, for instance, between the P-values 0.049 and 0.051, and it would be foolish to declare one case definitely a “real” effect and the other case definitely a “random” effect. In either case, the study results are roughly 5% likely by chance if there’s no actual effect.
Whether such a P-value is sufficient for us to reject a particular null hypothesis ultimately depends on the risk of making the wrong decision and the extent to which the hypothesized effect might contradict our prior experience or previous studies.
Example
Sample Size and Hypothesis Testing
Consider our earlier example about teenagers and Internet access. According to the Kaiser Family Foundation, 84% of U.S. children ages 8 to 18 had Internet access at home as of August 2009. Researchers wonder if this number has changed since then. The hypotheses we tested were:
- H0: p = 0.84
- Ha: p ≠ 0.84
The original sample consiste