Midterm_test_Mat_3375
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School
University of Ottawa *
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Course
3375
Subject
Mathematics
Date
Jan 9, 2024
Type
Pages
9
Uploaded by ghdjdhd
MAT 3375
Midterm test
October 19 2023
Professor M. Alvo
Time: 70 minutes
Student number
:
Given name
:
Family name
:
This is an open book exam. Calculators are permitted. Answer all questions
in the spaces provided. The test is out of 20. Each question is worth 2 points.
1
We are interested in studying the relationship between the degree of brand
liking
Y
and moisture content (
X
1
) and sweetness (
X
2
) of a certain product.
The data is given below
Liking
Moisture
Sweetness
1
64
4
2
2
73
4
4
3
61
4
2
4
76
4
4
5
72
6
2
6
80
6
4
7
71
6
2
8
83
6
4
9
83
8
2
10
89
8
4
11
86
8
2
12
93
8
4
13
88
10
2
14
95
10
4
15
94
10
2
16
100
10
4
Mean
81.75
7
3
Variance
131.1333
5.333
1.0667
2
a. What can you deduce from the scatter and Box plots below?
3
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b.
Indicate the usual assumptions made in fitting the following simple
linear model
Y
i
=
β
0
+
β
1
X
i
1
+
ε
i
, i
= 1
, ...,
16
c. Complete the analysis of variance table below whereby the degree of
brand liking
Y
is regressed against moisture content (
X
1
) .
Response: Liking
Sum of Squares
df
Mean Square
F
p-value
Moisture
1566.45
Residuals
400.55
Total
15
4
d. From the output of the coefficients table below, indicate the estimated
regression function for the model in part b.
Compute the test statistic to test the null hypothesis
H
0
:
β
1
= 0 against
the alternative
H
1
:
β
1
̸
= 0.
(Coefficients)
Estimate
Std. Error
t-value
p-value
Intercept
50
.
775
4
.
395
11
.
554
10
e
−
08
Moisture
4
.
425
5
e. For the model in part b. compute a 95% confidence interval for the
mean of the response when Moisture =5.
f. Using the extra sum of squares principle, conduct a formal test to see
if the variable sweetness (
X
2
) should be included in the regression function
using level
α
= 0
.
05. Use the information in the table below.
Model1 Liking˜Moisture
Model2 Liking˜Moisture+Sweetness
Model
Residual Sum of Squares
Res. d.f.
Sum of Squares
F
p-value
1
400.55
2
94.30
6
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g. Estimate
β
1
, β
2
jointly by the Bonferroni procedure using 95% confi-
dence intervals using the results in the table below when the full model is
fitted
(Coefficients)
Estimate
Std. Error
t-value
p-value
Intercept
37
.
6500
2
.
9961
12
.
566
1
.
20
e
−
08
Moisture
4
.
425
0
.
3011
14
.
695
1
.
78
e
−
09
Sweetness
4
.
3750
0
.
6733
6
.
498
2
.
01
e
−
05
h. Express in matrix notation an interval estimate at a 95% confidence
of the mean service time for a new observation when
x
h
1
= 5 and
x
h
2
= 4
7
i. What do the attached plots reveal about the assumptions of constant
variance and normality when the full model is fitted?
8
j) What are the % of variation explained by each of the two regressions
fitted? The anova table below is a summary when both Moisture and Sweet-
ness are fitted in the regression model.
Response: Liking
Sum of Squares
df
Mean Square
F
p-value
Moisture
1566
.
45
Sweetness
306
.
25
Residuals
Total
15
9
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