STAT177 Homework sheet #8 Solutions
1) The samples are independent and each one is normal. The appropriate test is one-way ANOVA.
2) Here is the output:
|
One-way
ANOVA |
|
Ho: all
the means are equal |
|
Ha: at least
two means are not equal |
|
Reject Ho
if test statistic > 2.866 |
|
Test
statistic = 4.371 |
|
P-value =
0.01 |
|
Reject Ho |
|
Conclude
at least two means are not equal |
Since the p-value is less than the level of significance, we reject Ho and conclude there is a significant difference in the average grades of at least 2 of the schools.
3) We look at the simultaneous confidence intervals:
|
95%
Simultaneous Confidence Intervals |
|
||
|
Interval |
Lower limit |
Upper
limit |
Significant
difference? |
|
School A
- School B |
-12.285 |
19.685 |
No |
|
School A
- School C |
-24.185 |
7.785 |
No |
|
School A
- School D |
-31.185 |
0.785 |
No |
|
School B
- School C |
-27.885 |
4.085 |
No |
|
School B
- School D |
-34.885 |
-2.915 |
Yes |
|
School C
- School D |
-22.985 |
8.985 |
No |
We see that the only schools that are significantly different are schools B and D.
4) The samples are dependent and each sample is normal. The appropriate test is 2-way ANOVA.
5) Here is the output:
|
Two-way
ANOVA |
|
Ho: all
the factor means are equal |
|
Ha: at
least two factor means are not equal |
|
Reject Ho
if test statistic > 4.459 |
|
Test
statistic = 5.755 |
|
P-value =
0.028 |
|
Reject Ho |
|
Conclude at
least two factor means are not equal |
Since the p-value is less than the level of significance, we reject Ho and conclude there is a significant difference among the shifts.
6) We look at the confidence intervals:
|
95% Simultaneous
Confidence Intervals |
|
||
|
Interval |
Lower
limit |
Upper
limit |
Significant
difference? |
|
Day -
Afternoon |
-0.446 |
8.446 |
No |
|
Day -
Night |
-5.046 |
3.846 |
No |
|
Afternoon
- Night |
-9.046 |
-0.154 |
Yes |
We see that the only shifts that are significantly different are afternoon and night.
7) Here is the output:
|
Ho: all
the block means are equal |
|
Ha: at
least two block means are not equal |
|
Reject Ho
if test statistic > 3.838 |
|
Test
statistic = 1.552 |
|
P-value =
0.276 |
|
Do not
reject Ho |
|
Conclude
the block means are equal |
Under the general rule of thumb, since the p-value is greater than 10%, we do not reject Ho and conclude there is no significant difference among the lines.
8) Since the data is ordinal and samples are dependent, the appropriate test is the Friedman test.
9) Here is the output:
|
Friedman
test |
|
Ho: all
the medians are equal |
|
Ha: at
least two medians are not equal |
|
Reject Ho
if test statistic > 5.991 |
|
Test statistic
= 9.15 |
|
P-value =
0.01 |
|
Reject Ho |
|
Conclude
at least two medians are not equal |
Since the p-value is on the borderline under the general rule of thumb, we reject Ho and conclude there is a significant difference in how the 3 brands are rated.
10) There are 2 ways we could approach this. The first way is to note that the among the average ranks, Brand A has the highest average rank of 2.65 while Brand C has the lowest at 1.3. The other way is to compute the average rating for each brand. Under this approach, Brand A has the highest average rating of 7.9 while Brand C has the lowest at 4.3.
11) Here is the output:
|
Kruskal-Wallis
test |
|
Ho: all
the medians are equal |
|
Ha: at least
two medians are not equal |
|
Reject Ho
if test statistic > 5.991 |
|
Test
statistic = 18.492 |
|
P-value =
0 |
|
Reject Ho |
|
Conclude
at least two medians are not equal |
Since the p-value is zero, we reject Ho and conclude there is a significant difference in the completion rates of the three firms.
12) For the follow-up analysis, we are looking at three pairs of Mann-Whitney tests. It appears that all three firms are significantly different based on the average ranks for each firm. I will first compare firms A and B and then Firms B and C. If these are significantly different, I can then conclude that Firms A and C are significantly different. All tests will be conducted as 2-tail tests.
Comparing Firms A and B, let Firm A be group 1:
|
Mann-Whitney
test for two medians |
|
Ho:
median(Firm A) - median(Firm B) equals 0 |
|
Ha:
median(Firm A) - median(Firm B) does not equal 0 |
|
Reject Ho
if test statistic < 79 or > 131 |
|
Sample
sizes = 10 and 10 |
|
Test statistic
= 75.5 |
|
Reject Ho |
|
Conclude
median(Firm A) - median(Firm B) does not equal 0 |
Comparing Firms B and C, let Firm B be group 1:
|
Mann-Whitney
test for two medians |
|
Ho:
median(Firm B) - median(Firm C) equals 0 |
|
Ha: median(Firm
B) - median(Firm C) does not equal 0 |
|
Reject Ho
if test statistic < 79 or > 131 |
|
Sample
sizes = 10 and 10 |
|
Test
statistic = 66 |
|
Reject Ho |
|
Conclude
median(Firm B) - median(Firm C) does not equal 0 |
Since Firms A and B are significantly different, as well as Firms B and C, we can conclude that Firms A and C are significantly different since A has the lowest average rating and C has the highest. Thus, all three firms are significantly different. Since Firm C has the highest average, it should be chosen.