CHAPTER 9: TESTING HYPOTHESIS WITH CATEGORICAL DATA

SOLUTIONS

1. The following table reflects data on shoplifting from a local store:

Shoplifter (S) / Not Shoplifter(NS) / Total
Male(M) / 6 / 12 / 18
Female(F) / 25 / 7 / 32
Total / 31 / 19 / 50

Use probabilities to determine if these two variables are independent or dependent.

P(A) = P(A | B) is the test for independence.

P(S) = 31/50 (.62) P(S | M) = 6/18 (.33) P(S | F) = 25/32 (.78)

Since P(A) ≠ P(A | B), the variables are not independent.

Conduct a hypothesis test using an alpha of .01 that these variables are independent versus the alternative that they are related.

Step 1: H0: χ² = 0 [gender and shoplifting are independent/ no relationship]

H1: χ² > 0 [gender and shoplifting are dependent / there is a relationship]

Step 2: Chi-square test for independence

Step 3: α = .01 df = (2-1)(2-1) = 1 χ²crit = 6.635 If χ²obt > 6.635, reject null

Step 4:

Cell / fo / / fo – fe / (fo – fe)² /
A / 6 / (18*31)/50 = 11.16 / -5.16 / 26.63 / 2.39
B / 12 / (18*19)/50 = 6.84 / 5.16 / 26.63 / 3.89
C / 25 / (32*31)/50 = 19.84 / 5.16 / 26.63 / 1.34
D / 7 / (32*19)/50 = 12.16 / -5.16 / 26.63 / 2.19
S / 50 / 0 / 9.81

χ²obt = 9.81

Step 5: I reject my null as my χ²obt of 9.81 is greater than the χ²crit of 6.635. There is sufficient evidence to suggest that one’s gender is related to their likelihood of being a shoplifter.

What is the strength of the relationship between sex and shoplifting?

Using gamma or Yule’s Q, the strength of the relationship is:

This implies that there is a strong inverse or negative relationship between gender and shoplifting. Females are more likely to shoplift than males.

2. You want to know if being the victim of crime causes a person to have greater fear of crime. You take a random sample of 75 people and ask them about the number of times they have been criminally victimized and their fear of crime. This is your data:

OBSERVED FREQUENCIES

Fear of Crime
Victimizations / Low / Medium / High / Total
0 / 13 / 7 / 3 / 23
1+ / 12 / 16 / 24 / 52
Total / 25 / 23 / 27 / 75

a). What are the independent and dependent variables? How are each measured?

Answer: IV: Victimization (Ordinal)

DV: Fear of Crime (Ordinal)

b). Compare the probability of someone having high fear of crime given that she has been victimized to the probability of high fear of crime given no victimization. Does there appear to be a relationship between victimization and fear of crime?

Answer: P(High | Victim) = 24/52 = .462

P(High | No Victim) = 3/23 = .13: Victimizations appear related to fear of crime

c). Calculate a table of expected frequencies. Assume that the null hypothesis is that the victimization and fear of crime are independent (not related to one another).

EXPECTED FREQUENCIES

Fear of Crime
Vicitmizations / Low / Medium / High / Total
0 / 7.67 / 7.05 / 8.28 / 23
1+ / 17.33 / 15.95 / 18.72 / 52
Total / 25 / 23 / 27 / 75

d). Test the null hypothesis that the number of victimizations is independent of fear of crime against the alternative hypothesis that they are significantly related to one another. Use an alpha of .05. What would you conclude from this test?

Answer:

Step 1: H0: χ2 = 0

H1: χ2 > 0

Step 2: χ2 Distribution; χ2 Test for Independence for 2 Categorical Variables

Step 3: α = .05 df = (2-1)(3-1) = 2 χ2crit = 5.99 Reject if c2obt > 5.99

Step 4:

Cell / fo / fe / fo – fe / (fo – fe) 2 /
A / 13 / 7.67 / 5.33 / 28.41 / 3.70
B / 7 / 7.05 / -0.05 / 0.00 / 0.00
C / 3 / 8.28 / -5.28 / 27.88 / 3.37
D / 12 / 17.33 / -5.33 / 28.41 / 1.64
E / 16 / 15.95 / 0.05 / 0.00 / 0.00
F / 24 / 18.72 / 5.28 / 27.88 / 1.49
Σ = 10.2

Step 5: 10.2 > 5.99 so we reject the null. I can conclude that number of victimizations is related to fear of crime.

e). This suggests a moderate to strong, negative relationship between victimization and fear of crime. Being victimized significantly increases one’s fear of crime.

f). Using gamma, the strength of the relationship is:

This implies that there is a moderate inverse or negative relationship between experiencing victimizations and fear of crime. Those who are victimized have a higher level of fear of crime.