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8.3 Carrying Out a Chi-Square Test for Goodness of Fit

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Complete GOF Test: From Data to Conclusion

Problem: A spinner has 4 equal sectors (A, B, C, D). After 120 spins:

  • A: 22, B: 38, C: 31, D: 29

Test whether the spinner is fair at α=0.05{\alpha = 0.05}.

H: H0{H_0}: All sectors equally likely (pA=pB=pC=pD=0.25{p_A = p_B = p_C = p_D = 0.25}) vs. Ha{H_a}: The spinner is not fair.

A: Random ✓; 120<10%{120 < 10\%} of all spins ✓; E=120(0.25)=30≥5{E = 120(0.25) = 30 \geq 5} for all categories ✓

N: Chi-square goodness-of-fit test, df=4−1=3{df = 4 - 1 = 3}

T:
χ2=(22−30)230+(38−30)230+(31−30)230+(29−30)230\chi^2 = \frac{(22-30)^2}{30} + \frac{(38-30)^2}{30} + \frac{(31-30)^2}{30} + \frac{(29-30)^2}{30}
=64+64+1+130=13030≈4.333= \frac{64+64+1+1}{30} = \frac{130}{30} \approx 4.333

O: p=P(χ32≥4.333)≈0.228{p = P(\chi^2_3 \geq 4.333) \approx 0.228}

M/S: 0.228>0.05{0.228 > 0.05} → Fail to reject H0{H_0}. We do not have convincing evidence that the spinner is unfair.

In this unit

6 lessons

  1. 8.1Introducing Statistics: Are My Results Unexpected?Preview lesson
  2. 8.2Setting Up a Chi-Square Goodness of Fit TestPreview lesson
  3. 8.3Carrying Out a Chi-Square Test for Goodness of FitThis lesson
  4. 8.4Expected Counts in Two-Way TablesPreview lesson
  5. 8.5Setting Up a Chi-Square Test for Homogeneity or IndependencePreview lesson
  6. 8.6Carrying Out a Chi-Square Test for Homogeneity or IndependencePreview lesson

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