Null and Alternative Hypotheses
In short: Null and alternative hypotheses are the two competing claims in a significance test: the null states no change or no difference, and the alternative states the effect the researcher suspects.
AP StatisticsStatisticsAP Statistics Unit 3
What is Null and Alternative Hypotheses?
The null and alternative hypotheses are the two rival statements that every significance test pits against each other. The null \(H_0\) is the baseline, always with an equals sign: “nothing unusual is going on.” The alternative \(H_a\) is the finding the data would need to back up.
Both describe a population parameter, never a sample statistic. The alternative sets the type of test: \(H_a: \mu < \mu_0\) or \(H_a: \mu > \mu_0\) is one-sided, while \(H_a: \mu \neq \mu_0\) is two-sided. Choose the direction from the research question before looking at any numbers.
How to spot it
- A statement such as “the true mean is 500 mL” or “2% are defective” is being checked with sample data, so the null and alternative hypotheses must be written.
- Wording like “less than”, “greater than” or “different from” signals the form of the alternative.
- You must write hypotheses before any numbers are computed.
- A question asks to define the parameter in context, such as p equals the true proportion of all customers.
The method
- Before writing the null and alternative hypotheses, name the parameter in words and with a symbol (\(p\), \(\mu\), and so on).
- Write \(H_0\) with an equals sign: parameter = claimed value.
- Pick \(H_a\) from the research question: \(<\), \(>\) or \(\neq\); never use the sample result.
- Use the same symbol and the same number in both statements.
Null and Alternative Hypotheses: solved example
Problem
A juice company prints 500 mL on every label, but a consumer group suspects the bottles hold less. Write the two hypotheses.
Solution
Let \(\mu\) be the true mean volume, in mL, of all bottles the company fills.
The claim being tested is no shortfall, so \(H_0: \mu = 500\). The suspicion is underfilling, which points one way: \(H_a: \mu < 500\).
Only unusually low sample means would be evidence for \(H_a\), so the test is one-sided on the left.
Try it
Question 1
A factory says 2% of its parts are defective. An inspector wants to know whether the rate is different from 2%. Write the hypotheses.
Show the answer
With \(p\) the true proportion defective: \(H_0: p = 0.02\) and \(H_a: p \neq 0.02\), a two-sided test.
Question 2
Why is \(H_a: \bar{x} > 52\) written incorrectly?
Show the answer
Hypotheses must describe a population parameter such as \(\mu\), not a sample statistic such as \(\bar{x}\), which is known from the data.
Null and Alternative Hypotheses vs. Hypothesis Testing
Writing the null and alternative hypotheses is just the opening move of hypothesis testing; verifying conditions, calculating the statistic and concluding come later. A slip here, like picking the wrong direction, spoils everything after it.
Keep going
Lessons
Worksheets and quizzes
- Significance Tests for a Proportion – Worksheet
- Significance Tests for a Proportion – Quiz
- t Intervals and t Tests for a Mean – Worksheet
- t Intervals and t Tests for a Mean – Quiz
Related words
Whole dictionary: A–Z · AP Statistics · Statistics