Key takeaway: The null and alternative hypotheses are a pair of competing claims about a population parameter: the null states no change or no effect, and the alternative states the change the researcher suspects.
Grade 12 – AP StatisticsAP Statistics Unit 3Statistics and Data
Definition of Null and Alternative Hypotheses
The null and alternative hypotheses frame every significance test. The null hypothesis, \(H_0\), is the default claim that nothing special is happening, and it always includes equality. The alternative, \(H_a\), is the claim that the data are meant to support.
Both statements are about a parameter, such as \(\mu\) or \(p\), never about a sample statistic. The direction of \(H_a\) comes from the question asked before looking at data: greater than, less than, or simply different from.
Formula and notation
With a claimed value \(\mu_0\), the three possible pairs are:
\[ H_0:\mu=\mu_0\ \text{ versus }\ H_a:\mu\gt\mu_0,\quad H_a:\mu\lt\mu_0,\quad H_a:\mu\neq\mu_0. \]
The first two are one-sided; the last is two-sided. The same pattern applies to proportions with \(p\).
Key facts
- Write both statements about parameters, using Greek letters or p, never x-bar or p-hat.
- Choose the direction of the alternative from the research question, not from the data.
- A two-sided alternative splits the rejection area between both tails.
- The null and alternative hypotheses together must cover every possible value of the parameter.
Null and Alternative Hypotheses example
A bakery says its muffins weigh 4.5 ounces on average. An inspector suspects they are lighter. Write the hypotheses, then do the same for a customer who just wants to check whether the average differs from 4.5.
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The parameter is \(\mu\), the true mean weight of all muffins.
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Inspector, suspecting lighter muffins: \(H_0:\mu=4.5\) and \(H_a:\mu\lt 4.5\).
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Customer, checking for any difference: \(H_0:\mu=4.5\) and \(H_a:\mu\neq 4.5\).
Both start from \(H_0:\mu=4.5\). The inspector’s alternative is \(H_a:\mu\lt 4.5\); the customer’s is \(H_a:\mu\neq 4.5\).
Common mistake
Watch out
A sample of muffins averages 4.3 ounces, so a student writes \(H_a:\,\overline{x}\lt 4.5\).
Why it is wrong
Hypotheses describe the population, so the parameter \(\mu\) belongs in them, not the sample mean 4.3. The data are used afterward to test the claim. Also, the direction should have been chosen before seeing 4.3, from the question of interest. Null and alternative hypotheses are written before the data are examined.
Related terms and lessons
Browse the glossary: A–Z list · AP Statistics vocabulary · Statistics and Data