Statistical Modelling Exam 22/02/2024 Exercise 1 At admission to a college an entrance test is administered to 20 randomly selected students. The study aims to determine whether a student's grade point average (GPA) at the end of the first year (y) can be predicted from the entrance test score (x). Assume that a Gaussian linear model Y₁ = ẞ₁ + B₂xi + is fitted. The observed values of the entrance test and of the GPA for the students are: unit 1 2 3 4 5 6 7 8 9 10 x 5.50 4.80 4.70 3.90 4.50 6.20 6.00 5.20 4.70 4.30 y 3.10 2.30 3.00 1.90 2.50 3.70 3.40 2.60 2.80 1.60 unit 11 12 13 14 15 y 2.00 x 4.90 5.40 5.00 2.90 2.30 4.60 4.30 5.00 18 6.30 5.90 3.20 1.80 1.40 2.00 3.80 16 17 19 20 4.10 4.70 2.20 1.50 From the data it is possible to compute the following useful quantities: 20 Σ i=1 20 Ii = 100.00 Σ ε = 50 Στ i=1 20 Σε = 509.12 Σ i=1 Moreover, from fitting the model we know that var (B₁(Y)) = 0.72682 == 20 20 20 Στ:1 = 257.66 Σε = 134.84 i=1 i=1 var (B2(Y)) = 0.14402 Σ(1 - 1)2 = 3.4063 i=1 a) State the assumptions on εi, i = 1,..., 20. === b) Obtain the maximum likelihood estimates of (31, 32) and interpret them. c) Write the expression of the estimated regression function and plot it (in the figure below). d) Obtain a 0.99 confidence interval for 32. Does it include zero? Why might one be interested in whether the confidence interval includes zero? e) Is 32 statistically significant using a significance level of 1%? f) Two new students "A" and "B" take the test. The result of the entrance test of student "A" is 5.0, the result of the entrance test of student "B" is 6.5. Imagine you want to estimate their mean GPA using a 0.95 confidence interval. What student do you expect to have the wider confidence interval? Why? т 2 3 1 2 5 6
Statistical Modelling Exam 22/02/2024 Exercise 1 At admission to a college an entrance test is administered to 20 randomly selected students. The study aims to determine whether a student's grade point average (GPA) at the end of the first year (y) can be predicted from the entrance test score (x). Assume that a Gaussian linear model Y₁ = ẞ₁ + B₂xi + is fitted. The observed values of the entrance test and of the GPA for the students are: unit 1 2 3 4 5 6 7 8 9 10 x 5.50 4.80 4.70 3.90 4.50 6.20 6.00 5.20 4.70 4.30 y 3.10 2.30 3.00 1.90 2.50 3.70 3.40 2.60 2.80 1.60 unit 11 12 13 14 15 y 2.00 x 4.90 5.40 5.00 2.90 2.30 4.60 4.30 5.00 18 6.30 5.90 3.20 1.80 1.40 2.00 3.80 16 17 19 20 4.10 4.70 2.20 1.50 From the data it is possible to compute the following useful quantities: 20 Σ i=1 20 Ii = 100.00 Σ ε = 50 Στ i=1 20 Σε = 509.12 Σ i=1 Moreover, from fitting the model we know that var (B₁(Y)) = 0.72682 == 20 20 20 Στ:1 = 257.66 Σε = 134.84 i=1 i=1 var (B2(Y)) = 0.14402 Σ(1 - 1)2 = 3.4063 i=1 a) State the assumptions on εi, i = 1,..., 20. === b) Obtain the maximum likelihood estimates of (31, 32) and interpret them. c) Write the expression of the estimated regression function and plot it (in the figure below). d) Obtain a 0.99 confidence interval for 32. Does it include zero? Why might one be interested in whether the confidence interval includes zero? e) Is 32 statistically significant using a significance level of 1%? f) Two new students "A" and "B" take the test. The result of the entrance test of student "A" is 5.0, the result of the entrance test of student "B" is 6.5. Imagine you want to estimate their mean GPA using a 0.95 confidence interval. What student do you expect to have the wider confidence interval? Why? т 2 3 1 2 5 6
Big Ideas Math A Bridge To Success Algebra 1: Student Edition 2015
1st Edition
ISBN:9781680331141
Author:HOUGHTON MIFFLIN HARCOURT
Publisher:HOUGHTON MIFFLIN HARCOURT
Chapter4: Writing Linear Equations
Section: Chapter Questions
Problem 12CR
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For context, the images attached below (question and related graph) are from a February 2024 past paper in statistical modeling
Transcribed Image Text:Statistical Modelling
Exam 22/02/2024
Exercise 1
At admission to a college an entrance test is administered to 20 randomly selected students. The
study aims to determine whether a student's grade point average (GPA) at the end of the first
year (y) can be predicted from the entrance test score (x). Assume that a Gaussian linear model
Y₁ = ẞ₁ + B₂xi + is fitted. The observed values of the entrance test and of the GPA for the
students are:
unit
1
2
3
4
5
6
7
8
9
10
x
5.50
4.80
4.70 3.90
4.50
6.20
6.00
5.20
4.70
4.30
y
3.10
2.30
3.00
1.90
2.50
3.70
3.40
2.60 2.80 1.60
unit
11
12
13
14
15
y 2.00
x 4.90 5.40 5.00
2.90 2.30
4.60 4.30 5.00
18
6.30
5.90
3.20 1.80 1.40 2.00 3.80
16
17
19 20
4.10 4.70
2.20
1.50
From the data it is possible to compute the following useful quantities:
20
Σ
i=1
20
Ii
=
100.00
Σ ε = 50
Στ
i=1
20
Σε = 509.12
Σ
i=1
Moreover, from fitting the model we know that
var (B₁(Y)) = 0.72682
==
20
20
20
Στ:1 = 257.66
Σε = 134.84
i=1
i=1
var (B2(Y)) = 0.14402
Σ(1 - 1)2 = 3.4063
i=1
a) State the assumptions on εi, i = 1,..., 20.
===
b) Obtain the maximum likelihood estimates of (31, 32) and interpret them.
c) Write the expression of the estimated regression function and plot it (in the figure below).
d) Obtain a 0.99 confidence interval for 32. Does it include zero? Why might one be interested
in whether the confidence interval includes zero?
e) Is 32 statistically significant using a significance level of 1%?
f) Two new students "A" and "B" take the test. The result of the entrance test of student
"A" is 5.0, the result of the entrance test of student "B" is 6.5. Imagine you want to
estimate their mean GPA using a 0.95 confidence interval. What student do you expect to
have the wider confidence interval? Why?
Transcribed Image Text:т
2
3
1
2
5
6
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