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An important application of chi-square involves using sample data to test for independence of two variables. Let us explain the concept by an example.

In an analysis of the election results inclination for the three categories of parties who are contesting the elections: Republicans, Democrats and Others, a news agency raised the question of whether preferences for the three categories differ among male and female voters.

Do the voting preferences differ significantly from the according to the gender? Use a 0.05 level of significance. Sample results for voting preferences of male and female voters (observed frequencies)

Step 1:State the hypotheses.State the null hypothesis and an alternative hypothesis.

H0: Gender and voting preferences are independent.

H1: Gender and voting preferences are not independent.

Step 2:Analyse sample data.Degree of freedom and expected frequency is calculated.

DF = (r - 1) * (c - 1) = (2 - 1) * (3 - 1) = 2

E_{r,c} = (n_{r} * n_{c}) / n

E_{1,1} = (400 * 450) / 1000 = 180000/1000 = 180

E_{1,2} = (400 * 450) / 1000 = 180000/1000 = 180

E_{1,3} = (400 * 100) / 1000 = 40000/1000 = 40

E_{2,1} = (600 * 450) / 1000 = 270000/1000 = 270

E_{2,2} = (600 * 450) / 1000 = 270000/1000 = 270

E_{2,3} = (600 * 100) / 1000 = 60000/1000 = 60

X^{2} = Σ [ (O_{r,c} - E_{r,c})^{2} / E_{r,c} ]

X^{2} = (200 - 180)^{2}/180 + (150 - 180)^{2}/180 + (50 - 40)^{2}/40
+ (250 - 270)^{2}/270 + (300 - 270)^{2}/270 + (50 - 60)^{2}/60

X^{2} = 400/180 + 900/180 + 100/40 + 400/270 + 900/270 + 100/60

X^{2} = 2.22 + 5.00 + 2.50 + 1.48 + 3.33 + 1.67 = 16.2

We use the Chi-Square Distribution value at 5% significance level = 5.991

Step 4: Interpret results. Since the chi-value (5.991) is less than the calculated value of 16.2, we cannot accept the null hypothesis. Thus, we conclude that gender influences the decision of voting.

We use the Chi-Square Distribution value at 5% significance level = 5.991

Step 4: Interpret results. Since the chi-value (5.991) is less than the calculated value of 16.2, we cannot accept the null hypothesis. Thus, we conclude that gender influences the decision of voting.