Question

For each exercise, answer the following along with any additional questions.  Select and justify the best test(s). The chi-square, Phi, Yates, or Lambda (or even a combination) might be best for a problem given the data and research question. Do not assume the independent is always on the row.  Provide the null and alternative hypotheses in formal and plain language for the appropriate test at the 0.05 significance level.  Do the math and reject/retain null at a=.05. State your critical value.  Explain the results in plain language.

1. A local United Way chapter is interested in testing if employment status relates to volunteering. They collect a random sample of 150 local adult residents. (C15PROB1.SAV)

Employment Volunteers Does Not Volunteer   

Full Time 25 20

part time 13 9

unemployed 8 20

Retired 35 20

Answer 1

• Let denote the observed and expected frequencies for the given cells of the cross tab data.

Employment	Volunteer	Does not volunteer	Total
Full time	25	20	45
Part time	13	9	22
Unemployed	8	20	28
Retired	35	20	55
Total	81	69	150

Here, E_i, i = 1,2,..,n are the cell frequencies we would have obtained when the variables Employment status and Volunteering were unrelated/independent.

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

• To test:

H₀: There is no association/relationship between Employment status and Volunteering Vs H_a: There is a significant association/elationship between Employment status and Volunteering

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 counts are the same as the expected, and if yes, we would fail to establish a relationship between the two variables.. 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 = 4 and c = 2. Hence, df = (4-1)(2-1) = 3 degrees of freedom.

• We may reject the null hypothesis if the observed test statistic ; From chi square table,

Hence, we reject H₀ if

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

Employment	Volunteer	Does not volunteer
Full time	81x45/150=24.30	69x45/150=20.70
Part time	81x22/150=11.88	69x22/150=10.12
Unemployed	81x28/150=15.12	69x28/150=12.88
Retired	81x55/150=29.70	69x55/150=25.30

Oi	Ei	(Oi-Ei)2	(Oi-Ei)2 / Ei
25	24.3	0.49	0.02
20	20.7	0.49	0.02
13	11.88	1.25	0.11
9	10.12	1.25	0.12
8	15.12	50.69	3.35
20	12.88	50.69	3.94
35	29.7	28.09	0.95
20	25.3	28.09	1.11
			9.62

Hence, the test statistic obtained is:

9.62

Since, 9.62 > 7.81 lies in the rejection region, we may reject H₀ at 5% level. We may conclude that the data provide sufficient evidence e to support the claim that employment status relates to volunteering.