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Exam Paper: Regression & ANOVA (MA27210) - Univ. of Wales, Aberystwyth - May 2008, Exams of Data Analysis & Statistical Methods

This is an examination paper for the course ma27210 - regression & anova, offered by the university of wales, aberystwyth in may 2008. The paper contains various questions related to regression and analysis of variance (anova) concepts, including linear regression, anova tables, tukey multiple comparison analysis, two-way anova, contrasts, and one-way analysis of variance model. Students are expected to demonstrate their understanding and ability to apply these statistical concepts and techniques in different scenarios.

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PRIFYSGOL CYMRU / UNIVERSITY OF WALES
ABERYSTWYTH
INSTITUTE OF MATHEMATICS AND PHYSICS
SEMESTER 2 EXAMINATIONS, MAY 2008
MA27210 – Regression & ANOVA
Time allowed – 2 hours
All questions may be attempted
Marks gained from questions in Section B will be given greater consideration in
assessing a first class performance.
Calculators are permitted, provided they are silent, self-powered, without
communications facilities, and incapable of holding text or other material that could
be used to give a candidate an unfair advantage. They must be made available on
request for inspection by invigilators, who are authorised to remove any suspect
calculators.
Statistical Tables will be provided
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Download Exam Paper: Regression & ANOVA (MA27210) - Univ. of Wales, Aberystwyth - May 2008 and more Exams Data Analysis & Statistical Methods in PDF only on Docsity!

PRIFYSGOL CYMRU / UNIVERSITY OF WALES

ABERYSTWYTH

INSTITUTE OF MATHEMATICS AND PHYSICS

SEMESTER 2 EXAMINATIONS, MAY 2008

MA27210 – Regression & ANOVA

Time allowed – 2 hours

 All questions may be attempted

 Marks gained from questions in Section B will be given greater consideration in

assessing a first class performance.

 Calculators are permitted, provided they are silent, self-powered, without

communications facilities, and incapable of holding text or other material that could

be used to give a candidate an unfair advantage. They must be made available on

request for inspection by invigilators, who are authorised to remove any suspect

calculators.

 Statistical Tables will be provided

Section A

1A Three random variables have respective expectations of θ, θ+2 and 2θ. All three

observations turn out to be 5. Show that the least squares estimate of θ is 3. [8]

2A In a study of the squid eaten by sharks, 22 squid were weighed. Also noted was the

“notch to wing” length, a measure of the beak size of the squid. A model was then

fitted to the data giving the results shown on page 3.

(a) List the assumptions underlying this analysis.

(b) The MINITAB output contains the line S = 0.949435. What does this mean

and how was it calculated?

(c) The output contains an Analysis of Variance table that contains “P-value”

for a hypothesis test. What is being tested here, and what is the conclusion?

(d) The model fitted is a cubic regression. Would you say that the cubic gives a

significantly better fit than a quadratic model? Justify your answer.

(e) The table of fitted values and residuals contains two missing entries for the

second observation. Calculate the appropriate values.

(f) Explain what the solid and dotted curves on the graph mean.

(g) MINITAB points out three unusual observations, quoting two different

reasons for noting them. Explain this.

[3]

[3]

[3]

[3]

[4]

[3]

[4]

3A Twenty chickens had their water consumption noted while taking part in an

experiment involving five different diets: A,B,C,D,E. Each diet was fed to four of

the chickens The following ANOVA table was computed:

Source SS DF MS F-ratio P

Within diets 0.

Total (corr) 19.

(a) Copy and complete the table. State a model underlying the analysis and

carry out the usual hypothesis test.

(b) A Tukey multiple comparison analysis was also carried out: Interpret the

output below..

Sample averages A 9.050 B 8.400 C 7.750 D 6.900 E 6.

Tukey 95% Simultaneous Confidence Intervals All Pairwise Comparisons among Levels of diet Individual confidence level = 99.25%

diet = A subtracted from: diet Lower Center Upper --+---------+---------+---------+------- B -1.6186 -0.6500 0.3186 (----------) C -2.2686 -1.3000 -0.3314 (----------) D -3.1186 -2.1500 -1.1814 (----------) E -3.4436 -2.4750 -1.5064 (-----------) --+---------+---------+---------+------- -3.2 -1.6 -0.0 1.

diet = B subtracted from: diet Lower Center Upper --+---------+---------+---------+------- C -1.6186 -0.6500 0.3186 (----------) D -2.4686 -1.5000 -0.5314 (----------) E -2.7936 -1.8250 -0.8564 (-----*-----) --+---------+---------+---------+------- -3.2 -1.6 -0.0 1.

diet = C subtracted from: diet Lower Center Upper --+---------+---------+---------+------- D -1.8186 -0.8500 0.1186 (----------) E -2.1436 -1.1750 -0.2064 (----------) --+---------+---------+---------+------- -3.2 -1.6 -0.0 1.

diet = D subtracted from: diet Lower Center Upper --+---------+---------+---------+------- E -1.2936 -0.3250 0.6436 (-----*-----) --+---------+---------+---------+------- -3.2 -1.6 -0.0 1.

[7]

[5]

4A A study was conducted to investigate the effects of three different types of

phosphor and two types of face-plate glass on the light output in a television tube.

The light output (in coded units) are shown below:

Phosphor A Phosphor B Phosphor C

Glass 1 4

Glass 2 -

The data are to be analysed using the model

Y ijk μ αi βj γij

E

(a) List the usual restrictions on the parameters in this model.

(b) Give the values of Y 223 , Y12• and Y•2•

(c) Give the least squares estimates of α 2 and γ 12.

(d) Interpret the following ANOVA table

Two-way ANOVA: light versus glass, phosphor Source DF SS MS F glass 1 512 512.000 180. phosphor 2 49 24.500 8. Interaction 2 1 0.500 0. Error 12 34 2. Total 17 596

(e) Would you say that any dependence of light output on the different types

of phosphor is similar for both types of glass? Justify your answer.

[3]

[3]

[3]

[5]

[3]

distribution of the ratio of the mean squares.

(f) What happens to your result in (e) if these conditions do not

hold? Explain how this is exploited in the usual test procedure

carried out in such analyses.

[3]

[3]

8B The following results were reported in a study of factors affecting the

salary of high school teachers of commercial subjects. Each teacher was

classified according to sex (male or female), size of school (small,

medium, large) and years in service (<3, 3-6, 7-10, 11-14, 15-19, 20+).

Each of the resulting 36 groups contained the same number of

teachers.

Source SS DF

Sex 240 1

Size of school 420 2

Years in post 560 5

Sex x Size 20

Sex x Years 90

Size x Years 220

Sex x Size x Years 150

Within groups 1440 144

(a) How many teachers were there in each group?

(b) If the data was analysed as a one-way classification into 12 classes

according to “sex” and “years in post”, what would be the

Between Classes sum of squares?

(c) Report briefly on what the ANOVA table shows.

[2]

[2]

[8]

9B A measurement of the purity of a chemical compound varies from cask

to cask produced, and there is also variation inside individual casks.

The measurements have a mean 840; variability between casks has a

standard deviation of 4 and repeat observations inside a cask have

standard deviation 3.

Write down a one-way random effects model for this and calculate the

standard deviation of the average of nine purity measurements if

(i) all nine measurements relate to one randomly chosen cask;

(ii) the nine measurements all come from different casks.

The average of nine measurements in a particular consignment, three

from each of three casks, turns out to be 850. Would you say that this

suggests that the mean has changed? [10]