MGMT2263 Tutorial Sheet 4 Solutions
Question 1
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.
Question 2
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.
Question 3
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 4
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.
Question 5
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.
Question 6
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.
Question 7
The t value is 2.131 since we have 15 degrees of freedom.
Lower limit = (16.68 15.95) 2.131sqrt(3.9576/6 + 3.9576/6) = 0.73 2.45 = -1.72
Upper limit = 0.73 + 2.45 = 3.18
-1.72 < m3 - m1 < 3.18
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 8
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.
d) The t value is 2.306 since we have 8 degrees of freedom.
Lower limit = (30.6 26) 2.306sqrt(5.4333/5 + 5.4333/5) = 4.6 3.4 = 1.2
Upper limit = 4.6 + 3.4 = 8.0
1.2 < mnight - mafternoon < 8; the afternoon shifts produces between 1.2 and 8 more widgets per hour on average than the afternoon shift
Question 9
First, here is the ANOVA table:
|
Source of Variation |
SS |
df |
MS |
F |
|
Education |
587.6763 |
2 |
293.8381 |
26.3252 |
|
Age |
437.6763 |
2 |
218.8381 |
19.6059 |
|
Interaction |
292.2170 |
4 |
73.0543 |
6.5450 |
|
Error |
200.9133 |
18 |
11.1619 |
|
|
|
|
|
|
|
|
Total |
1518.483 |
26 |
|
|
a) Ho: age is not significant
Ha: age is significant
The degrees of freedom are 2 and 18. Reject Ho if F > 3.55
F = 19.6059
Reject Ho and conclude that income depends on age.
b) Ho: education is not significant
Ha: education is significant
The degrees of freedom are 2 and 18. Reject Ho if F > 3.55
F = 26.3252
Reject Ho and conclude that income depends on education levels.
c) Ho: no significant interaction
Ha: significant interaction
The degrees of freedom are 4 and 18. Reject Ho if F > 2.93
F = 6.545
Reject Ho and conclude there is significant interaction between age and education levels.