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Chi-square: intelligence vs economic condition (1650)

Short answer

Chi-square test yields χ²=149.513 with 9 d.f., critical value 16.919 at 5%. Reject H0; there is a highly significant association between intelligence and economic condition.

In a recent survey 1650 Engineers were classified according to their intelligence (GPA in Bachelor) and economic condition after graduation. Test whether there is any association between intelligence and economic condition. [χ² value for 9 degree of freedom = 16.919]

Economic Condition after graduationExcellentGoodMediocreDull
Good4819918182
Average80736589
Below average86515184
Not good81185190105

Chi-square test of independence for the 4 × 4 table. Grand total = 1650.

  1. H0: the two attributes are independent (no association). H1: the two attributes are associated (dependent).
  2. Level of significance: α = 0.05
  3. Test statistic: χ² = Σ (O − E)²/E, where E = (row total × column total)/grand total.

Observed frequencies with totals:

               Excellent      Good  Mediocre      Dull     Total
Good                  48       199       181        82       510
Average               80        73        65        89       307
Below avg             86        51        51        84       272
Not good              81       185       190       105       561
Total                295       508       487       360      1650

Expected frequencies:

               Excellent      Good  Mediocre      Dull
Good               91.18    157.02    150.53    111.27
Average            54.89     94.52     90.61     66.98
Below avg          48.63     83.74     80.28     59.35
Not good          100.30    172.72    165.58    122.40

Terms (O − E)²/E:

               Excellent      Good  Mediocre      Dull
Good              20.450    11.225     6.169     7.701
Average           11.489     4.899     7.239     7.238
Below avg         28.717    12.802    10.680    10.243
Not good           3.714     0.873     3.602     2.474

χ² = Σ (O − E)²/E = 149.513

  1. Degrees of freedom = (r − 1)(c − 1) = (4 − 1)(4 − 1) = 9. Critical value: χ²(0.05; 9) = 16.919. Reject H0 if χ² > 16.919.
  2. Decision: calculated χ² > tabulated χ² (149.513 > 16.919, p-value < 0.0001), so reject H0.

Conclusion: there is a significant association between the two attributes; they are not independent.

✓ Answer checked by Claude Sonnet 5.5

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