STATISTICS 401B
Fall 2014
Laboratory Assignment 9
1. An experiment was performed to study the effect of two variables, reaction temperature and reactant
concentration, on the percent conversion of a chemical process. The data appear below and in the
file process.jmp. You may use the JMP output to perform as much of the computation as you can.
Temperature ◦ C(x1 )
200
200
200
225
225
225
225
225
225
250
250
250
Concentration % (x2 )
15
20
25
15
20
20
20
20
25
15
20
25
Percent Conversion(y)
45
54
60
65
74
70
73
72
74
76
73
78
(a) The first order model
y = β0 + β1 x1 + β2 x2 + was fitted to the data. Using the JMP output, construct an analysis of variance table and test
the hypothesis of H0 : β1 = β2 = 0 vs. Ha : at least one of β1 or β2 6= 0. Give the R2 value.
Should we accept the first order model and not proceed further because R2 is fairly high? Why
or why not?
(b) Use the residual plots to detect possible violations of the model assumptions. By examining
the residual plots, are there reasons to believe that the first order model is not adequate? Also
is there evidence that interaction exists between the two factors? As a result of your anallysis
what other terms would you consider adding to the model. Explain why?
(c) Based on the above residual analysis, the model
y = β0 + β1 x1 + β2 x2 + β3 x21 + β4 x1 x2 + β5 x21 x2 + was fitted to the data. Using the sums of squares from the JMP output, construct an F statistic
to test the hypothesis H0 : β4 = β5 = 0. Use a percentile from the F table to test this hypothesis
at α = .05. What does the result of this test tell you ?
(d) Use the residual plots to verify the assumptions needed to fit the model in part (c). What
improvements do these show over those from the first order model?
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2. Asphalt pavements undergo water associated deteriorations such as cracking, potholes, and surface
raveling. Research is directed to find improved asphalt pavement mixes more resistant to deterioration. An experiment is performed to compare the effects of the aggregate type used in the asphalt
mixture and the compaction methods used during construction. Combinations of two compaction
methods (Static, Kneading) and two aggregate types (Silicious Rock, Basalt) are assigned to the test
specimens completely at random and tensile bond strength (psi) measured. The four combinations
(labeled 1, 2, 3, and 4, below) are tested on 20 specimens in random order so that each combination
is replicated 5 times. The model yij = µi + ij , i = 1, . . . , 4; j = 1, . . . , 5 was used to analyze the
resulting data:
Mix (Aggregate Type, Compaction Method)
1 (Silicious, Static)
2 (Silicious, Kneading)
3 (Basalt, Static)
4 (Basalt, Kneading)
58
70
72
70
Tensile Strength (psi)
62
61
64
62
65
64
75
68
70
77
75
78
66
66
67
73
In Problem 2, compute answers to parts (b), (c), and (d), using hand calculations showing work. You
may use the JMP output obtained using the file pavement.jmp from the coursepage to check your
answers to these parts. Turn in your JMP output.
(a) Construct an analysis of variance using numbers from the JMP output. Test H0 : µ1 = µ2 =
µ3 = µ4 vs. Ha : at least one µi is different from the others, using α = .05. Use the p-value
from the JMP output to make a decision. (Note: Use s2 and its degrees of freedom from this
table in the following parts.)
(b) Use the LSD procedure to test all possible differences H0 : µi − µj = 0 vs. Ha : µi − µj 6= 0 using
α = .05. Report the results using the underlining display. Summarize your conclusions from
this procedure.
(c) Use the TUKEY procedure to test all possible differences H0 : µi − µj = 0 vs. Ha : µi − µj 6= 0
using α = .05. Report the results using the underlining display. Point out any conclusions that
are different from those made using the LSD procedure.
(d) It would be more useful to test pre-planned comparisons than testing pairwise differences among
the 4 treatment means. In this experiment the investigator could have planned to
(i) compare the average tensile strength of pavement made with silicious aggregate type with
the average tensile strength of pavement made with basalt aggregate type,
(µ1 +µ2 )
2
4)
− (µ3 +µ
2
(ii) compare the average tensile strength of pavement made using static compaction method
with the average tensile strength of pavement made using kneading compaction method,
(µ1 +µ3 )
2
4)
− (µ2 +µ
2
(iii) compare the difference between the average tensile strength of pavement made with silicious
aggregate type using each of the two compaction methods,
µ1 − µ2
(iv) compare the difference between the average tensile strength of pavement made with basalt
aggregate type using each of the two compaction methods,
µ3 − µ4
Compute appropriate t-statistics to test the corresponding comparisons given, assuming that
controlling the individual Type I error rate at α = .05 was only of interest. Write a statement
in words of your conclusion in each case. You must not use any model parameters (µ’s) in any
discussion of your results.
Due Thursday, December 11, 2014 (turn-in during lab)
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