Hypothesis testing is a way to use sample data to judge whether a claim about a population is believable. The null hypothesis is the starting assumption, such as no difference or no effect. A test statistic measures how far the sample result is from what the null hypothesis predicts.
Critical values mark the boundary between results that are considered ordinary under the null and results that are unusual enough to reject it.
The decision depends on the sampling distribution of the test statistic, the significance level, and whether the test is one-tailed or two-tailed. If the test statistic falls in a rejection region beyond a critical value, the result is statistically significant and the null hypothesis is rejected. If it does not, the data do not provide enough evidence to reject the null.
For example, in a two-tailed z test with alpha = 0.05, the critical values are about z = -1.96 and z = 1.96, so a test statistic of z = 2.30 leads to rejecting the null hypothesis.
Key Facts
- Test statistic = (sample estimate - null value) / standard error
- For a z test of a mean with known population standard deviation, z = (x̄ - μ0) / (σ / √n)
- For a one-sample t test, t = (x̄ - μ0) / (s / √n)
- In a two-tailed z test with alpha = 0.05, the critical values are approximately -1.96 and 1.96
- Reject H0 if the test statistic falls in the rejection region
- The significance level alpha is the probability of rejecting H0 when H0 is actually true
Vocabulary
- Null hypothesis
- The starting claim in a hypothesis test, usually stating no effect, no difference, or a specific population value.
- Test statistic
- A standardized number that shows how far the sample result is from the null hypothesis in standard error units.
- Critical value
- A cutoff point on the test statistic scale that separates the non-rejection region from the rejection region.
- Rejection region
- The part of the sampling distribution where test statistic values are unlikely enough under the null hypothesis to reject it.
- Significance level
- The chosen probability of a Type I error, commonly written as alpha and often set to 0.05.
Common Mistakes to Avoid
- Comparing the raw sample mean to the critical value is wrong because critical values are on the standardized test statistic scale, not the original measurement scale.
- Using a two-tailed critical value for a one-tailed test is wrong because the rejection area is split differently depending on the alternative hypothesis.
- Saying fail to reject H0 proves H0 is true is wrong because the test only shows that the sample evidence was not strong enough to reject it.
- Ignoring the sign of the test statistic in a one-tailed test is wrong because the rejection region may be only on the left or only on the right side of the distribution.
Practice Questions
- 1 A z test has H0: μ = 100, sample mean x̄ = 106, population standard deviation σ = 12, and n = 36. Compute the test statistic z.
- 2 For a two-tailed z test with alpha = 0.05, the critical values are -1.96 and 1.96. If the test statistic is z = -2.14, should you reject H0?
- 3 A study reports a test statistic that is close to 0 and lies near the center of the null distribution. Explain what this suggests about the evidence against the null hypothesis.