Cogito
AP Statistics · Chapter 7 · Lesson 1
Chi-Square Goodness of Fit
Do the counts match the claimed distribution?
12 problems · about 22 minutes · VAR-8.A, DAT-3.M
What this lesson teaches
The student carries out a chi-square goodness-of-fit test.
- Goodness of fit compares observed counts against a claimed distribution.
- The statistic sums (observed − expected)² ÷ expected.
- Degrees of freedom are categories minus 1, and every expected count must reach 5.
Warm Up
Straightforward practice. Get the method working first.
5 problems150 observations across 5 equally likely categories. Each expected count?
Answer 30
Why 30.
Why is each term divided by the expected count?
Answer It scales the gap, so a fixed difference matters more in a small category.
Why It measures the gap relative to what was expected.
200 observations across 5 equally likely categories. Each expected count?
Answer 40
Why 200 ÷ 5.
Six categories. Degrees of freedom?
Answer 5
Why Categories minus 1.
Observed 30, expected 25. What is (observed − expected) squared over expected?
Answer 1
Why 25 ÷ 25.
Build It Up
The same ideas with more to keep track of.
3 problemsEvery expected count must be at least what number?
Answer 5
Why The condition.
300 observations, claimed proportion 0.2. Expected count?
Answer 60
Why 300 × 0.2.
Should expected counts be rounded to whole numbers? 1 yes, 0 no.
Answer 0
Why They are expectations, not counts.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsPut the goodness-of-fit procedure in order.
Answer 1. State hypotheses about the claimed distribution. 2. Compute the expected count for each category. 3. Check that all expected counts reach 5. 4. Compute the chi-square statistic and the p-value.
Why The condition is checked on the expected counts.
80 observations across 4 equally likely categories. Each expected count?
Answer 20
Why 80 ÷ 4.
The Expected Count: 100 observations across 4 equally likely categories. What is each expected count?
Answer 25
Why 25.
The Degrees of Freedom: Four categories in a goodness-of-fit test. Degrees of freedom?
Answer 3
Why 3.