STATISTICS 402B
Spring 2016
Homework Set#2
1. A study is performed on the effect of temperature on the strength of plastic welds. Currently a temperature
of 170 ◦ C is used for welding plastic parts. There are seven different temperatures (140, 150, 160, 170, 180,
190 and 200 ◦ C. At each temperature five specimens of the plastic material are welded. The welds are then
subjected to a load until the weld breaks. The maximum load (in Newtons) before the weld breaks is the
strength of the weld. The thirty five welds (5 specimens at each of 7 temperatures) are run in a random
order, with a specimen being assigned to a temperature level completely at random. All welds are made
with the same equipment by the same operator. All strengths are measured with the same equipment by
the same operator.
(a) Why is this study an experiment and not an observational study?
(b) What is the response variable? Is this variable categorical or numerical?
(c) What is the factor under study? What are the treatments? Be specific.
(d) Identify an external source of variability that is controlled in the study and indicate how that variability
is controlled.
(e) Is there a control/check/comparison group in the study? Explain briefly.
(f) How is replication within this study accomplished?
(g) Explain clearly how you would estimate the error variance (σ 2 ) in this experiment.
2. In each of the following examples identify the experimental units and the treatment(s) applied (or the
experimental run and the experimental condition under study, as the case may be). Note carefully that, in
each case, the observations were obtained from replication of the treatment. Use one of these examples to
explain what is meant by experimental error and show how to measure it from the data. Write down at
least 2 possible sources of experimental error in each example (as categorized in p.8 of Note Set #2). No
calculations required.
(a) To estimate the lifetime of car batteries (in weeks) of a certain brand, 5 batteries of the same brand
were tested under similar conditions with the following results: 108, 100, 96, 98, and 100.
(b) In order to obtain an idea of the moisture retention properties of a packaging material, 10 samples
of the same food product are wrapped with the material and stored for a fixed period of time and
moisture loss of the contents measured. The results were: 201, 218, 198, 252, 243, 241, 238, 211, 227, 232
(c) Two different methods (A and B) were compared for determining dissolved oxygen concentration in
water (in mg/ltr). Twelve samples of water from the same source were tested in random order using
each method and the following results obtained:
Sample
Method A
Method B
1
2.62
2.73
2
2.65
2.80
3
2.79
2.87
4
2.83
2.95
5
2.91
2.99
6
3.57
3.67
3. Four specimens each of five brands of a synthetic wood veneer material are subjected to a friction test.
This type of testing is done in machines that simulate wear of material by artificially subjecting various
material to abrasion. A measure of wear (yij , usually weight loss) is determined for each specimen. All
tests are made on the same machine using randomly selected samples of each brand in completely random
order. The data are:
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Acme
2.3 (16)
2.4 (13)
2.1 (17)
2.5 ( 6)
Brand of Veneer
Champ
Ajax
Tuffy
2.2 (19) 2.2 ( 3) 2.4 ( 7)
2.3 ( 4) 2.0 (12) 2.7 ( 9)
2.4 ( 8) 1.9 (14) 2.6 (15)
2.6 (11) 2.1 ( 1) 2.7 ( 5)
Xtra
2.3 ( 2)
2.5 (18)
2.3 (10)
2.4 (20)
Use JMP to perform the analysis needed. However, extract numbers from the JMP output to present your
own written answers to the parts below. Assume the model yij = µi + ij , i = 1, . . . , 5; j = 1, . . . , 4 where
ij are iid N (0, σ 2 ).
(a) Construct an analysis of variance table and test the hypothesis H0 : µ1 = µ2 = µ3 = µ4 = µ5 using
the p-value and α = .05. What are the best unbiased estimates of µi , i = 1, . . . , 5 and σ 2 .
(b) Calculate the residuals from fitting this model to the data. Present these in a table like the data table.
Obtain plots needed for model adequacy checking as in Section 3.4 of the text. Do these indicate any
departures from the model assumptions?
(c) Compute 95% confidence intervals for the differences in population means for all pairs of treatments
(there are 10 such differences).
(d) Compute the LSD at α = .05 and conduct the underscoring procedure entirely by hand. You may use
the JMP output to obtain the statistics needed to calculate the LSD. Also perform the LSD procedure
as a part of the JMP output. Make a concluding statement from your analysis of the above results.
(e) Repeat the previous part but using the Tukey HSD procedure including the hand computation. Are
there any differences from the conclusions you made from the LSD procedure?
4. The effect of 4 different electrolytes (labeled 1,2,3, and 4 below) on the lifetime of automobile batteries is
being studied. Electrolyte 1 and 2 are quite similar in chemical composition, 3 and 4 are also somewhat
similar but 1 and 2 differ considerably from 3 and 4. Thus it is reasonable to plan comparisons of
• Electrolyte 1 vs. Electrolyte 2
• Electrolyte 3 vs. Electrolyte 4
• Electrolyte 1 & 2 vs. Electrolyte 3 & 4
A completely randomized design is employed for the experiment with 5 replications of each type of electrolyte and following data (with lifetime in months) are obtained:
Electrolytes
1
2
3
4
34 35 45 41
32 36 46 44
35 33 44 43
33 34 49 41
35 34 47 42
(a) Use JMP to analyze this data. State an appropriate model and give best estimates of all parameters,
Report the ANOVA table and use the F-statistics and the p-value to test whether the mean lifetime
from the 4 electrolytes were the same.
(b) To make the above comparisons the following hypotheses may be tested:
i. H0 : µ1 = µ2 vs. Ha : µ1 6= µ2 .
ii. H0 : µ3 = µ4 vs. Ha : µ3 6= µ4 .
iii. H0 : (µ1 + µ2 )/2 = (µ3 + µ4 )/2 vs.
Ha : (µ1 + µ2 )/2 6= (µ3 + µ4 )/2.
Write the contrast coefficients appropriate for testing each of the above 3 comparisons.
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(c) Check to verify that these comparisons are pairwise orthogonal. Use JMP to obtain the estimates
C1 , C2 , and C3 of the above comparisons and their standard errors, respectively.
(d) Use t-statistics to test each of the above hypotheses. State the hypotheses, extract the corresponding
t-value and the p-value from the JMP output and include in your answer. Make a decision from each
test and write your conclusion.
(e) Expand the ANOVA table incoporating the sums of squares corresponding to each contrast (extracted
from the JMP output) as illustrated for the Plasma Etching experiment (see table in page 6 of Note
Set #4 or Table 3.11 in the text). Test each comparison using an F-statistic that you compute in
your ANOVA table (compute p-values using an internet F distribution calculator). Verify that the
sum of the 3 single degree of freedom sums of squares equals the treatment sum of squares (say, upto
2 decimal places accuracy).
Note: Need to present written answers to each part, with calculation shown for parts that you are required to do
hand computation. Use the JMP output to obtain numbers for answering other parts. Attach edited JMP output
when you use the JMP output to extract numbers as part of the analysis. The JMP data files Wood-Veneer.jmp
and electrolyte.jmp are available to download.
Due Monday, February 15, 2016 (turn-in at the beginning of class)
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