MGMT2263 Tutorial Sheet 4 Solutions
Question 1
a) Let the members be group 1.
Ho: p1 £ p2
Ha: p1 > p2
Reject Ho if Z > 1.645
p-hat = (144 + 115)/(400) = 0.6475. As well, p1 = 144/200 = 0.72, p2 = 0.575
Z = (0.72 0.575)/sqrt(0.6475(0.3525)/200 + 0.6475(0.3525)/200) = 3.035
Reject Ho. Conclude the members are more likely to visit new exhibits than non-members.
b) P-value = P(Z > 3.04) = 0.5 0.4988 = 0.0012. Since this is less than 1%, we have strong support for Ha. The answer is no.
c) Lower limit = (0.72 0.575) 1.96sqrt(0.72(0.28)/200 + 0.575(0.425)/200) = 0.145 0.0926 = 0.0524 = 5.24%
Upper limit = 0.145 + 0.0926 = 0.2376 = 23.76%
The percentage of members who will visit new exhibits ranges from 5.24% to 23.76% above the corresponding percentage of non-members.
Question 2
a) Ho: m1 = m2 = m3
Ha: not all the means are equal
The degrees of freedom are 2 and 15. Reject Ho if F > 3.68.
Here is the ANOVA table:
|
Source
of Variation |
SS |
df |
MS |
F |
|
Between
Groups |
1.8678 |
2 |
0.9339 |
0.2360 |
|
Within
Groups |
59.3640 |
15 |
3.9576 |
|
|
Total |
61.2318 |
17 |
|
|
F = 0.236. Do not reject Ho. Conclude there is no significant difference among the 3 counters in average sales.
b) Ho: s1 = s2 = s3
Ha: not all the standard deviations are equal
Reject Ho if H > 10.8
H = 8.083937/0.794057 = 10.18
Do not reject Ho.
Conclude there is no significant difference among the standard deviations.
c) Q = 3.67; D = 3.67sqrt(3.9576/6) = 2.9806. The largest mean is 16.68 from counter 3, the smallest, 15.95 from counter 1. Since 16.68 15.95 = 0.73 is less than D, we can conclude there is no significant difference between any of the pairs of means which is consistent with the results of ANOVA in which we concluded that none of the means are significantly different.
Here is the Megastat output for this section of the question:
|
Tukey simultaneous comparison t-values (d.f. = 15) |
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|
Counter
1 |
Counter
2 |
Counter
3 |
|||
|
15.948 |
16.058 |
16.680 |
|||
|
Counter
1 |
15.948 |
|
|
|
|
|
Counter
2 |
16.058 |
0.10 |
|
|
|
|
Counter
3 |
16.680 |
0.64 |
0.54 |
|
|
|
critical
values for experimentwise error rate: |
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|
0.05 |
2.60 |
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|
0.01 |
3.42 |
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The 5% critical value is 2.60. The values of the test statistics are:
Counters 1 and 2: 0.10
Counters 1 and 3: 0.64
Counters 2 and 3: 0.54
Since all these tests statistics are less than the critical value of 2.60, there is no significant difference between any of the pairs of means.
d) Lower limit = (16.68 15.95) 2.9806 = 0.73 2.98 = -2.25
Upper limit = 0.73 + 2.98 = 3.71
-2.25 < m3 - m1 < 3.71
Since m3 - m1 = 0 falls inside the 95% confidence interval, we conclude there is no significant difference between m1 and m3. This is consistent with the results of ANOVA in which we concluded that none of the means are significantly different.
Question 3
a) Ho: m1 = m2 = m3
Ha: not all the factor means are equal
The degrees of freedom are 2 and 8. Reject Ho if F > 4.46
Here is the ANOVA table:
|
Source
of Variation |
SS |
df |
MS |
F |
P-value |
|
Rows |
33.7333 |
4 |
8.4333 |
1.5521 |
0.2762 |
|
Columns |
62.5333 |
2 |
31.2667 |
5.7546 |
0.0283 |
|
Error |
43.4667 |
8 |
5.4333 |
||
|
Total |
139.7333 |
14 |
|
|
|
F = 5.7546. Reject Ho. Conclude that at least 2 of the shifts are significantly different.
b) Ho: m1 = m2 = m3 = m4 = m5
Ha: not all the block means are equal
The degrees of freedom are 4 and 8. Reject Ho if F > 3.84
F = 1.5521. Do not reject Ho. Conclude there is no significant difference among the lines means.
c) Q = 4.04; D = 4.04sqrt(5.4333/5) = 4.2114. The means of the day, afternoon and night shifts are 30, 26 and 30.6 respectively. The difference between the night and afternoon shifts is 4.6 which is significant. However, the other differences are less than 4.2114 and so are not significant.
Here is the Megastat output for this section of the question:
|
Tukey simultaneous comparison t-values (d.f. = 8) |
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|
Afternoon |
Day |
Night |
|
||
|
26.000 |
30.000 |
30.600 |
|
||
|
Afternoon |
26.000 |
|
|
|
|
|
Day |
30.000 |
2.71 |
|
|
|
|
Night |
30.600 |
3.12 |
0.41 |
|
|
|
|
|||||
|
critical
values for experimentwise error rate: |
|
||||
|
0.05 |
2.86 |
|
|||
|
0.01 |
3.98 |
|
|||
The 5% critical value is 2.86. The values of the test statistics are:
Afternoon/Day: 2.71
Afternoon/Night: 3.12
Day/Night: 0.41
The only pairing in which the test statistic is greater than the critical value is Afternoon/Night.
d) Lower limit = (30.6 26) 4.2114 = 4.6 4.2114 = 0.4
Upper limit = 4.6 + 4.2114 = 8.8
0.4 < mnight - mafternoon < 8.8; the afternoon shifts produces between 0.4 and 8.8 more widgets per hour on average than the afternoon shift