Question

2. A lawyer believes that a certain judge imposes prison sentences for property crimes that are longer than the state average 11.7 months. He randomly selects 36 of the judge’s sentences and obtains mean 13.8 and standard deviation 3.9 months.

a) Test the hypothesis at 1% significance level.

b) Construct a 99% confidence interval for the true average length of sentences im- posed by this judge.

c) Construct a 95% confidence interval for the true average length of sentences im- posed by this judge.

d) Compare the margins of error from b) and c).

Answer 1

Part a)

To Test :-
H0 :- µ = 11.7
H1 :- µ > 11.7

Test Statistic :-
t = ( X̅ - µ ) / (S / √(n) )
t = ( 13.8 - 11.7 ) / ( 3.9 / √(36) )
t = 3.2308

Test Criteria :-
Reject null hypothesis if t > t(α, n-1)
Critical value t(α, n-1) = t(0.01 , 36-1) = 2.438 ( From t table )
t > t(α, n-1) = 3.2308 > 2.438
Result :- Reject null hypothesis

Decision based on P value
P - value = P ( t > 3.2308 ) = 0.0013
Reject null hypothesis if P value < α = 0.01 level of significance
P - value = 0.0013 < 0.01 ,hence we reject null hypothesis
Conclusion :- Reject null hypothesis

There is sufficient evidence to support the claim that a certain judge imposes prison sentences for property crimes that are longer than the state average 11.7 months.

Part b)

Confidence Interval
X̅ ± t(α/2, n-1) S/√(n)
t(α/2, n-1) = t(0.01 /2, 36- 1 ) = 2.724
13.8 ± t(0.01/2, 36 -1) * 3.9/√(36)
Lower Limit = 13.8 - t(0.01/2, 36 -1) 3.9/√(36)
Lower Limit = 12.0294
Upper Limit = 13.8 + t(0.01/2, 36 -1) 3.9/√(36)
Upper Limit = 15.5706
99% Confidence interval is ( 12.0294 , 15.5706 )

Margin of Error = t(α/2, n-1) S/√(n) = 1.7706

Part c)
Confidence Interval
X̅ ± t(α/2, n-1) S/√(n)
t(α/2, n-1) = t(0.05 /2, 36- 1 ) = 2.03
13.8 ± t(0.05/2, 36 -1) * 3.9/√(36)
Lower Limit = 13.8 - t(0.05/2, 36 -1) 3.9/√(36)
Lower Limit = 12.4805
Upper Limit = 13.8 + t(0.05/2, 36 -1) 3.9/√(36)
Upper Limit = 15.1195
95% Confidence interval is ( 12.4805 , 15.1195 )

Margin of Error = t(α/2, n-1) S/√(n) = 1.3195

Part d)

As level of confidence decreases, margin of error also decreases.