Hypothesis Tests involving Two Population Means or Proportions Report 2022

Chapter 10: Hypothesis Tests involving Two Population Means or Proportions
1. Some criminologists argue there is a relationship between “impulsivity” and criminal
offending. The idea is that impulsive people act on immediate gratification and that since crime
involves quick pleasure and only the long-term possibility of any cost (getting caught and
punished), it should be highly attractive to them. To test this notion you take a random sample of
120 people, you give them a personality test that includes a measure of impulsivity. Based on
this test, you divide your sample into two groups: (1) the non-impulsive group (n = 80) and, (2)
the impulsive group (n = 40). You then ask each person to report the number of criminal offenses
they have committed in the last year. Finally, you calculate the mean number of self-reported
offenses for each group, and here is the data you get:
Impulsive Non-Impulsive
n1 = 40 n2 = 80
̄X1 = 13.5 ̄X2 = 10.3
s1 = 4.9 s2 = 4.0
a) What are the independent and dependent variables in your study? How are they measured
(nominal, ordinal, interval, ratio)?
Answer: IV: Impulsivity - Nominal
DV: Number of criminal offences - Ratio
b) Test the null hypothesis that there is no difference between the two groups versus the
alternative hypothesis that those who are impulsive commit more criminal offenses. Use
an alpha of .01 and assume that the two population standard deviations are equal (σ1 = σ2),
and make sure to properly interpret your results.
Answer:
Step 1:
H0: μ1 = μ2 (There is no difference between the two groups)
HA: μ1 > μ2 (Those who are impulsive commit more criminal
offenses)
Step 2:
This is right-tailed test.
Step 3: α = 0.01
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df = (n1 – 1) + (n2 – 1) = n1 + n2 – 2 = 40 + 80 – 2 = 118
tcrit = t0.01, 118 = 2.358
Reject the Null if the test statistic is greater than the tcrit
Step 4:
tobt= ̄X1− ̄X2
( √ ( n1−1 ) s12+( n2−1)s22
n1+ n2−2 ) ( √ n1+n2
n1⋅n2 )
=
13.5−10.3
( √ ( 40−1 ) 4.92+ ( 80−1 ) 4.02
40+80−2 ) ( √ 40+80
40∗80 )
=
3.2
( √ 936.39+1264
118 )( √ 120
3200 ) = 3.2
( 4.318 ) ( 0.194 )
= 3.2
0.8377 = 3.820
Step 5: We reject the null hypothesis because the test statistic is 3.820 > 2.358. Therefore, we
can conclude that there is sufficient statistical evidence, at 1% significance level, to support
the claim that that those who are impulsive commit more criminal offenses.
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