1)
In
a deli, 3 checkouts were examined to see if there was any significant
difference in the amount people spent at each checkout; 6 receipts were sampled
from each checkout. These were the results:
|
Counter
1 |
Counter
2 |
Counter
3 |
|
14.00 |
17.25 |
17.45 |
|
11.56 |
16.48 |
16.89 |
|
15.36 |
14.68 |
14.25 |
|
17.94 |
16.24 |
15.82 |
|
17.75 |
15.43 |
19.42 |
|
19.08 |
16.27 |
16.25 |
Analysis of the data indicates the three groups
are normally distributed with equal variances.
a)
Is
there any significant difference in the average amount spent at each counter?
Test at a 5% level of significance.
(F = 0.236; conclude no significant difference)
b)
Conduct
the appropriate test to confirm the equality of variances among the 3 counters
at a 5% level of significance. (H = 10.18; conclude the variances are equal)
c)
Conduct
Tukey’s test on the data to confirm the ANOVA results
at a 5% level of significance. Note: Q = 3.67
(D = 2.9806; all differences between the sample
means are less than this)
d)
Construct
a 95% simultaneous confidence interval of the average difference between the
counter with the largest average and that with the smallest, rounding the
limits to the nearest cent.
2)
A
factory wanted to compare production between its day, afternoon and night
shifts. It has 5 lines. They sampled 5 hours worth of production from each line
and measured the number of widgets built per hour. These were the results:
|
Day |
Afternoon |
Night |
|
|
Line 1 |
31 |
25 |
35 |
|
Line 2 |
33 |
26 |
33 |
|
Line 3 |
28 |
24 |
30 |
|
Line 4 |
30 |
29 |
28 |
|
Line 5 |
28 |
26 |
27 |
Analysis of the data indicates it is normally
distributed with equal variances.
a)
Is
there any significant difference in production between the 3 shifts at a 5%
level of significance? (F = 5.7546; conclude there is a significant difference)
b)
Is
there any significant difference in production between the 5 lines at a 5%
level of significance? (F = 1.5521; conclude there is no significant
difference)
c)
Between
which shifts is there a significant difference at a 5% level of significance? Note:
Q = 4.04 (afternoon and night)
d)
Construct
a 95% simultaneous confidence interval of the average difference in production
between the night and afternoon shifts, rounding the limits to 1 decimal. (0.4
< mnight - mafternoon < 8.8)
3)
Steel
hardness was measured using three different methods of fabrication. These were
the results:
|
Method
1 |
Method
2 |
Method
3 |
|
95 |
61 |
75 |
|
74 |
77 |
89 |
|
95 |
100 |
75 |
|
102 |
80 |
64 |
|
123 |
95 |
86 |
|
110 |
69 |
77 |
|
112 |
86 |
95 |
|
82 |
90 |
90 |
Analysis of the data indicates the groups are
normally distributed with equal variances.
a)
Is
there any significant difference in the methods in average hardness? Test at a
5% level of significance. (F = 4.4556; conclude there is a significant
difference)
b)
For
which levels of significance between 1% and 10% would the opposite conclusion
be reached? (1% to 2.44%)
c)
Between
which methods is there a significant difference at a 5% level of significance? Note:
Q = 3.58 (A and C)
d)
Construct
a 95% simultaneous confidence interval of the average difference in hardness
between A and C. Why is this interval consistent with the ANOVA results?
(0.7815 < m(A) - m(C) < 34.7185)
4)
A
set of 3 steel plants were measured for sulphur
dioxide emissions (kg) on a set of 8 dates:
|
|
Plant
#1 |
Plant
#2 |
Plant
#3 |
|
June
1 |
292.5 |
383.8 |
274.7 |
|
June
4 |
268.1 |
349 |
248 |
|
June
7 |
306.1 |
404.7 |
290.4 |
|
June
10 |
331.9 |
333.7 |
280.7 |
|
June
13 |
330 |
338.1 |
325.4 |
|
June
16 |
343.3 |
343.4 |
300.7 |
|
June
19 |
245.4 |
332.5 |
300.1 |
|
June
22 |
294.1 |
341.9 |
269.3 |
Analysis of the data indicates it is normally
distributed with equal variances.
a)
Is
there any significant difference between the plants in the average amount of sulphur dioxide emissions? Test at a 5% level of
significance. (F = 13.801; conclude there is a significant difference between
at least 2 of the plants)
b)
Between
which plants is there a significant difference at a 5% level of significance? Note:
Q = 3.70 (Plant #2 and the other plants)
c)
Construct
a 95% simultaneous confidence interval of the average difference in sulphur dioxide emissions between Plant #2 and Plant #3.
Why is this interval consistent with the results of ANOVA?
(32.125 < m(Plant #2) - m(Plant #3) < 102.325)