A manager wishes to determine whether
the mean times required to complete a certain task differ for the three
levels of employee training. He randomly selected 10 employees with each
of the three levels of training (Beginner, Intermediate and Advanced).
Do the data provide sufficient evidence to indicate that the mean times
required to complete a certain task differ for at least two of the three
levels of training? The data is summarized in the table.
Level of Training | n | s^{2} | |
Advanced | 10 | 24.2 | 21.54 |
Intermediate | 10 | 27.1 | 18.64 |
Beginner | 10 | 30.2 | 17.76 |
H_{o}: The mean times required to complete a certain task do not differ the three levels of training. ( µ_{B} = µ_{I} = µ_{A})
Assumptions: The samples were drawn independently and randomly from the three populations. The time required to complete the task is normally distributed for each of the three levels of training. The populations have equal variances.
Test Statistic:
RR: or
Calculations: = 10(24.2 - 27.16...)^{2} + 10(27.1 - 27.16...)^{2} + 10(30.2 - 27.16...)^{2} = 180.066....
=
9(21.54) + 9(18.64) + 9(17.76) = 521.46
Source | df | SS | MS | F |
Treatments | 2 | 180.067 | 90.033 | 4.662 |
Error | 27 | 521.46 | 19.313 | |
Total | 29 | 702.527 |
Decision: Reject H_{o}.
Conclusion:
There is sufficient evidence to indicate that the mean times required to
complete a certain task differ for at least two of the three levels of
training.
The Bonferroni Test is done for all possible pairs of means.
Decision rule: Reject H_{o}, if the interval does not contain 0.
c = # of pairs c = p(p-1)/2 = 3(2)/2 = 3
t_{.0083} = 2.554
(This value is not in the t table;
it was obtained from a computer program.)
Since t_{.010} < t_{.0083}
< t_{.0050} (2.473 < t_{.0083} < 2.771), use t_{.005}
when using a table. If you reject the null hypothesis when t = 2.771; you
will also reject it for t_{.0083}.
There is sufficient evidence to indicate that the mean response time for the advanced level of training is less than the mean response time for the beginning level. There is not sufficient evidence to indicate that the mean response time for the intermediate level differs from the mean response time of either of the other two levels.