Question

Correctional services were interested in evaluating a cognitive-behavioural based intervention designed to reduce violence and offending. A group of researchers randomly assigned 18 offenders into either a control group or a treatment group. Offenders in the treatment group received the intervention for one year prior to their release while those in the control group received no intervention prior to their release. The researchers then gathered data on the number of violent incidents perpetrated by offenders over the five years after their release from the institution. The researchers are interested in testing whether the treatment had any effect on the outcome.

Control 4 7 4 6 8 5 9 7 4

Treatment 3 5 2 7 5 5 6 4 2

Please bold answers

a) What is the appropriate model of the population(s)?

b) What are the appropriate hypotheses for this analysis?

c) What is/are the critical value(s) for this test using an alpha of 0.05?

d) What is the observed value of the appropriate test statistic?

e) What is your decision regarding the stated hypotheses?

f) What is the estimated effect size using Cohen’s d?

g) What is the 95% confidence interval around the population mean difference?

h) Did the intervention have an effect on the outcome?

Answer 1

Given that,
mean(x)=6
standard deviation , s.d1=1.8708
number(n1)=9
y(mean)=4.3333
standard deviation, s.d2 =1.7321
number(n2)=9
null, Ho: u1 = u2
alternate, H1: u1 != u2
level of significance, α = 0.05
from standard normal table, two tailed t α/2 =2.306
since our test is two-tailed
reject Ho, if to < -2.306 OR if to > 2.306
we use test statistic (t) = (x-y)/sqrt(s.d1^2/n1)+(s.d2^2/n2)
to =6-4.3333/sqrt((3.49989/9)+(3.00017/9))
to =1.961
| to | =1.961
critical value
the value of |t α| with min (n1-1, n2-1) i.e 8 d.f is 2.306
we got |to| = 1.96119 & | t α | = 2.306
make decision
hence value of |to | < | t α | and here we do not reject Ho
p-value: two tailed ( double the one tail ) - Ha : ( p != 1.9612 ) = 0.086
hence value of p0.05 < 0.086,here we do not reject Ho
ANSWERS
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a.
the appropriate model of the population(s) is independent samples t test is used
b.
null, Ho: u1 = u2
alternate, H1: u1 != u2
d.
test statistic: 1.961
c.
critical value: -2.306 , 2.306
e.
decision: do not reject Ho
p-value: 0.086
we do not have enough evidence to support the claim that The researchers are interested in testing whether the treatment had any effect on the outcome.
f.
cohen's d Effective size
Effective size = mean difference /pooled standard deviation
pooled standard deviation = sqrt((s.d1^2+s.d2^2)/2)
pooled standard deviation = sqrt((1.8708^2 +1.7321^2)/2) = 1.8027
Effective size =(6-4.3333)/1.8027 =0.924
large effect
g.
TRADITIONAL METHOD
given that,
mean(x)=6
standard deviation , s.d1=1.8708
number(n1)=9
y(mean)=4.3333
standard deviation, s.d2 =1.7321
number(n2)=9
I.
standard error = sqrt(s.d1^2/n1)+(s.d2^2/n2)
where,
sd1, sd2 = standard deviation of both
n1, n2 = sample size
standard error = sqrt((3.5/9)+(3/9))
= 0.85
II.
margin of error = t a/2 * (standard error)
where,
t a/2 = t -table value
level of significance, α = 0.05
from standard normal table, two tailed and
value of |t α| with min (n1-1, n2-1) i.e 8 d.f is 2.306
margin of error = 2.306 * 0.85
= 1.96
III.
CI = (x1-x2) ± margin of error
confidence interval = [ (6-4.3333) ± 1.96 ]
= [-0.293 , 3.626]
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DIRECT METHOD
given that,
mean(x)=6
standard deviation , s.d1=1.8708
sample size, n1=9
y(mean)=4.3333
standard deviation, s.d2 =1.7321
sample size,n2 =9
CI = x1 - x2 ± t a/2 * Sqrt ( sd1 ^2 / n1 + sd2 ^2 /n2 )
where,
x1,x2 = mean of populations
sd1,sd2 = standard deviations
n1,n2 = size of both
a = 1 - (confidence Level/100)
ta/2 = t-table value
CI = confidence interval
CI = [( 6-4.3333) ± t a/2 * sqrt((3.5/9)+(3/9)]
= [ (1.667) ± t a/2 * 0.85]
= [-0.293 , 3.626]
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interpretations:
1. we are 95% sure that the interval [-0.293 , 3.626] contains the true population proportion
2. If a large number of samples are collected, and a confidence interval is created
for each sample, 95% of these intervals will contains the true population proportion
h.
the intervention have an effect on the outcome is
we do not have enough evidence to support the claim that The researchers are interested in testing whether the treatment had any effect on the outcome.