Question

11. It has always been claimed that being unemployed puts you at higher risk of committing a crime. To investigate this possibility, you examine the relationship between being unemployed in the past year and being arrested. You gather data from 170 persons and find out if they were employed at any time in the past year or not, and whether or not they were arrested in the past year. Here’s the data that you get:

OBSERVED FREQUENCIES

		Employed	Unemployed	Total
	No Arrest	65	25	90
	One or More Arrests	35	45	80
Total		100	70	170

1. What is the independent variable?

1. What is the dependent variable?

1. What is the unconditional risk of having one or more arrests?

1. What is the relative risk of having one or more arrests for those who were employed in the past year?

1. What is the relative risk of having one or more arrests for those who were unemployed in the past year?

1. Are the events of employment and arrest independent events? Explain.

1. Test the null hypothesis that employment/unemployment in the past year and arrest are independent against the alternative hypothesis that they are dependent or are related to one another. Use an alpha of .01 and state each step of your hypothesis test. What is your decision and what do you conclude? To help you out I have provided you with the table of expected frequencies below.

STEP 1:

STEP 2:

STEP 3:

STEP 4:

EXPECTED FREQUENCIES

		Employed	Unemployed	Total
	No Arrest			90
	One or More Arrests			80
Total		100	70	170

STEP 5:

1. Calculate an appropriate measure of association and describe the strength of the relationship between (if any) between unemployment and crime.

Answer 1

11.

Employment is the independent variable.

Arrest is the dependent variable.

Unconditional risk of having one or more arrests is

Relative risk of having one or more arrests for those who were employed in the past year is

Relative risk of having one or more arrests for those who were unemployed in the past year is

HYPOTHESIS TEST

We have to test for null hypothesis

against the alternative hypothesis

Under null hypothesis, expected frequencies are as follows.

Expecected frequencies	Employed	Unemployed	Row total
No arrest	90*100/170=52.94	90*70/170=37.06	90
One or more arrest	80*100/170=47.06	80*70/170=32.94	80
Column total	100.00	70.00	170

Our Chi-square test statistic is given by

Here,

Number of rows

Number of columns

denotes observed frequency of the cell in intersection of i-th row and j-th column.

Degrees of freedom

[Using R-code '1-pchisq(14.17788,1)']

Level of significance

We reject our null hypothesis if

Here, we observe that

So, we reject our null hypothesis.

Hence, based on the given data we can conclude that there is significent evidence that employment and arrest are not independent variable. So, these variables are dependent and related to one another.

MEASURE OF ASSOCIATION

An appropriate measure of association would be Yule's measure of association which van be calculated as follows.