Null and alternative hypotheses

Why this exists

A made-up surf school on Oahu advertises that 80\% of beginners stand up on their first lesson. Makoa, thinking of signing up, doubts it. He watches 50 beginners and sees 36 of them stand up, a sample proportion of \displaystyle \frac{36}{50} = 0.72. Is that evidence that the advertisement is wrong?

Not by itself. The 0.72 is a statistic, worked out from the 50 beginners he watched; the advertisement is about a parameter, the share of all the school’s beginners (see parameters and statistics). Even if the true share were exactly 0.80, a sample of 50 would rarely land on exactly 0.80, because statistics vary from sample to sample. So the question has to be asked more carefully: is 0.72 further below 0.80 than chance alone would usually carry a sample?

To ask that, Makoa first has to say precisely which claim he is testing and what he suspects instead. Those two statements are the null hypothesis and the alternative hypothesis, and every hypothesis test starts by writing them down.

Definition

A hypothesis in statistics is a claim about a parameter of a population, such as a population proportion p or a population mean \mu.

The null hypothesis, written H_0, is the claim of no effect, no change, no difference, or the value someone has claimed. It is always written as an equality: the parameter equals one particular number, called the null value.

The alternative hypothesis, written H_a, is the claim the study is looking for evidence of. It says the parameter differs from the null value in a stated way, and it uses one of three signs:

• < (the parameter is less than the null value), a one-sided alternative;
• > (the parameter is greater than the null value), also one-sided;
• \neq (the parameter is not equal to the null value, in either direction), a two-sided alternative.

Which sign is used comes from the research question, and it is chosen before the data are seen. Some books write H_A or H_1 for the alternative; the meaning is the same.

Every set of hypotheses comes with a sentence that defines the parameter in words, in context: which number, about which population. For the surf school: p is the proportion of all the school’s beginners who stand up on their first lesson. The same pattern covers each kind of parameter in this course.

Here p_0 and \mu_0 stand for the null value in the question at hand.

Intuition

Figure 1 draws each hypothesis as the set of values of p it names. The null hypothesis is a single point. Each alternative is a stretch of the line that leaves the null value out.

A hypothesis test works like a fair trial. The null hypothesis is the presumption of innocence: it is assumed true, and the data have to argue against it. The test asks: if H_0 were true, how surprising would data like ours be? Ordinary data make no case against H_0. Very unusual data, unusual in the direction H_a points, are evidence for H_a. The number that measures this surprise is the p-value, the subject of a later lesson.

This is the same logic as comparing treatments by rerandomizing: there, “the treatment does nothing” was assumed, the data were reshuffled to see what chance alone produces, and the real result was compared with that. The words “null hypothesis” give that assumption its name.

Results

Worked examples

Boundaries

Only parameters. A claim about the sample in hand is not a hypothesis; it is checked by looking. And the population in the hypotheses must be the one the sample came from: watching only Kekoa’s beginners says nothing directly about the whole school.

One value in H_0. The null must name a single value to compute from. Some books write a one-sided null as an inequality, such as H_0\colon p \geq 0.80 against H_a\colon p < 0.80; the test is still carried out at the boundary value p = 0.80, the null value closest to the alternative.

A one-sided test looks one way. With H_a\colon p < 0.80, a true share above 0.80 can never be detected, however clear the data (Example 6). And hypotheses chosen or changed after the data are seen do not carry the meaning a test gives them.

A test does not give cause. Evidence that p > 0.80 for Kekoa’s new method shows the share is above 0.80, not that the method raised it: a calmer season or a different crowd of beginners could do the same. A claim about cause needs beginners assigned to the methods at random, compared through p_1 - p_2.

Where this goes next

Once the hypotheses are written, the data are summarized by a test statistic, which says how many standard errors the statistic sits from the null value. The p-value turns that into the surprise this lesson only named: the chance, if H_0 were true, of data at least as far in the direction of H_a. A randomization test finds that chance by reshuffling instead of by formula. Comparing the p-value with a chosen cutoff gives the decision, to reject H_0 or not. A two-sided test and a confidence interval at the matching confidence level turn out to be closely linked.

Exercises

Writing hypotheses about one proportion

1. The school’s rental counter claims that 95\% of rented boards come back undamaged. Pualani, who runs the counter, suspects damage has risen this season. Define the parameter and write her hypotheses.
2. The school says 30\% of its beginners book a second lesson. Kekoa starts offering a discount on the second lesson and wants evidence that it raises that share. Define the parameter and write his hypotheses.
3. A forecast site says the waves at the school’s break are over 3 feet on 40\% of mornings. Makoa wants to know whether that is accurate, with no direction in mind. Write his hypotheses.
4. Each statement is about the school’s beginners, with null value 0.80. Write H_a for each: (a) “fewer than 80\% stand up”; (b) “more than 80\% stand up”; (c) “the share has changed since last year, when it was 80\%”; (d) “the advertised 80\% is too low”.

Means and differences

5. The school says a beginner lesson spends a mean of 90 minutes in the water. Makoa suspects lessons run short. Define the parameter and write his hypotheses.
6. Intermediate students at the school catch a mean of 12 waves per session. Pualani tries a new warm-up and wants to know whether it changes that mean, in either direction. Write her hypotheses.
7. Makoa suspects that Pualani’s beginners catch more waves per session, on average, than Kekoa’s. Let \mu_1 be the mean for all Kekoa’s beginners and \mu_2 for all Pualani’s. Write the hypotheses.
8. Repeat Exercise 7 with the labels swapped: \mu_1 for Pualani’s beginners and \mu_2 for Kekoa’s.
9. Kekoa thinks morning lessons, when the wind is lighter, give a higher share of beginners who stand up than afternoon lessons. Define two parameters and write the hypotheses.

Finding the fault

10. A student writes H_0\colon \hat{p} = 0.80, H_a\colon \hat{p} < 0.80 for Makoa’s question. What is wrong? Write the correct version.
11. A student writes H_0\colon p < 0.80, H_a\colon p = 0.80. What is wrong? Write the correct version.
12. A student writes H_0\colon \mu = 7, H_a\colon \mu > 8. What is wrong? Write the correct version.
13. Makoa watches 50 beginners and 41 stand up. He then writes H_0\colon p = 0.82. Find his sample proportion and explain what is wrong with this null hypothesis.
14. Pualani’s question is neutral, as in Example 3. In her sample, 38 of 50 beginners stand up, and she then chooses H_a\colon p < 0.80. Find her sample proportion and explain what is wrong.

Reasoning

15. Explain why the null hypothesis is written with an equals sign and the alternative never is.
16. Explain why hypotheses are written about p and not about \hat{p}.
17. Kekoa compares this season’s beginners, all taught the new pop-up, with the 80\% the school has advertised for years, and finds evidence that p > 0.80. Can he conclude the new method caused the rise? Describe a study that could support that claim, and its hypotheses.
18. Someone says: “The data gave no evidence against H_0\colon p = 0.80, so the school’s claim is proved.” Respond.
19. Makoa, Kekoa and Pualani all test the same advertised 80\%, yet write three different alternatives. Explain how all three can be correct.