You’ve been hired by a company who is interested in finding a way to decrease in
ID: 3216897 • Letter: Y
Question
You’ve been hired by a company who is interested in finding a way to decrease incidents of offensive interpersonal behavior at work. The company has identified one department that seems to have a big problem with offensive interpersonal behavior. The company hopes to resolve this problem by putting all of the employees in this department through a training course. The company measures their offensiveness (on a scale of 1 indicating no offensiveness to 20 indicating highly offensive) both before (pre) and after (post) the training course. They also recorded the person’s rank in the company (1 indicating manager and 2 indicating employee).
Family Name
Pre-Training
Post-Training
Rank
Taub
10
4
2
Hadley
12
9
2
Cameron
17
16
1
House
20
18
1
Wilson
8
6
1
Cuddy
12
11
1
Foreman
18
16
1
Volakis
14
9
2
Chase
11
4
2
The company wants to know whether the training was able to reduce offensiveness for its employees. Based on the data provided above, determine whether offensiveness after the training (Post-Training) was lower than offensiveness before the training (Pre-Training).
A.Calculate the appropriate statistical test for this scenario BY HAND. Be sure to show all work! (2 pts)
B.State formal test of the hypothesis for this scenario. (1.75 pts)
C.Report a formal conclusion for this scenario. (1 pt)
Family Name
Pre-Training
Post-Training
Rank
Taub
10
4
2
Hadley
12
9
2
Cameron
17
16
1
House
20
18
1
Wilson
8
6
1
Cuddy
12
11
1
Foreman
18
16
1
Volakis
14
9
2
Chase
11
4
2
Explanation / Answer
let X denotes the score pre-training and Y denotes the same post training for employees.
since the scores are calculated on the same people before and after training, hence
(X,Y)~Bivariate normal(ux,uy,sx2,sy2,r) where all the parameters are unknown.
we need to determine whether offensiveness after the training (Post-Training) was lower than offensiveness before the training (Pre-Training).
a) so H0:ux=uy vs H1:ux>uy
we have n=4 observations as
(X1,Y1)=(10,4) (X2,Y2)=(12,9) (X3,Y3)=(14,9) (X4,Y4)=(11,4)
to test the hypothesis let
T=X-Y then T~N(ux-uy,sx2+sy2-2r*sx*sy)
now under H0 T~N(0,st2) where st2=sx2+sy2-2r*sx*sy which is unknown and estimated by sample variance of T
so the test statistic is Z=(Tbar-0)*sqrt(n)/s where s is the sample variance of T which under H0 follows a t distribution with df n-1
b) now T takes the values T1=X1-Y1=6 T2=X2-Y2=3 T3=X3-Y3=5 T4=X4-Y4=7
so Tbar=(6+3+5+7)/4=5.25
s2=[(6-5.25)2+(3-5.25)2+(5-5.25)2+(7-5.25)2]/(4-1)=2.9167
so the observed value of the test statistic is Z=5.25*sqrt(4)/sqrt(2.9167)=6.148
let the level of significance be alpha=0.05
since the alternative hypothesis is right tailed hence H0 is rejected iff
observed value of Z>t0.05;3 where t0.05;3 is the upper 0.05 point of a t distribution with df 3
now t0.05;3=2.353363<6.148
so observed value of Z>t0.05;3 hence H0 is rejected
c) hence the conclusion at 5% level of significance is that
offensiveness after the training (Post-Training) was indeed lower than offensiveness before the training (Pre-Training).
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