Question

Party Affiliation and Opinion on Tax Reform example. 83 favor indifferent opposed total democrat 138 64 285 republican 64 67 84 215 total 202 150 148 500 Here are the formulae. Given that chi square = a (f. – fo)? /fe , and that fe= (row marginal*column marginal)/sample size, and that df = (# rows - 1) * (# columns - 1)

Answer 1

(a)

Number of rows $\tiny m=2$

Number of columns $\tiny n=3$

So, degrees of freedom

v = (m 1) (n − 1) = 2

Hence, degrees of freedom associated with the given table is 2.

(b)

We have to test for the null hypothesis $\tiny H_0:\text{Party affiliation and opinion on tax reform are independent}$

against the alternative hypothesis $\tiny H_1:\text{Party affiliation and opinion on tax reform are not independent}$

(c)

In the actual situation we have as follows.

Observed frequencies Row total Party Democrat affiliation Republican Column total Opinion on tax Favor Indifferent Opposed 138 83 64 64 67 84 202 150 148 285 215 500

Expected frequencies Row total Party Democrat affiliation Republican Column total Opinion on tax Favor Indifferent Opposed 115.14 85.50 84.36 86.86 64.50 63.64 202.00 150.00 148.00 285.00 215.00 500.00

Chi_square Row total Party Democrat affiliation Republican Column total Opinion on tax Favor Indifferent Opposed 4.5386 0.0731 4.9138 6.0163 0.0969 6.5137 10.5550 0.1700 11.4275 9.5256 12.6269 22.1525

$\tiny \therefore \chi^2_{calculated}=\sum\frac{\left(f_o-f_e\right)^2}{f_e}=22.1525$

In the moderated situation,

$\tiny f^*_o=\frac{f_o}{10}$

$\tiny f^*_e=\frac{f_e}{10}$

$\tiny \therefore \chi^2_{calculated}=\sum\frac{\left(f_o^*-f_e^*\right)^2}{f_e^*}=\sum\frac{\left(f_o/10-f_e/10\right)^2}{f_e/10}=\frac{1}{10}\sum\frac{\left(f_o-f_e\right)^2}{f_e}=2.21525$

Thus the value of chi square for the table is 2.21525.

(d)

Critical value is given by $\tiny \chi^2_{critical}=5.99$

We reject our null hypothesis if $\tiny \chi^2_{calculated}>\chi^2_{critical}$

Here, we observe that $\tiny \chi^2_{calculated}=2.21525\ngtr5.99=\chi^2_{critical}$

So, we cannot reject our null hypothesis.

Hence, based on the given data we can conclude that there is no significant evidence that party affiliation and opinion on tax reform are independent.