University of London The London School of Economics and Political Science
Homework exercises
ST 102

Spring 2015
ST107 Exercise 10
1. Explain the main types of variables which statistical methods are used to study.
What are the dierences between association and correlation as statistical terms?
2. A researcher into the use of computational aids conducts a survey in
University of London The London School of Economics and Political Science
Homework exercises
ST 102

Spring 2015
ST107 Outline solutions to Exercise 9
1. (a) This question requires the test of a single proportion. We test the hypotheses:
H0 : = 0.01
vs.
H1 : > 0.01.
We use the following test statistic:
Z=
P
(1 )/n
N (0, 1)
approximately, since n is large. Under H0
University of London The London School of Economics and Political Science
Homework exercises
ST 102

Spring 2017
ST102 Class 11 Additional exercises
1. Consider a sequence of random variables X1 , X2 , X3 , . . . that are independent and
normally distributed with mean 0 and variance 1. Using as many of these random
variables as you like construct a random variable t
University of London The London School of Economics and Political Science
ST 102

Spring 2016
ST102
Elementary Statistical Theory
Example workshop:
Sampling distributions of statistics
Dr James Abdey
Department
of Statistics
London School of Economics and Political Science
ST102 Elementary Statistical Theory
Dr James Abdey
LT 2017
Example worksho
University of London The London School of Economics and Political Science
ST 102

Spring 2016
ST102
Elementary Statistical Theory
Example workshop:
Point estimation
Dr James Abdey
Department of Statistics
London School of Economics and Political Science
ST102 Elementary Statistical Theory
Dr James Abdey
LT 2017
Example workshop: Point estimation
University of London The London School of Economics and Political Science
ST 102

Spring 2016
ST102 Outline solutions to Exercise 10
1. (a) The sum of n independent Poisson() random variables follows the Poisson(n)
distribution.
P
= P Xi /n are
(b) Since i Xi has possible values 0, 1, 2, . . . , the possible values of X
i
0/n, 1/n, 2/n, . . .
The
University of London The London School of Economics and Political Science
ST 102

Spring 2016
ST102
Elementary Statistical Theory
Example workshop:
Descriptive statistics
Dr James Abdey
Department
of Statistics
London School of Economics and Political Science
ST102 Elementary Statistical Theory
Dr James Abdey
MT 2016
Example workshop: Descriptive
University of London The London School of Economics and Political Science
ST 102

Spring 2016
ST102 Exercise 9
In this exercise you will practise various topics related to multivariate random variables.
Questions 1 and 2 cover discrete bivariate distributions. Questions 3, 4 and 5 concern
linear combinations of independent random variables. Questi
University of London The London School of Economics and Political Science
ST 102

Spring 2016
ST102 Exercise 12
In this exercise you will practise deriving expressions for the mean and variance of a linear
combination of random variables in Question 1. Question 2 requires you to explore the
properties of an estimator of the parameter of a Bernoull
University of London The London School of Economics and Political Science
ST 102

Spring 2016
ST102
Elementary Statistical Theory
Chapter 4:
Common distributions of random variables
Dr James Abdey
Department of Statistics
London School of Economics and Political Science
ST102 Elementary Statistical Theory
Dr James Abdey
MT 2015
Common distributio
University of London The London School of Economics and Political Science
ST 102

Spring 2016
ST102
Elementary Statistical Theory
Chapter 3:
Random variables
Dr James Abdey
Department of Statistics
London School of Economics and Political Science
ST102 Elementary Statistical Theory
Dr James Abdey
MT 2015
Random variables
1
Introduction
In Chapter
University of London The London School of Economics and Political Science
ST 102

Spring 2016
ST102
Elementary Statistical Theory
Chapter 5:
Multivariate random variables
Dr James Abdey
Department of Statistics
London School of Economics and Political Science
ST102 Elementary Statistical Theory
Dr James Abdey
MT 2015
Multivariate random variables
University of London The London School of Economics and Political Science
ST 102

Spring 2016
ST102 Class 5 Additional exercises
1. A discrete random variable X has possible values 0, 1, 2, . . . , n, where n is a known integer.
The probability function of X is:
(
n
x
nx
for x = 0, 1, 2, . . . , n
x (1 )
p(x) =
0
otherwise
where nx = n!/[x! (n x)
University of London The London School of Economics and Political Science
ST 102

Spring 2016
ST102 Class 4 Solutions to Additional exercises
1. Let:
i = P (the best candidate is hired  the hiring occurs in the ith interview).
With n candidates, after i 1 rejections there are n (i 1) = n i + 1 remaining
candidates. Since candidates are selected f
University of London The London School of Economics and Political Science
ST 102

Spring 2016
ST102
Elementary Statistical Theory
Example workshop:
Multivariate random variables
Dr James Abdey
Department of Statistics
London School of Economics and Political Science
ST102 Elementary Statistical Theory
Dr James Abdey
MT 2016
Example workshop: Mult
University of London The London School of Economics and Political Science
ST 102

Spring 2016
ST102
Elementary Statistical Theory
Example workshop:
Probability theory
Dr James Abdey
Department
of Statistics
London School of Economics and Political Science
ST102 Elementary Statistical Theory
Dr James Abdey
MT 2016
Example workshop: Probability the
University of London The London School of Economics and Political Science
ST 102

Spring 2016
ST102 Outline solutions to Exercise 9
1. (a) The marginal distributions are found by adding across rows and columns:
1
0.50
X=x
pX (x)
0
0.25
1
0.25
and:
0
0.45
Y =y
pY (y)
1
0.55
(b) For example, pX,Y (0, 0) = 0.10, pX (0) = 0.25 and pY (0) = 0.45. We ha
University of London The London School of Economics and Political Science
Homework exercises
ST 102

Spring 2017
ST102 Class 12 Additional exercises
1. Let cfw_X1 , X2 , . . . , Xn be a random sample of size n from the pf:
(
x (1 )1x for x = 0, 1
p(x; ) =
0
otherwise.
Find the method of moments estimator of .
2. Let cfw_X1 , X2 , . . . , Xn be a random sample of
University of London The London School of Economics and Political Science
Homework exercises
ST 102

Spring 2017
ST102 Class 10 Additional exercises
1. Suppose Xi N (0, 3), for i = 1, 2, 3, 4. Assume all these random variables are
independent. Derive the value of k in each of the following.
(a) P (X1 + 3X2 > 4) = k.
(b) P (X12 + X22 + X32 + X42 < k) = 0.99.
p
(c) P
University of London The London School of Economics and Political Science
ST 102

Spring 2016
ST102 Class 14 Solutions to Additional exercises
1. Since n is large, then approximately:
X
N
n
(1 )
,
n
.
Since:
P (0.67 < Z < 0.67) = 0.4971 < 0.50 = P (0 < Z < )
where Z N (0, 1), then:
!
r
r
X
(X/n) (1 X/n) X
(X/n) (1 X/n)
0.67
,
+ 0.67
n
n
n
n
ha
University of London The London School of Economics and Political Science
ST 102

Spring 2016
Chapter 11
Linear regression
11.1
Synopsis of chapter
This chapter covers linear regression whereby the variation in a continuous dependent
variable is modelled as being explained by one or more continuous independent
variables.
11.2
Learning outcomes
Aft
University of London The London School of Economics and Political Science
ST 102

Spring 2016
Chapter 9
Hypothesis testing
9.1
Synopsis of chapter
This chapter discusses hypothesis testing which is used to answer questions about an
unknown parameter. We consider how to perform an appropriate hypothesis test for a
given problem, determine error pro
University of London The London School of Economics and Political Science
ST 102

Spring 2016
Chapter 10
Analysis of variance (ANOVA)
10.1
Synopsis of chapter
This chapter introduces analysis of variance (ANOVA) which is a widelyused technique
for detecting differences between groups based on continuous dependent variables.
10.2
Learning outcomes
University of London The London School of Economics and Political Science
ST 102

Spring 2016
ST102 Class 16 Additional exercises
1. Suppose that two independent random samples, of sizes m and n, respectively, are drawn
from a normal distribution with variance 2 . Let S12 and S22 denote the two sample
variances. Use the fact that (n 1)S 2 / 2 2n1
University of London The London School of Economics and Political Science
ST 102

Spring 2016
ST102 Class 15 Solutions to Additional exercises
1. For n = 16, = 4 and = 0.05, H0 : = 60 should be rejected in favour of a twosided
H1 if:
x
60
x
60
< 1.96
> 1.96.
or
4/ 16
4/ 16
Rearranging, we reject H0 if:
4
x
< 1.96 + 60 = 58.04
16
or
4
x
> 1.
University of London The London School of Economics and Political Science
ST 102

Spring 2016
ST102 Class 17 Solutions to Additional exercises
1. If Y t8 , then X = Y 2 F1, 8 . Also:
0.2 = P (X > c) = P (Y >
Therefore, P (Y >
we have:
c) + P (Y < c) = 2 P (Y > c).
c) = 0.1. Using Table 7 of Murdoch and Barnes Statistical Tables,
P (Y > 1.397) = 0.
University of London The London School of Economics and Political Science
ST 102

Spring 2016
ST102 Class 16 Solutions to Additional exercises
1. Since the random samples are independent, and (n 1)S 2 / 2 2n1 , it follows that:
S12
(m 1)S12 /(m 1) 2
=
Fm1, n1 .
(n 1)S22 /(n 1) 2
S22
As m and n increase, the distributions of the unbiased estimator
University of London The London School of Economics and Political Science
ST 102

Spring 2016
ST102 Class 15 Additional exercises
1. Construct a power function for the = 0.05 test of H0 : = 60 vs. H1 : 6= 60 if the
data consist of a random sample of size 16 from a normal distribution having = 4.
2. A random sample of size 1 is taken from the proba
University of London The London School of Economics and Political Science
ST 102

Spring 2016
ST102 Class 17 Additional exercises
1. Suppose that a random variable X has the F distribution with 1 and 8 degrees of freedom.
Use the table of the t distribution to determine the value of c such that P (X > c) = 0.2.
2. What is the value of the median o
University of London The London School of Economics and Political Science
elementary statistics
ST 102

Winter 2014
Examiners commentaries 2015
Examiners commentary 2015
ST102 Elementary Statistical Theory
General remarks
Learning outcomes
By the end of this module you should:
be able to summarise the ideas of randomness and variability, and the way in which these lin