Question

Recent reports suggest that children who grow up with family members who use drugs are more likely to be arrested. To test this phenomenon, a researcher interviews a sample of n = 120 college students. Each student is asked about family member drug use during their childhood and about his or her criminal history. Number of drug using family members 0 1    2 or More No Arrest 22 50 18 90 Arrest 18 10 2 30 40 60 20 Do the data indicate a significant relationship between family member drug use and arrest? Test at the .05 level of significance. (Use 2 decimal places.) χ2-critical = A χ2 = B What is your decision regarding the null? Type either “ reject”> or “accept” in the box C Compute Cramér's V to measure the effect size (Use 3 decimal places). V = D How would you describe the effect size? Write small, medium, or large. E Please answer A, B, C, D, E. Thank you so much!

Answer 1

A

The degrees of freedom (DF) is equal to:

DF = (r - 1) * (c - 1)

where r and c is the number of levels for the categorical variable.

DF = (2 - 1) * (3 - 1) = 2

χ2-critical for .05 level of significance. and df = 2 is 5.99

B

The expected frequency counts are computed separately for each population at each level of the categorical variable, according to the following formula.

Er,c = (nr * nc) / n

where Er,c is the expected frequency count for population r at level c of the categorical variable, nr is the total number of observations from population r, nc is the total number of observations at treatment level c, and n is the total sample size.

Expected frequency count table is,

	0	1	2 or More	Total
No Arrest	(90 * 40)/120 = 30	(90 * 60)/120 = 45	(90 * 20)/120 = 15	90
Arrest	(30 * 40)/120 = 10	(30 * 60)/120 = 15	(30 * 20)/120 = 5	30
Total	40	60	20	120

χ2 = Σ [ (Or,c - Er,c)2 / Er,c ]

where Or,c is the observed frequency count in population r for level c of the categorical variable, and Er,c is the expected frequency count in population r for level c of the categorical variable.

χ2 = (22 - 30)^2 / 30 + (50 - 45)^2 / 45 + (18 - 15)^2 / 15 + (18 - 10)^2 / 10 + (10 - 15)^2 / 15 + (2 - 5)^2 / 5 = 13.16

C

As, χ2  > χ2-critical, we reject null hypothesis.

D

Cramér's V =

where n = Total number of observations and k = the smaller of the number of rows or columns.

n = 120

k - min(2, 3) = 2

Cramér's V = = 0.33

E

For df = 2, Cramer's V is less than 0.35, so, the effect size is medium.