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
University of London The London School of Economics and Political Science
Sample Surveys and Experiments
ST 205

Spring 2017
ST205
2016/17
Sample Surveys & Experiments
Course Notes
George Tzougas
This version is based heavily on the earlier editions by
Professor Chris Skinner
1
Weeks of course
Week 1. Introduction to Sample Surveys
p.3
Week 2. Probability Sampling and Estimatio
University of London The London School of Economics and Political Science
Sample Surveys and Experiments
ST 205

Spring 2017
ST203.2/ST205 Sample Surveys and Experiments
Third Exercise (for handin Week 5)  Solutions
1. A simple random sample of 100 water meters within a community is
monitored to estimate the average daily water consumption per household over a specified dry s
University of London The London School of Economics and Political Science
Sample Surveys and Experiments
ST 205

Spring 2017
Summer 2011 Examination
ST205
Sample Surveys and Experiments
(1/2 Unit)
This paper is suitable for all candidates
Instructions to candidates
Time allowed: 2 hours
This paper contains four questions. Answer three of the four questions, but include
question
University of London The London School of Economics and Political Science
Sample Surveys and Experiments
ST 205

Spring 2017
Summer 2012 Examination
ST205
Sample Surveys and Experiments
(1/2 Unit)
This paper is suitable for all candidates
Instructions to candidates
Time allowed: 2 hours
This paper contains four questions. Answer THREE of the four questions: Question 1
is compul
University of London The London School of Economics and Political Science
Sample Surveys and Experiments
ST 205

Spring 2017
ST203.2/ST205 Sample Surveys and Experiments
Second Exercise  to be handed in Week 4
This exercise is designed to help you learn material presented in the lectures
on Monday Week 3. Your answer should be posted into the coursework box
of your class teach
University of London The London School of Economics and Political Science
Sample Surveys and Experiments
ST 205

Spring 2017
ST205 Sample Surveys and Experiments 2016/17
Work for weeks 2 to 11
(dates to be conrmed in lecture and on Moodle)
Class Exercises (Always keep a copy of your work)
1. Exercise to be handed in before 12 noon on Monday 10th October (week
3), in the coursew
University of London The London School of Economics and Political Science
Sample Surveys and Experiments
ST 205

Spring 2017
ST203.2/ST205 week 2: Class Exercise and Solutions
1. Summation questions
Write each of the following expressions without the summation. Simplify as far as possible.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
10
X
i = (10) + (9) + . . . + 9 + 10 = 0
i=10
3
X
xk = x1
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
ST102 Exercise 17
In this exercise you will practise hypothesis tests involving F distributed test statistics.
Question 1 provides an opportunity to use the F distribution statistical table, i.e. Table 9
of Murdoch and Barnes Statistical Tables. Question
University of London The London School of Economics and Political Science
elementary statistics
ST 102

Winter 2014
ST102 Exercise 15
In this exercise you will practise aspects of hypothesis testing. Question 1 ensures you are
comfortable with using Murdoch and Barnes Statistical Tables.
Question 2 involves
hypothesis testing of the mean of a normal population. Questio
University of London The London School of Economics and Political Science
elementary statistics
ST 102

Winter 2014
ST102 Exercise 18
In this exercise you will practise oneway and twoway analysis of variance (ANOVA), least
squares estimation and the simple linear regression model. Question 1 requires you to explain
how a oneway ANOVA table would be produced (note no
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
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
elementary statistics
ST 102

Winter 2014
Chapter 1
Data visualisation and descriptive
statistics
1.1
Synopsis of chapter
Graphical representations of data provide us with a useful view of the distribution of
variables. In this chapter, we shall cover a selection of approaches for displaying data
University of London The London School of Economics and Political Science
elementary statistics
ST 102

Winter 2014
ST102: Text for the gaps in the course pack Chapter 7
This document contains the text which has been omitted from the course pack and filled in
during the lectures, referenced by page number.
Page 192:
We define the bias of an estimator as:
b = E()
b .
Bi
University of London The London School of Economics and Political Science
elementary statistics
ST 102

Winter 2014
ST102: Text for the gaps in the course pack Chapter 9
This document contains the text which has been omitted from the course pack and filled in
during the lectures, referenced by page number.
Page 230:
Remember:
not reject 6= accept.
Page 231:
2. Compute
University of London The London School of Economics and Political Science
elementary statistics
ST 102

Winter 2014
ST102 Exercise 1
Practicalities: Your class teacher will explain to you the dates when solutions to future
exercises should be handed in, and when they will be marked and returned to you, during your
first class. You should spend a reasonable amount of ti
University of London The London School of Economics and Political Science
elementary statistics
ST 102

Winter 2014
ST102
Elementary Statistical Theory
Chapter 2:
Probability theory
Dr James Abdey
Department
of Statistics
London School of Economics and Political Science
ST102 Elementary Statistical Theory
Dr James Abdey
MT 2016
Probability theory
1
Introduction
Consid