STAT 400 Chap.11 Power and Chi-Square Tests
Power and Chi-Square Tests
Power: how often a test finds a real effect
The power of a test is the probability of rejecting the null hypothesis when a specific alternative is true, which equals one minus the Type II error probability. Computing it takes two stages. First find the rejection region using the null hypothesis. Then compute the probability of landing in that region using the alternative value.
Plotted against the true parameter, the power function equals α at the null value and rises toward one as the true value moves away.
What controls power
Power increases with the sample size, with the significance level, and with the distance between the null and the true value, and it decreases as the population standard deviation grows.
Solving the power condition for n gives a sample size formula for a target power, which is how studies are planned before data are collected. The course objectives ask you both to compute power for a given test and to explain what the result means.
Chi-square tests for count data
The last topics of the semester test hypotheses about categorical data using observed and expected counts.
The statistic sums (O − E)²/E across categories and is compared with a chi-square distribution, provided every expected count is at least five.
A goodness-of-fit test compares one categorical variable with specified probabilities, using k − 1 degrees of freedom, reduced by one for each parameter estimated from the data.
Independence versus homogeneity
Both tests use a two-way table, expected counts equal to row total times column total divided by the grand total, and (r − 1)(c − 1) degrees of freedom. The difference is the design.
A test of independence uses one sample classified by two variables and asks whether the variables are related. A test of homogeneity uses separate samples from several populations, with the group sizes fixed in advance, and asks whether the populations share the same distribution.
The arithmetic is identical; the hypotheses and conclusions are worded differently.
Power and planning together
Power connects the two halves of inference. A confidence interval that is too wide and a test with low power are symptoms of the same problem, too little information, and the cure is also the same: a larger sample or less noisy measurement.
Reporting power alongside a non-significant result tells a reader whether the study could realistically have detected an effect of the size that matters.
Reading the result
Large values of the chi-square statistic are evidence against the null hypothesis, so these tests are upper-tailed.
When a result is significant, comparing observed with expected counts cell by cell shows which categories drive the difference.
What this chapter covers
- 01
Power and Type II error
- 02
Power functions
- 03
Sample size for a target power
- 04
Chi-square goodness of fit
- 05
Tests of independence
- 06
Tests of homogeneity
Worked example · free
Goodness of fit with three categories
- 1Expected counts under H0 are 50, 30 and 20, all at least 5.
- 2Q = 4/50 + 1/30 + 9/20 = 0.080 + 0.033 + 0.450 = 0.563.
- 2Degrees of freedom = 3 − 1 = 2, and the upper 5% chi-square critical value with 2 degrees of freedom is 5.991.
- 10.563 is far below 5.991, so do not reject: the choices fit the expected ratio.
Key terms
- Power
- The probability that a test rejects the null hypothesis when a specified alternative is true, equal to 1 − β.
- Power Function
- The probability of rejection viewed as a function of the true parameter value.
- Goodness-of-Fit Test
- A chi-square test of whether observed category counts match probabilities specified by a hypothesis.
- Test of Homogeneity
- A chi-square test of whether several separately sampled populations share the same categorical distribution.
Power and Chi-Square Tests FAQ
What is the difference between a test of independence and a test of homogeneity?
Independence uses one sample classified by two variables; homogeneity uses separate samples from several populations. The expected counts and degrees of freedom are computed the same way.
How can I increase the power of a test?
Take a larger sample, accept a larger significance level, reduce measurement variability, or target a larger effect. Each makes it more likely that a real difference produces a rejection.
Why must expected counts be at least five?
The chi-square distribution is only an approximation to the true sampling distribution of the statistic. When expected counts are small, the approximation becomes unreliable and p-values can be badly wrong, so combine sparse categories or use an exact method.
Exam move
For power, always draw two curves, one centred at the null value and one at the alternative, and shade the rejection region under the second. For count data, practise deciding the test from the sampling design before computing any expected count.
Working through Power and Chi-Square Tests in STAT 400? Sia is AskSia’s AI Statistics tutor — ask any STAT 400 Power and Chi-Square Tests question and get a clear, step-by-step explanation grounded in how STAT 400 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
