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Null Hypothesis

The null hypothesis is the starting assumption in a hypothesis test that there is no real effect, no difference, or no relationship. It is written as H0H_0 and represents the claim you are trying to find evidence against.

In statistical hypothesis testing, the null hypothesis (H0H_0) is a statement that asserts no effect, no difference, or no association between variables in a population. It serves as the default position and is assumed true unless sample data provides sufficient evidence to reject it. The burden of proof lies on demonstrating that the observed data would be unlikely under H0H_0, typically measured by a p-value compared against a chosen significance level α\alpha.

Key Formula

H0:μ=μ0H_0: \mu = \mu_0
Where:
  • H0H_0 = the null hypothesis
  • μ\mu = the true population mean being tested
  • μ0\mu_0 = the hypothesized value of the population mean (often a known or assumed value)

Worked Example

Problem: A company claims that the average weight of its cereal boxes is 500 grams. A quality inspector suspects the boxes are underfilled and collects a sample of 36 boxes with a mean weight of 496 grams and a standard deviation of 12 grams. Set up the null and alternative hypotheses, then calculate the test statistic.
Step 1: Write the null hypothesis. The company's claim is that the mean weight equals 500 g, so this is the default assumption.
H0:μ=500H_0: \mu = 500
Step 2: Write the alternative hypothesis. The inspector suspects underfilling, so the alternative is one-sided (less than).
Ha:μ<500H_a: \mu < 500
Step 3: Calculate the standard error of the sample mean.
SE=sn=1236=126=2SE = \dfrac{s}{\sqrt{n}} = \dfrac{12}{\sqrt{36}} = \dfrac{12}{6} = 2
Step 4: Compute the test statistic (z-score) by comparing the sample mean to the hypothesized value.
z=xˉ−μ0SE=496−5002=−42=−2.0z = \dfrac{\bar{x} - \mu_0}{SE} = \dfrac{496 - 500}{2} = \dfrac{-4}{2} = -2.0
Step 5: Interpret: A z-score of −2.0 has a p-value of about 0.0228. At a significance level of α=0.05\alpha = 0.05, this p-value is less than 0.05, so you would reject H0H_0. The data provides sufficient evidence that the mean weight is less than 500 g.
Answer: With a test statistic of z=−2.0z = -2.0 and a p-value of approximately 0.023, we reject H0:μ=500H_0: \mu = 500 at the 0.05 significance level. There is statistically significant evidence that the boxes are underfilled.

Why It Matters

The null hypothesis is the foundation of every hypothesis test you encounter in AP Statistics and beyond. Researchers in medicine, psychology, economics, and engineering all rely on this framework — they formulate H0H_0, collect data, and determine whether the evidence is strong enough to reject it. Without a clearly stated null hypothesis, there is no objective way to measure whether a result is statistically meaningful or just due to chance.

Common Mistakes

Mistake: Saying you "accept" the null hypothesis when the data fails to reject it.
Correction: You never prove the null hypothesis is true. When the evidence is insufficient, the correct language is "we fail to reject H0H_0." The data simply didn't provide strong enough evidence against it.
Mistake: Writing the null hypothesis with an inequality (e.g., H0:μ<500H_0: \mu < 500).
Correction: The null hypothesis always contains an equality — it states a specific value or "no difference." Inequalities belong in the alternative hypothesis HaH_a.

Related Terms

  • Probability — P-values measure probability under the null hypothesis
  • Expected Value — The null often specifies an expected population parameter