3.10. A product developer is investigating the tensile strength of a new synthet
ID: 3044906 • Letter: 3
Question
3.10. A product developer is investigating the tensile strength of a new synthetic fiber that will be used to make cloth for men's shirts. Strength is usually affected by the percentage of cotton used in the blend of materials for the fiber. The engineer conducts a completely randomized experiment with five levels of cotton content and replicates the experiment five times. The data are shown in the following table Cotton Weight Percent Observations 15 20 25 30 35 7 15 12 17 12 8 18 14 19 25 22 19 23 7 10 5 19 19 18 18 (a) Obtain the ANOVA table and do the test of Ho of equal means. (b) Test 2 at level of significance = (c) Obtain a 95% CI, for the contrast in (b). (d) Give the tot al number of (meaningful) contrasts (e) Give a maximal set of mutually orthogonal contrasts (f) How many possible contrasts are there if you use Scheffe's method for multiple comparisons'met hods? (g) How many possible pairs can be obtained if you want to use Tukey's method? (h) Compare all possible pairs by Tukey's method, Scheffe's method, and Bonferroni's method at 90% C.1. Which method gives the shortest interval? Which gives the worst? (i) What is your recommendation?Explanation / Answer
a)
ANOVA : Single Factor as COTTON WEIGHT %
SUMMARY
COTTON WEIGHT %
Count
Sum
Average
Variance
15
5
49
9.8
11.2
20
5
77
15.4
9.8
25
5
88
17.6
4.3
30
5
108
21.6
6.8
35
5
54
10.8
8.2
ANOVA
Source of Variation
SS
df
MS
F
P-value
F critical
Between Groups
475.76
4
118.94
14.75682
9.13E-06
2.866081
Within Groups
161.2
20
8.06
Total
636.96
24
One-way ANOVA: Strength versus Weight
Analysis of Variance for Stength
Source DF SS MS F P
Weight 4 475.76 118.94 14.76 0.000
Error 20 161.20 8.06
Total 24 636.96
Individual 95% CIs For Mean
Based on Pooled StDev
Level N Mean StDev ------+---------+---------+---------+
15 5 9.800 3.347 (-----*----)
20 5 15.400 3.130 (----*----)
25 5 17.600 2.074 (----*----)
30 5 21.600 2.608 (----*----)
35 5 10.800 2.864 (-----*----)
------+---------+---------+---------+
Pooled StDev = 2.839 10.0 15.0 20.0 25.0
Tukey's pairwise comparisons
Family error rate = 0.0500
Individual error rate = 0.00722
Critical value = 4.23
Intervals for (column level mean) - (row level mean)
15 20 25 30
20 -10.971
-0.229
25 -13.171 -7.571
-2.429 3.171
30 -17.171 -11.571 -9.371
-6.429 -0.829 1.371
35 -6.371 -0.771 1.429 5.429
4.371 9.971 12.171 16.171
From above two outputs of one way ANOVA with Strength as a factor it is seen that p-value = 9.13E-06 < < 0.05 indicates that tensile strength of new synthetic fiber product varies with % cotton weight. The highest tensile strength is observed when cotton weight % is 35.
b) To test the hypothesis H0: (1+ 5) /2 =(2 + 3 + 4)/3, we will convert the observed data accordingly.
ANOVA : Single Factor
SUMMARY
Groups
Count
Sum
Average
Variance
A
(Average of 15 and 35 % cotton weight)
5
51.5
10.3
7.2
B
(Average of 20. 25 and 25 % cotton weight)
5
91
18.2
4.311111
ANOVA
Source of Variation
SS
df
MS
F
P-value
F crit
Between Groups
156.025
1
156.025
27.10859
0.000816
5.317655
Within Groups
46.04444
8
5.755556
Total
202.0694
9
From above ANOVA table it is clear that the proposed hypothesis is rejected as p-value = 0.000816 < < 0.05.
One-way ANOVA: Tensile Strength versus Cotton Weight
Analysis of Variance for Tensile Strength
Source DF SS MS F P
Weight 1 156.03 156.03 27.11 0.001
Error 8 46.04 5.76
Total 9 202.07
Individual 95% CIs For Mean
Based on Pooled StDev
Level N Mean StDev --------+---------+---------+--------
1 5 10.300 2.683 (------*------)
2 5 18.200 2.076 (------*------)
--------+---------+---------+--------
Pooled StDev = 2.399 10.5 14.0 17.5
Tukey's pairwise comparisons
Family error rate = 0.0500
Individual error rate = 0.0500
Critical value = 3.26
Intervals for (column level mean) - (row level mean)
1
2 -11.399
-4.401
From above ANOVA output it is seen that the equal mixture of 20, 25 and 30 % cotton weight gives better tensile strength as compared to that when 15 and 35 % weight of cotton is mixed equally.
ANOVA : Single Factor as COTTON WEIGHT %
SUMMARY
COTTON WEIGHT %
Count
Sum
Average
Variance
15
5
49
9.8
11.2
20
5
77
15.4
9.8
25
5
88
17.6
4.3
30
5
108
21.6
6.8
35
5
54
10.8
8.2
ANOVA
Source of Variation
SS
df
MS
F
P-value
F critical
Between Groups
475.76
4
118.94
14.75682
9.13E-06
2.866081
Within Groups
161.2
20
8.06
Total
636.96
24
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