Chi-Square Test Math Example 1

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Example 1

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A die is rolled 60 times. Observed: 1→8, 2→12, 3→9, 4→11, 5→13, 6→7. Conduct a chi-square goodness-of-fit test at α=0.05\alpha=0.05.

Solution

  1. 1
    Expected under H0H_0 (fair die): E=60/6=10E = 60/6 = 10 for each outcome
  2. 2
    χ2=∑(O−E)2E=(8−10)210+(12−10)210+(9−10)210+(11−10)210+(13−10)210+(7−10)210\chi^2 = \sum \frac{(O-E)^2}{E} = \frac{(8-10)^2}{10} + \frac{(12-10)^2}{10} + \frac{(9-10)^2}{10} + \frac{(11-10)^2}{10} + \frac{(13-10)^2}{10} + \frac{(7-10)^2}{10}
  3. 3
    =4+4+1+1+9+910=2810=2.8= \frac{4+4+1+1+9+9}{10} = \frac{28}{10} = 2.8
  4. 4
    df =6−1=5= 6-1 = 5; critical value χ0.05,52=11.07\chi^2_{0.05,5} = 11.07; since 2.8<11.072.8 < 11.07, fail to reject H0H_0

Answer

χ2=2.8<11.07\chi^2 = 2.8 < 11.07. Fail to reject H0H_0. No evidence the die is unfair.
The chi-square goodness-of-fit test compares observed frequencies to expected frequencies under a null model. Large χ2\chi^2 values (in the critical region) indicate the observed distribution differs from expected. Degrees of freedom = (categories - 1).

About Chi-Square Test

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

Learn more about Chi-Square Test →

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