Semester 1 Assessment, 2018
School of Mathematics and Statistics
MAST30025 Linear Statistical Models
Writing time: 3 hours
Read
- Length:Load 4 0.13575 0.30167 -91.345
- Length:Amplitude 4 0.40116 0.56707 -74.304
Step: AIC=-105.21
log(Cycles) ~ Length + Amplitude + Load + Length:Amplitude +
Length:Load
Df Sum of Sq RSS AIC
- Length:Load 4 0.13575 0.31626 -98.069
- Length:Amplitude 4 0.40116 0.58167 -81.618
> summary(model4)
Call:
lm(formula = log(Cycles) ~ Length + Amplitude + Load + Length:Amplitude +
Length:Load, data = wool)
Residuals:
Min 1Q Median 3Q Max
-0.153728 -0.055232 -0.008017 0.067786 0.175706
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 6.384806 0.091416 69.843 < 2e-16 ***
Length300 0.913780 0.129282 7.068 1.30e-05 ***
Length350 1.963516 0.129282 15.188 3.37e-09 ***
Amplitude9 -0.449946 0.100142 -4.493 0.000735 ***
Amplitude10 -1.232398 0.100142 -12.307 3.65e-08 ***
Load45 -0.401464 0.100142 -4.009 0.001734 **
Load50 -0.649468 0.100142 -6.485 3.00e-05 ***
Length300:Amplitude9 -0.001114 0.141622 -0.008 0.993851
Length350:Amplitude9 -0.614678 0.141622 -4.340 0.000961 ***
Length300:Amplitude10 0.064964 0.141622 0.459 0.654638
Length350:Amplitude10 -0.152966 0.141622 -1.080 0.301328
Length300:Load45 0.083463 0.141622 0.589 0.566565
Length350:Load45 0.145059 0.141622 1.024 0.325914
Length300:Load50 -0.133655 0.141622 -0.944 0.363913
Length350:Load50 -0.273658 0.141622 -1.932 0.077269 .
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MAST30025 Semester 1, 2018
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.1226 on 12 degrees of freedom
Multiple R-squared: 0.9922, Adjusted R-squared: 0.9831
F-statistic: 109.3 on 14 and 12 DF, p-value: 1.968e-10
> qt(0.975,20:27)
[1] 2.085963 2.079614 2.073873 2.068658 2.063899 2.059539 2.055529 2.051831
(a) Identify the features in the diagnostic plots which support the use of the logarithmic trans-
formation on the Cycles variable.
(b) From the additive model, calculate a 95% confidence interval for the average ratio of the
number of cycles to failure for 50g loads against 40g loads. (Hint: The logarithm of the ratio
is the difference of the logarithms.)
(c) From the additive model, test whether length has an effect on wool strength, at the 5%
significance level.
(d) Calculate the change in AIC if the amplitude variable was removed from the additive model.
(e) Test for the presence of 2-way interaction between the factors.
(f) Is your answer above consistent with the results of the variable selection? Why or why not?
(g) Using the model resulting from variable selection, calculate a point estimate for the average
number of cycles to failure for a wool specimen of length 350mm, loading cycle amplitude
8mm, with 45g load.
Question 6 (14 marks) Consider the general linear model, y = Xβ+ ε. This model may be of full
or less than full rank.
(a) Explain the difference between an error and a residual.
(b) Define and explain the purpose of the standardised residual of a point.
(c) When is a model with fewer explanatory variables more desirable than a model with more
explanatory variables? When is it less desirable?
(d) State the general linear hypothesis and explain how it is tested.
(e) Define a treatment contrast and explain its usage.
(f) Explain what randomisation is and its use in experimental design.
(g) Explain what a Latin square is and its use in experimental design.
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MAST30025 Semester 1, 2018
Question 7 (8 marks) An experiment compares four different mixtures of the components of a
rocket propellant; the mixtures contain different proportions of oxidizer, fuel, and binder. To
compare the mixtures, five different samples of propellant are prepared for each mixture. Each
of five investigators is randomly assigned one sample of each of the four mixtures and is asked to
measure the propellant thrust. The data is given below:
Mixture Investigator Mixture
1 2 3 4 5 Total
A 2340 2355 2362 2350 2348 11755
B 2658 2650 2665 2640 2653 13266
C 2449 2458 2432 2437 2445 12221
D 2403 2410 2418 2397 2405 12033
Investigator
Total 9850 9873 9877 9824 9851
(a) What type of experimental design is described above?
(b) Which are the treatment and blocking variables in this experiment?
(c) Is it better to analyse this data as a complete block design or completely randomised design?
Justify your answer.
(d) A larger experiment is planned with the goal of testing whether mixture D, a newly developed
formula, is more effective than industry standard mixtures A and B. This experiment has
resources to prepare 100 samples of propellant. Calculate the best number of samples for
each mixture. (Hint: In a completely randomised design with treatment effects τi, we have
var τi =
σ2
ni
. To minimise a function f(x) under the constraint g(x) = c, minimise f(x, λ) =
f(x) + λ(g(x)− c).)
End of Exam—Total Available Marks = 90.
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