L-REG-4: Solutions — Chi-Square Test of Independence

A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Critical values are rounded to three decimals, expected counts to two; values written with are rounded, and every decision is made at full precision.

Example 1 — Smoking and Exercise

The complete five-step solution is in the lesson: with , above 3.841, so reject (); , small.

Example 2 — Commuting by Age

Expected counts :

The smallest is ; the sample is random. Contributions : 3.701, 1.110, 3.528 / 4.318, 1.295, 4.116.

: reject (). There is enough evidence of an association between age group and main way to work; younger commuters bike or walk more and drive less than no link would predict. , small. A survey, so association only.

Example 3 — Cramér’s V

, so : medium.

Example 4 — A Test That Should Not Have Run

and are below 5, so the chi-square approximation is unreliable: combine categories that belong together or use Fisher’s exact test. And even a valid non-significant test says “not enough evidence of an association”, never “independent”.

A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Problem 1 — Reading a Two-Way Table

Compare the rows as percentages: the share of the first row in a column against the share of the second row in the same column. One cell on its own, the row totals, or “are the counts equal?” cannot show a link, because “no link” means the rows split in the same proportions, not that the counts are equal.

Problem 2 — Fill In the No-Link Table

Each cell is , to two decimals. Worked instance: 238 students, rows First year 97 and Upper year 141; columns Library 88, Home 104, Café 46.

Each row adds back to its total (97 and 141) and each column to its total. Two typed values have their own feedback: (an equal split: “no link means equal counts”) and (the row total used twice).

Problem 3 — May the Test Run? How Many Degrees of Freedom?

Condition. The expected counts are shown. The test may run only if every one is at least 5; the observed counts and the sample size do not decide it. The smallest expected count is always the smallest row total times the smallest column total, over . If it is below 5: combine categories that genuinely belong together, or use an exact test.

Degrees of freedom. : 1 for a table, 2 for or , 4 for . The typed values , , and each have their own feedback.

Problem 4 — Reading the Chi-Square Table

(a) . (b) . (c) but : reject at 0.05, fail to reject at 0.01. (d) In the row, , so .

Problem 5 — The Full Five-Step Test

The table is one of six curated tables. Hypotheses about independence; the smallest expected count (all are at least 5); to three decimals (adding contributions rounded to three or four decimals is also accepted); and ; then “sufficient” or “insufficient evidence of an association”, never “independent” and never “causes”.

A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Problem 1 — Is There a Link?

The table, the sample size (80 to 400, never a multiple of 5) and the significance level (, or , stated before the data) all change.

1. Hypotheses. : the two variables are independent; : they are not.
2. Condition. Compute every expected count. In about one table in six, one is below 5: then the right report is “no χ² p-value from this table: combine categories or use an exact test”, and the problem stops there.
3. Statistic. , to three decimals. Sums of contributions rounded before adding, or computed from expected counts rounded to two decimals, are also accepted.
4. df and the critical value. ; read the row and the column of the stated .
5. Decision. Reject when is above the critical value.

Worked instance. 238 students, :

Expected counts as in §5 Problem 2 (smallest 18.75). Contributions: 0.735, 0.684, 0.003 / 0.506, 0.471, 0.002, so ; and . Since , fail to reject (software: ).

A table that may not be tested. 118 commuters with column totals 61, 48 and 9, row totals 54 and 64: the smallest expected count is , below 5. Stop, whatever would come out as.

Problem 2 — What May the Report Say?

Five blanks: the critical value (the row , the column of the stated ); whether is above or below it; the decision (“accept” is never right); “enough” or “not enough evidence of an association” (“evidence that there is no association” is never right); and the design. In a survey, an association cannot by itself show that one variable affects the other. In a randomized experiment, where one variable was randomly assigned, an association can be read as its effect. The test is the same in both: the design decides.

For the 238 students at : critical value 4.605 (2 degrees of freedom); is below it; fail to reject; not enough evidence of an association; a survey, so even a link would not show cause.

Problem 3 — Find the Error

The workflow item corrupts one of five claims: df written as , dividing by instead of , checking the observed counts instead of the expected ones, “accept : independent”, or “causes”. The conclusion item corrupts the decision (backwards against the critical value), or the conclusion (“independent” after failing to reject, or “causes” after rejecting a survey result).

Problem 4 — How Strong?

with , to three decimals; the tables span all four bands. Worked instance: a table with and : , so , small. Typed values with their own feedback: without the root, , and .

Problem 5 — Which Analysis Fits?

Classify both variables first. Two categorical variables call for a χ² test (postal-code region and grocery chain; a 1–5 satisfaction label and campus; blood type and flu; area code and carrier). Two quantitative variables call for correlation and regression (sleep and exam score; weight and fuel use; temperature and cones sold; experience and salary). One of each needs another method (smoking and lung capacity; jersey number and points; province and rent; coffee size and minutes). Codes, labels and jersey numbers are categories even when they are written as numbers.

Problem 6 — Association or Cause?

A significant χ² from a survey shows association. Some cases have a lurking variable (stress behind coffee and insomnia; age behind gaming and glasses; hot weather behind ice cream and drownings); others have reverse causation (income makes owning a car affordable; people who are overweight may switch to diet soda; struggling students sign up for tutoring).

Problem 7 — Pooled or Split?

Severe cases: A , B . Mild cases: both 90%. A looks better pooled (78% against 73%) only because 80 of its 100 patients were mild cases, which recover more often whatever the treatment: severity is a lurking variable. The records suggest B is no worse, but because doctors chose the treatment, only a randomized trial would compare the treatments fairly.

Problem 8 — Physical Activity and Stress

Expected counts 21.54, 26.92, 21.54 / 18.46, 23.08, 18.46, all at least 5. , , above 5.991: reject (). , small. The reply: evidence of an association, of small strength; a requirement for every employee would need evidence of an effect (a randomized study) and of its benefits, harms and costs.

Mixed Review

Review 1 (L-INF-5). is right-sided because the suspicion came before the data. , compared with . Each context keeps realistic values (a city’s mean commute between 24 and 40 minutes, weekly study hours between 12 and 22).

Review 2 (L-INF-6). Two-sided; check and with the claimed ; , compared with 1.960.

A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Question 1 — Feynman Test

The model answer is in the lesson: a p-value measures evidence against independence, not strength; grows with , so report Cramér’s V too.

Question 2 — Expected Counts, and May the Test Run?

(a) Each cell to two decimals, as in §5 Problem 2; about one table in six has an expected count below 5 (the explanation says so). (b) Compute the smallest expected count yourself: the smallest row total times the smallest column total, over . At least 5: the test may run. Below 5: combine categories or use an exact test.

Question 3 — The Statistic and Its Degrees of Freedom

Solve on paper, then type to three decimals; the steps, if you open them, ask for the condition, , , the critical value at the stated , and the decision. The worked instance in §6 Problem 1 shows the method.

Question 4 — The Conclusion

As in §6 Problem 2: critical value, comparison, decision, evidence, design. The study design (survey or randomized experiment) is stated in the stem.

Question 5 — Which Analysis?

Classify both variables. A department number, a postal code or a rating used as a label is categorical; a measured amount is quantitative. Two categorical: χ² test. Two quantitative: correlation and regression. One of each: neither, in this module.

A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Path A — The Admissions Audit

The college is fictional and its records are generated. Worked instance:

1. The pooled gap. Sex and admission are both categorical: a χ² test on the table. , , far above 3.841 (; ). Not chance.
2. Inside each programme. X: 60% of men and 64% of women, . Y: 20% of men and 22% of women, . Neither is significant, and women’s rates are higher in both.
3. Who applied where. 250 of the 300 women (83%) applied to Y, which admits about one applicant in five of either sex; most men applied to X, which admits most. The programme is a lurking variable: Simpson’s paradox, as at Berkeley.
4. Verdict. No sign of a gap in the admission decisions; the records cannot by themselves prove or rule out bias in individual files.

In about one college in three, programme Y also admits women at a significantly lower rate: then part of the pooled gap is real, and Y’s decisions deserve a closer look.

Path B — The Fact-Checker

Claim 1 (a correlation on codes). The numbers only label categories, so means nothing: changing the codes would change it. Test the table with a χ² test. For example, region codes 1–3 against two grocery chains with counts 34, 28 / 41, 22 / 19, 37 (): , , above 5.991, so reject .

Claim 2 (a huge sample, a “strong link”). . For example, from a table with : , negligible. The link is real, but “strong” is wrong: the sample is huge.

Claim 3 (a cause from a survey). The data show an association; because they are observational, they cannot show cause. The clearest alternative is named in the claim’s context: a lurking variable (regular routines behind breakfast and grades; hot weather behind ice cream and drownings) or reverse causation (back pain keeping people from standing desks; struggling students signing up for tutoring).

A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Challenge 1 — The Same Percentages, a Bigger Survey

: 8.333 at , 16.667 at 400, 33.333 at 800. at every size. Every test rejects at 0.05, but the strength never changes: report V beside .

Challenge 2 — A 3×3 Table

, , far above 9.488; : small with the standard bands, medium with the bands for . The full solution is in the lesson.

Challenge 3 — The Lady Tasting Tea

All expected counts are : no χ² test. If she were guessing, each of the ways to choose four cups would be equally likely, so : evidence she can tell (Fisher’s exact test).