Question

In: Statistics and Probability

A political researcher wishes to know if political affiliation and age are related. He has collected...

A political researcher wishes to know if political affiliation and age are related. He has collected data on 380380 persons in three age categories. Is there evidence that age and political affiliation are related?

Age Democrat Republican Independent Total
1818-3131 4646 4949 2626 121121
3232-5151 5656 5555 3131 142142
5252-6868 4646 4646 2525 117117
Total 148148 150150 8282 380380

Copy Data

Step 4 of 8:

Find the value of the test statistic. Round your answer to three decimal places

Solutions

Expert Solution

Let denote the observed and expected frequencies in the given cells of the cross tab data. Here, Ei, i = 1,2,..,n are the cell frequencies we would have obtained when the variables Political affiliation and Age were independent.

We are asked to test whether the observed frequencies are the same as what is expected.

To test:

H0: There is no association between political affiliation and age Vs Ha: There is a significant association between political affiliation and age

Vs

The appropriate statistical test to test the existence of relationship between the two categorical variables would be a Chi-square test of Association, where we test whether the observed distribution is the same as the expected. The test statistic is given by:

where r = No. of rows and c = No. of columns; with rejection region of the test given by,

Here, r = 3 and c = 3.

To obtain the expected frequency off a cell, we may divide the product of its corresponding row and column totals by the grand total:

Age Democratic Republican Independent
18-31 (148x121)/380 = 47.126 (150x121)/380 = 47.763 (82x121)/380 = 26.111
32-51 (148x142)/380 = 55.305 (150x142)/380 = 56.053 (82x142)/380 = 30.642
52-68 (148x117)/380 = 45.568 (150x117)/380 = 46.184 (82x117)/380 =25.247

Substituting the values in the test statistic:

Oi Ei (Oi-Ei)2 (Oi-Ei)2 / Ei
46 47.126 1.268 0.027
56 55.305 0.483 0.009
46 45.568 0.187 0.004
49 47.763 1.530 0.032
55 56.053 1.109 0.020
46 46.184 0.034 0.001
26 26.111 0.012 0.000
31 30.642 0.128 0.004
25 25.247 0.061 0.002
0.099

Hence, the test statistic obtained is:

0.099


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