Chi-square: intelligence vs economic condition (1072)

A chi-square test of independence is used to test association between intelligence (GPA) and economic condition. Total = 1072.

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
Not good              81       185       190       106       562
Total                129       384       371       188      1072

Expected frequencies:

               Excellent      Good  Mediocre      Dull
Good               61.37    182.69    176.50     89.44
Not good           67.63    201.31    194.50     98.56

Terms (O − E)²/E:

               Excellent      Good  Mediocre      Dull
Good               2.913     1.457     0.115     0.619
Not good           2.644     1.322     0.104     0.562

χ² = Σ (O − E)²/E = 9.735

4. Degrees of freedom = (r − 1)(c − 1) = (2 − 1)(4 − 1) = 3. Critical value: χ²(0.05; 3) = 7.815. Reject H0 if χ² > 7.815.
5. Decision: calculated χ² > tabulated χ² (9.735 > 7.815, p-value = 0.0210), so reject H0.

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

Note: the value 5.991 printed in the question is the χ² value for 2 d.f. For a 2 × 4 table, d.f. = (2−1)(4−1) = 3 and the correct table value at 5% is 7.815. The decision is the same with either value because the calculated χ² is larger than both.