Question

The following contingency table represents the voters’ opinion concerning a new tax

reform bill and their level of income.

Levels of Income

Opinion Low Medium High

For 213 203 182

Against 138 110 154

Apply chi-square (χ2

) test to examine whether the opinion of people on new tax bill is

independent from their income level. [Table value of chi- square (χ2

) at 5% level of

significance and for 2 degrees of freedom is 5.99

Answer 1

The null hypothesis to be tested is,

$H_0:$ The tax opinion is independent from the income level

against the alternative,

$H_1:$ The tax opinion is not independent from the income level

The expected counts in the 6 possible combinations are derived from the formula,

$E_{ij}={r_i\times c_j\over n}$ where $r_i$ is the total for row $i$, $c_j$ is the total for column $j$, $n=1000$ is the sample size for $i=1,2$ and $j=1,2,3$

The expected counts are,

$E_{11}={r_1\times c_1\over 1000}={598\times351\over1000}=209.898$

$E_{21}={r_2\times c_1\over1000}={402\times351\over1000}=141.102$

$E_{12}={r_1\times c_2\over 1000}={598\times313\over1000}=187.174$

$E_{22}={r_2\times c_2\over1000}={402\times313\over1000}=125.826$

$E_{13}={r_1\times c_3\over1000}={598\times336\over1000}=200.928$

$E_{23}={r_2\times c_3\over1000}={402\times336\over1000}=135.072$

The test statistic is given as,

$\chi^2_c=\sum\sum{(O_{ij}-E_{ij})^2\over E_{ij}}$

Therefore,

$\chi^2_c={(213-209.898)^2\over209.898}+{(182-200.928)^2\over200.928}+{(203-187.174)^2\over187.174}+{(154-135.072)^2\over 135.072}+{(138-141.102)^2\over141.102}+{(110-125.826)^2\over 125.826}=7.87821$

The $\chi^2_{0.05,2}=5.99$ as given above and the null hypothesis is rejected if $\chi^2_c\gt\chi^2_{0.05,2}$.

Since $\chi^2_{c}=7.87821\gt\chi^2_{0.05,2}=5.99$, we reject the null hypothesis and conclude that there is no sufficient evidence to show that the opinion of people on new tax bill is independent from their income level at 5% level of significance.