Question

A study of people who refused to answer survey questions provided the randomly selected sample data shown in the table below (based on data from “I Hear You Knocking But You Can’t Come In,” by Fitzgerald and Fuller, Sociological Methods and Research,Vol. 11, No. 1). At the 0.01 significance level, test the claim that the cooperation of

the subject (response or refusal) is independent of the age category. Does any particular age group appear to be particularly uncooperative?

	Age
	18-21	22-29	30-39	40-49	50-59	60 and over
Responded	73	255	245	136	138	202
Refused	11	20	33	16	27	49

Short answer

The cooperation of the subject (response or refusal) is dependent on the age category.

The highest category of uncooperative subjects was in the age group of 60 and over.

Step-by-step solution

Given information

The data for the subject’s cooperation (response or refusal) and of the age category is provided.

Check the requirements of the test

Assume the subjects are randomly selected for the study

Use theformula for expected frequency as stated below,

\(E = \frac{{\left( {row\;total} \right)\left( {column\;total} \right)}}{{\left( {grand\;total} \right)}}\)

The observed frequency table along with row and column total is represented as,

	18-21	22-29	30-39	40-49	50-59	60 and over	Row total
Responded	73	255	245	136	138	202	1049
Refused	11	20	33	16	27	49	156
Column total	84	275	278	152	165	251	1205

Theexpected frequency tableis represented as,

	18-21	22-29	30-39	40-49	50-59	60 and over
Responded	73.125	239.398	242.010	132.322	143.639	218.505
Refused	10.875	35.602	35.990	19.678	21.361	32.495

Here, all expected frequencies are greater than 5, which implies the requirements of the test are satisfied.

State the null and alternate hypothesis

The hypotheses are formulated as follows:

\({H_0}:\)The cooperation of the subject (response or refusal) is independent of age.

\({H_1}:\)The cooperation of the subject (response or refusal) is dependent of age.

Compute the test statistic

The value of the test statisticis computed as,

\[\begin{aligned}{c}{\chi ^2} = \sum {\frac{{{{\left( {O - E} \right)}^2}}}{E}} \\ = \frac{{{{\left( {73 - 73.125} \right)}^2}}}{{73.125}} + \frac{{{{\left( {255 - 239.398} \right)}^2}}}{{239.398}} + ... + \frac{{{{\left( {49 - 21.361} \right)}^2}}}{{21.361}}\\ = 20.271\end{aligned}\]

Therefore, the value of the test statistic is 20.271.

Compute the degrees of freedom

The degrees of freedomare computed as,

\(\begin{aligned}{c}\left( {r - 1} \right)\left( {c - 1} \right) = \left( {2 - 1} \right)\left( {6 - 1} \right)\\ = 5\end{aligned}\)

Therefore, the degrees of freedom are 5.

Compute the critical value

From the chi-square table, the critical value for row corresponding to 5 degrees of freedom and at 0.01 level of significance 15.086.

The p-value is obtained as 0.0011.

State the decision

Since the critical value (15.086) is less than the value of the test statistic (20.271). In this case, the null hypothesis is rejected.

Therefore, the decision is to reject the null hypothesis.

State the conclusion

There is insufficient evidence to support claimthat the cooperation of the subject (response or refusal) is independent of age.

Thus, the cooperation of a subject is dependent on this age.

The proportions of subjects who were uncooperative in each category are stated below.

	18-21	22-29	30-39	40-49	50-59	60 and over
Refused	0.1309\(\left( {\frac{{11}}{{84}}} \right)\)	0.0727 \(\left( {\frac{{20}}{{275}}} \right)\)	0.1187 \(\left( {\frac{{33}}{{278}}} \right)\)	0.1053 \(\left( {\frac{{16}}{{152}}} \right)\)	0.1636 \(\left( {\frac{{27}}{{165}}} \right)\)	0.1952 \(\left( {\frac{{49}}{{251}}} \right)\)

From the results, the most proportion of uncooperative subjects were in the age category 60 and over.