Question

A city council suspects a judge of being a "hanging judge" because s/he is perceived as imposing harsher penalties for the same sentence. To investigate this, a random sample of 47 cases is taken from the judge's prior cases that resulted in a guilty verdict for a certain crime. The average jail sentence s/he imposed for the sample is 26 months. The average jail sentence for the same type of crimes is 25 months with a standard deviation of 13 months. What can be concluded with an α of 0.05?

a) What is the appropriate test statistic?
---Select--- na z-test one-sample t-test independent-samples t-test related-samples t-test

b)
Population:
---Select--- cases same type of crimes city council judge's prior cases months
Sample:
---Select--- cases same type of crimes city council judge's prior cases months

c) Compute the appropriate test statistic(s) to make a decision about H₀.
(Hint: Make sure to write down the null and alternative hypotheses to help solve the problem.)
critical value =  ; test statistic =
Decision:  ---Select--- Reject H0 Fail to reject H0

d) If appropriate, compute the CI. If not appropriate, input "na" for both spaces below.
[  ,  ]

e) Compute the corresponding effect size(s) and indicate magnitude(s).
If not appropriate, input and select "na" below.
d =  ;   ---Select--- na trivial effect small effect medium effect large effect
r² =  ;   ---Select--- na trivial effect small effect medium effect large effect

f) Make an interpretation based on the results.

For the same sentence, the judge imposed significantly harsher penalties than other judges.

For the same sentence, the judge imposed significantly milder penalties than other judges.   

For the same sentence, the judge's penalties did not significantly differ from other judges.

Answer 1

Given that,
population mean(u)=25
standard deviation, σ =13
sample mean, x =26
number (n)=47
null, Ho: μ=25
alternate, H1: μ!=25
level of significance, α = 0.05
a.
we use Z test for single mean because they given population standard deviation
from standard normal table, two tailed z α/2 =1.96
since our test is two-tailed
reject Ho, if zo < -1.96 OR if zo > 1.96
we use test statistic (z) = x-u/(s.d/sqrt(n))
zo = 26-25/(13/sqrt(47)
zo = 0.527
| zo | = 0.527
critical value
the value of |z α| at los 5% is 1.96
we got |zo| =0.527 & | z α | = 1.96
make decision
hence value of |zo | < | z α | and here we do not reject Ho
p-value : two tailed ( double the one tail ) - ha : ( p != 0.527 ) = 0.598
hence value of p0.05 < 0.598, here we do not reject Ho
ANSWERS
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b.
population:
cases same type of crimes city council judge's prior cases months
null, Ho: μ=25
alternate, H1: μ!=25
c.
test statistic: 0.527
critical value: -1.96 , 1.96
decision: do not reject Ho
p-value: 0.598
we do not have enough evidence to support the claim that the judge's penalties did not significantly differ from other judges
d.
TRADITIONAL METHOD
given that,
standard deviation, σ =13
sample mean, x =26
population size (n)=47
I.
standard error = sd/ sqrt(n)
where,
sd = population standard deviation
n = population size
standard error = ( 13/ sqrt ( 47) )
= 1.896
II.
margin of error = Z a/2 * (standard error)
where,
Za/2 = Z-table value
level of significance, α = 0.05
from standard normal table, two tailed z α/2 =1.96
since our test is two-tailed
value of z table is 1.96
margin of error = 1.96 * 1.896
= 3.717
III.
CI = x ± margin of error
confidence interval = [ 26 ± 3.717 ]
= [ 22.283,29.717 ]
-----------------------------------------------------------------------------------------------
DIRECT METHOD
given that,
standard deviation, σ =13
sample mean, x =26
population size (n)=47
level of significance, α = 0.05
from standard normal table, two tailed z α/2 =1.96
since our test is two-tailed
value of z table is 1.96
we use CI = x ± Z a/2 * (sd/ Sqrt(n))
where,
x = mean
sd = standard deviation
a = 1 - (confidence level/100)
Za/2 = Z-table value
CI = confidence interval
confidence interval = [ 26 ± Z a/2 ( 13/ Sqrt ( 47) ) ]
= [ 26 - 1.96 * (1.896) , 26 + 1.96 * (1.896) ]
= [ 22.283,29.717 ]
-----------------------------------------------------------------------------------------------
interpretations:
1. we are 95% sure that the interval [22.283 , 29.717 ] contains the true population mean
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 mean
e.
effctive size = (sample mean -population mean)/standard deviation
effctive size = (26-25)/13
effctive size = 0.0769 small
trivial effect
f.
the judge's penalties did not significantly differ from other judges.