Hypothesis testing is a method for using sample data to judge between two competing claims about a population. The null hypothesis, written H0, usually represents no effect, no difference, or the standard value being tested. The alternative hypothesis, written H1 or Ha, represents the change, effect, or difference that the researcher is looking for.
Stating these hypotheses clearly matters because every later step in the test depends on them.
Key Facts
- H0 is the default claim tested against the data, often using equality: μ = μ0, p = p0, or μ1 = μ2.
- Ha is the competing claim and uses an inequality: μ ≠ μ0, μ > μ0, μ < μ0, p ≠ p0, p > p0, or p < p0.
- A two-sided test uses Ha: parameter ≠ value and looks for evidence in both tails of the sampling distribution.
- A right-tailed test uses Ha: parameter > value, while a left-tailed test uses Ha: parameter < value.
- The test statistic measures how far the sample result is from H0 in standard error units, such as z = (x̄ - μ0)/(σ/√n).
- If p-value ≤ α, reject H0; if p-value > α, fail to reject H0.
Vocabulary
- Null hypothesis
- The null hypothesis is the claim that there is no effect, no difference, or that a population parameter equals a specified value.
- Alternative hypothesis
- The alternative hypothesis is the claim that contradicts H0 and represents the effect or difference being tested for.
- Significance level
- The significance level, α, is the chosen cutoff probability for deciding when sample evidence is strong enough to reject H0.
- P-value
- The p-value is the probability, assuming H0 is true, of getting a result at least as extreme as the observed sample result.
- One-sided test
- A one-sided test checks for evidence in only one direction, either greater than or less than the null value.
Common Mistakes to Avoid
- Putting the equality sign in Ha is wrong because the null hypothesis is the hypothesis that contains equality, such as =, ≤, or ≥.
- Saying accept H0 is misleading because a large p-value means the data did not provide enough evidence to reject H0, not that H0 has been proven true.
- Choosing a one-sided test after seeing the data is wrong because the direction of Ha must be chosen before collecting or analyzing the sample.
- Confusing the sample statistic with the population parameter is wrong because hypotheses are statements about population values such as μ or p, not sample values such as x̄ or p̂.
Practice Questions
- 1 A cereal box is labeled as having a mean fill weight of 500 g. A quality inspector wants to test whether the true mean fill weight is less than 500 g. Write H0 and Ha using μ.
- 2 A school claims that 60% of students ride the bus. A survey of 200 students finds that 108 ride the bus. Write H0 and Ha for testing whether the true proportion is different from 0.60, then compute the sample proportion p̂.
- 3 A new study wants to know whether a tutoring program improves average exam scores compared with the old method. Should the alternative hypothesis be one-sided or two-sided, and what claim should it make?