Alternative Hypothesis — Definition, Formula & Examples

The alternative hypothesis is the claim you are trying to find evidence for in a hypothesis test. It states that a population parameter differs from the value assumed by the null hypothesis — either by being greater, less, or simply not equal.

Denoted $H_a$ (or $H_1$ ), the alternative hypothesis is the statement that the population parameter satisfies a strict inequality relative to the null value $\mu_0$ (or $p_0$ ). It takes one of three forms: $\neq$ , $>$ , or $<$ , corresponding to two-tailed or one-tailed tests, respectively.

Key Formula

H_a: \mu \neq \mu_0 \quad \text{or} \quad H_a: \mu > \mu_0 \quad \text{or} \quad H_a: \mu < \mu_0

Where:

$H_a$ = The alternative hypothesis
$\mu$ = The true population mean being tested
$\mu_0$ = The hypothesized value stated in the null hypothesis

How It Works

Before collecting data, you set up two competing hypotheses: the null hypothesis

H_0

and the alternative hypothesis

H_a

. The null assumes no effect or no difference, while the alternative captures the specific change or effect you suspect. You then gather data and calculate a test statistic. If the data are sufficiently unlikely under

H_0

, you reject

H_0

in favor of

H_a

. The direction of your alternative hypothesis determines whether you run a one-tailed or two-tailed test.

Worked Example

Problem: A coffee shop claims its large cups contain 16 oz of coffee on average. You suspect they are underfilling. Write the null and alternative hypotheses.

Identify the claim to test: You suspect the true mean is less than the advertised 16 oz, so the alternative should reflect a 'less than' direction.

Write the null hypothesis: The null assumes the mean equals the claimed value.

H_0: \mu = 16

Write the alternative hypothesis: Since you suspect underfilling, use a left-tailed alternative.

H_a: \mu < 16

Answer: The alternative hypothesis is

H_a: \mu < 16

, indicating a one-tailed (left-tailed) test.

Why It Matters

Every hypothesis test on the AP Statistics exam requires you to state

H_a

correctly before performing any calculations. Choosing the wrong direction (one-tailed vs. two-tailed) changes your p-value and can lead to an incorrect conclusion. In fields like pharmaceutical research, a properly stated alternative hypothesis determines what kind of evidence is needed to approve a new drug.

Common Mistakes

Mistake: Writing the alternative hypothesis with an equals sign, such as

H_a: \mu = 18

Correction: The alternative hypothesis always uses a strict inequality (

\neq

>

, or

<

). The equals sign belongs only in the null hypothesis

H_0