Practice Chi-Square Test in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

A hypothesis test that compares observed frequencies to expected frequencies using the chi-square statistic to assess independence or goodness of fit.

You expect a die to land on each face about 16\frac{1}{6} of the time. You roll it 600 times and compare what you observed to what you expected. If the differences are small, the die is probably fair. If they're large, something is off. The chi-square statistic measures 'how far off are the observed counts from what we expected?'

Showing a random 20 of 50 problems.

Example 1

medium
A chi-square test produces χ2=12.3\chi^2=12.3 with df=5df=5, critical value 11.0711.07 at α=0.05\alpha=0.05. State the decision.

Example 2

medium
A goodness-of-fit test claims four candies should appear in ratio 1:1:1:11:1:1:1 among n=120n=120. What is the expected count per color?

Example 3

medium
A goodness-of-fit test on 4 candy colors with n=400n=400, equal expected. Observed: 90,110,95,10590,110,95,105. Compute χ2\chi^2.

Example 4

easy
A category had observed count O=30O = 30 and expected count E=25E = 25. Compute its contribution (O−E)2E\frac{(O-E)^2}{E} to the chi-square statistic.

Example 5

medium
Observed counts: 40,30,3040,30,30. Expected: 33.3,33.3,33.333.3,33.3,33.3. Compute χ2\chi^2.

Example 6

challenge
A genetics experiment expects ratio 9:3:3:19:3:3:1 in 160 plants. Observed: 84,33,30,1384,33,30,13. Compute χ2\chi^2.

Example 7

medium
In a 2×22 \times 2 table, the expected count for a cell is (row total)(column total)grand total\frac{(\text{row total})(\text{column total})}{\text{grand total}}. If row total =40= 40, column total =50= 50, grand total =200= 200, find the expected count.

Example 8

medium
For n=240n=240 split into 6 categories under uniform null. What is the expected count per category?

Example 9

hard
Why is the chi-square distribution right-skewed for small dfdf?

Example 10

easy
As the chi-square statistic gets larger, does the evidence against the null hypothesis get stronger or weaker?

Example 11

medium
A test of independence and a test of homogeneity use the same chi-square formula. What distinguishes them?

Example 12

medium
A goodness-of-fit test compares observed counts O=(20,30,50)O = (20, 30, 50) to expected counts E=(30,30,40)E = (30, 30, 40). Compute the chi-square statistic.

Example 13

easy
Chi-square tests are appropriate for which kind of data: categorical counts or continuous measurements?

Example 14

challenge
In a 2×32\times 3 table the smallest expected count comes out to 4.24.2. All others exceed 55. Why might the chi-square p-value be unreliable, and what is one fix?

Example 15

easy
For a chi-square test with observed=15, expected=20 for one category, calculate that category's contribution to the χ2\chi^2 statistic.

Example 16

easy
For one cell of a chi-square test, O=24O=24 and E=20E=20. Compute (O−E)2E\frac{(O-E)^2}{E}.

Example 17

easy
Two cells contribute (O−E)2E\frac{(O-E)^2}{E} values of 1.21.2 and 0.80.8. If these are the only cells, what is the chi-square statistic?

Example 18

medium
A claimed distribution is 50%,30%,20%50\%, 30\%, 20\% over n=200n = 200. Find the expected counts for the three categories.

Example 19

medium
One cell of a chi-square test has O=18O = 18, E=12E = 12. Compute that cell's contribution.

Example 20

hard
A 2×2 table: Men: 30 prefer A, 20 prefer B. Women: 15 prefer A, 35 prefer B. Test independence of gender and preference at α=0.05\alpha=0.05.
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