Chi-square: opinion change before election

A chi-square test of homogeneity (equivalently independence) is used: the opinion distribution is compared across the two polls (3 × 2 table, N = 400).

1. H0: the distribution of opinion is the same in both polls (no change in opinion). H1: the distribution of opinion differs between the polls (there has been a change).
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:

                  2 wk    4 wk   Total
Republican          79      91     170
Democratic          84      66     150
Undecided           37      43      80
Total              200     200     400

Expected frequencies:

                  2 wk    4 wk   
Republican       85.00   85.00
Democratic       75.00   75.00
Undecided        40.00   40.00

Terms (O − E)²/E:

                  2 wk    4 wk   
Republican       0.424   0.424
Democratic       1.080   1.080
Undecided        0.225   0.225

χ² = Σ (O − E)²/E = 3.457

4. Degrees of freedom = (r − 1)(c − 1) = (3 − 1)(2 − 1) = 2. Critical value: χ²(0.05; 2) = 5.991. Reject H0 if χ² > 5.991.
5. Decision: calculated χ² < tabulated χ² (3.457 < 5.991, p-value = 0.1775), so fail to reject H0.

Conclusion: there is no significant change in opinion between the two polls; the difference is due to sampling variation.