Understanding Hypothesis Tests

Much like the terminology, notation, and definitions relating to sampling we saw previously, there’s a lot of terminology, notation, and definitions related to hypothesis testing as well. Learning it may seem like a very difficult task at first; however, with practice, anyone can become proficient in it.

• First, a hypothesis is a statement about the value of an unknown population parameter. In our résumé activity, our population parameter of interest is the difference in population proportions $p_m$− $p_f$. Hypothesis tests can involve any of the population parameters.

• Second, a hypothesis test consists of a test between two competing hypotheses: a null hypothesis $H_0$ (pronounced “H-naught”) vs. an alternative hypothesis $H_A$ (also denoted $H_1$).

Generally, the null hypothesis is a claim that there’s no effect or no difference of interest. In many cases, the null hypothesis represents the status quo or a situation in which nothing interesting is happening. Furthermore, generally, the alternative hypothesis is the claim the experimenter or researcher wants to establish or find evidence to support. It’s viewed as a challenger hypothesis to the null hypothesis $H_0$. In our résumé activity, an appropriate hypothesis test would be:

• $H$₀ : Men and women are promoted at the same rate.

• $H$_A : Men are promoted at a higher rate than women.

Note some of the choices we’ve made. First, we set the null hypothesis $H_0$ to be that there’s no difference in promotion rate, and the challenger alternative hypothesis $H_A$ to be that there’s a difference. While it wouldn’t be wrong in principle to reverse the two, it’s a convention in statistical inference that the null hypothesis is set to reflect a null situation where nothing is going on. As we discussed earlier, in this case, $H_0$ corresponds to there being no difference in promotion rates. Furthermore, we set $H_A$ to be that men are promoted at a higher rate, a subjective choice reflecting a prior suspicion we have that this is the case. We call such alternative hypotheses one-sided alternatives. If someone else, however, doesn’t share such suspicions and only wants to investigate that there is a difference, whether higher or lower, they would set what is known as a two-sided alternative.

We can reexpress the formulation of our hypothesis test using the mathematical notation for our population parameter of interest, the difference in population proportions $p_m$ − $p_f$: