Independence would mean the proportion favouring the policy is the same for both genders; the chi-square test compares the actually observed cell counts against the counts we would expect if that were exactly true. …

Step 1 - Hypotheses:

$H_0$: Attitude towards the policy is independent of gender.

$H_1$: Attitude towards the policy is associated with (not independent of) gender.

Step 2 - Expected frequencies, $E = \dfrac{\text{Row Total}\times\text{Column Total}}{\text{Grand Total}}$:

$E_{11} = \dfrac{50\times55}{100} = 27.5$ (Male, For)

$E_{12} = \dfrac{50\times45}{100} = 22.5$ (Male, Against)

$E_{21} = \dfrac{50\times55}{100} = 27.5$ (Female, For)

$E_{22} = \dfrac{50\times45}{100} = 22.5$ (Female, Against)

Step 3 - Test statistic:

Check (cross-verify with the 2x2 shortcut formula): with $a=30, b=20, c=25, d=25$, $N=100$:

$\chi^2 = \dfrac{N(ad-bc)^2}{(a+b)(c+d)(a+c)(b+d)} = \dfrac{100\times(30\times25-20\times25)^2}{50\times50\times55\times45} = \dfrac{100\times250^2}{6{,}187{,}500} = \dfrac{6{,}250{,}000}{6{,}187{,}500} \approx 1.01$ …

The 2x2 shortcut formula chi-square = N(ad-bc)^2 / [(a+b)(c+d)(a+c)(b+d)] gives the same result directly from the observed table without computing expected frequ …

A frequent error is using the observed row/column totals as if they were the expected frequencies, or forgetting to square the (O-E) difference before dividing by E. Also, degrees o …