Null vs Alternative Hypothesis: How to Write Them

Null vs alternative hypothesis at a glance

A significance test compares two claims about a population, never about your sample alone. The null hypothesis, written $H_0$ (read "H-naught"), is the no-effect claim: it sets a population parameter equal to one specific value. The alternative hypothesis, written $H_a$, is the claim you actually want evidence for.

You reject $H_0$ when the p-value is small enough, meaning below your significance level. The p-value is the probability of getting a result as extreme or more extreme than what you observed, assuming $H_0$ is true. Think of $H_0$ as the default position that stays in place until the evidence pushes hard enough to abandon it. Getting the two hypotheses right is the first step of every test, so it pays to slow down here, because a wrong parameter or a flipped inequality will send the rest of your work in the wrong direction.

What the null hypothesis (H0) says

The null hypothesis is a statement of no difference or no effect, and it is always about a population parameter. For a proportion that parameter is $p$, the population proportion. For a mean it is $\mu$ (the Greek letter mu), the population mean.

$H_0$ always contains equality, so it looks like $H_0: p = p_0$ or $H_0: \mu = \mu_0$, where $p_0$ and $\mu_0$ are the specific hypothesized values. That hypothesized value usually comes from a claim, a target, or the status quo you are checking. If a label says a box holds 500 grams, then $\mu_0 = 500$ is the number you place in $H_0$.

What the alternative hypothesis (Ha) says

The alternative hypothesis is the claim you are collecting data to support. It uses a strict inequality or the not-equal sign, never an equals sign, so it is written with $<$, $>$, or $\neq$ (not equal to).

Use a one-sided alternative when the question points in a single direction. "Is the on-time rate lower than claimed?" gives $H_a: p < p_0$, while "did the mean increase?" gives $H_a: \mu > \mu_0$. Use a two-sided alternative, $H_a: p \neq p_0$ or $H_a: \mu \neq \mu_0$, when you care whether the parameter differs from the hypothesized value in either direction.

Pick the direction from the question wording before you look at the data. Choosing the side after you see which way the sample came out inflates your error rate, a Type I error problem.

Why hypotheses use parameters, not statistics

Every hypothesis is a guess about a fixed but unknown number that describes the whole population, called a parameter. The sample proportion $\hat{p}$ (read "p-hat") and the sample mean $\bar{x}$ (read "x-bar") are statistics: numbers you compute from data that change from one sample to the next. Because a statistic is already known once you have the data, there is nothing left to test about it.

So a hypothesis like $H_0: \hat{p} = 0.5$ has no meaning, since the value of $\hat{p}$ is simply whatever your sample gave you. The unknown you are actually asking about is $p$, the population proportion, which is why $H_0: p = 0.5$ is the correct form. Keep parameters ($p$, $\mu$, and $\mu_d$, the population mean difference) inside the hypotheses and statistics ($\hat{p}$, $\bar{x}$) outside them.

Symbols and words for proportions and means

The table below lists the parameter, the null, and the common alternatives for the three setups you meet most in AP Statistics. Every row keeps the population parameter in the hypotheses and leaves the sample statistic out.

For paired data you first compute a difference for each pair, then treat $\mu_d$, the population mean of those differences, like any other mean.

How to write H0 and Ha in four steps

1. Name the population parameter in words and symbols, such as $p$ = the true proportion of voters who approve, or $\mu$ = the mean battery life of all units.
2. Find the hypothesized value from the claim or target, and write the null as parameter = value.
3. Read the question to decide the direction, then write the alternative with $<$, $>$, or $\neq$.
4. Check that both hypotheses use the parameter, that only $H_0$ has an equals sign, and that you fixed the direction before seeing the data.

This is the same setup work the AP course labels "Setting Up a Test." If you are unsure which parameter and test apply, see which statistical test to use.

Common mistakes when writing hypotheses

Writing hypotheses about a statistic. $H_0: \bar{x} = 500$ is wrong because $\bar{x}$, the sample mean, is something you calculate, not an unknown you test. Hypotheses are always about the parameter, so write $H_0: \mu = 500$ instead.

Putting the equality in Ha. The alternative never contains $=$, $\leq$, or $\geq$; only $H_0$ carries the equals sign. If you find yourself writing $H_a: p \geq 0.5$, move the equality back to $H_0$ and leave a strict inequality in $H_a$.

Choosing the direction after seeing the data. Deciding on $<$ versus $>$ once you notice which way $\hat{p}$ landed makes your test look stronger than it really is. Set the sidedness from the research question first, then collect and analyze the data.