Answer to Question #152035 in Statistics and Probability for JG

Question

The following table represents a sample of 500 people and their opinion on Tax Reform. Do we have enough evidence to conclude that Party Affiliation and Opinion on Tax Reform are dependent on 5% level of significance.

Affiliation/Opinion on Tax | Reforms | Favor | Indifferent | Opposed
---------------------------------------------------------------------------------------------------------------
Democrat
---------------------------------------------------------------------------------------------------------------
Republican

Answer 1

To test independence of two attributes we use chi-square test.

The given information is:

number of rows m=2

number of column n=3

Hypothesis:

H₀ - The Party Affiliation and Opinion on Tax Reform are independent.

H₁ - The Party Affiliation and Opinion on Tax Reform are dependent.

Test statistic:

$χ^2 = \sum_{n}^{j=1} \sum_{m}^{i=1} \frac{(O_{ij}-E_{ij})^2}{E_{ij}} \\ χ^2 = \sum_{n}^{j=1} \sum_{m}^{i=1} \frac{O_{ij}^2}{E_{ij}} - N$

Is follow χ² distribution with (m-1)(n-1) degrees of freedom.

O​​​​​_ij - observed frequency i​​​​​​^throw and j​​​​​​^th column

E​​​​​​_ij - expected frequency i​​​​​​^th row and j​​​​​​^th column

$E_{ij} = \frac{(i^{th} \; row \; total) \times (j^{th} \;column \;total)}{N} \\ E_{11} = \frac{R_1 \times C_1}{N} \\ = \frac{285 \times 202}{500} = 115.14 \\ E_{12} = \frac{R_1 \times C_2}{N} \\ = \frac{285 \times 150}{500} = 85.5 \\ E_{13} = \frac{R_1 \times C_3}{N} \\ = \frac{285 \times 148}{500} = 84.36 \\ E_{21} = \frac{R_2 \times C_1}{N} \\ = \frac{215 \times 202}{500} = 86.86 \\ E_{22} = \frac{R_2 \times C_2}{N} \\ = \frac{215 \times 150}{500} = 64.5 \\ E_{23} = \frac{R_2 \times C_3}{N} \\ = \frac{215 \times 148}{500} = 63.64 \\ χ^2 = \frac{138^2}{115.14} + \frac{83^2}{85.5} + \frac{64^2}{84.36} + \frac{64^2}{86.86} + \frac{67^2}{64.5} + \frac{84^2}{63.64} - 500 \\ = 22.0147$

Critical value or table value at

$(m-1)(n-1)=(2-1)(3-1)= 1 \times 2=2$ degrees of freedom and 5 % level of significance.

$Tab \; χ^2 = 5.991$

$Cal \; χ^2 > Tab \; χ^2$

Reject null hypothesis at 5 % level of significance.

There is enough evidence to conclude that the Party Affiliation and Opinion on Tax Reform are dependent at 5 % level of significance.