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MSc Regulation and Competition
Quantitative Techniques
QT week 6: Sampling distributions
In this lecture
Inference
Population versus sample
Sampling distributions
Distribution of the sample mean
Likelihoods and probabilities
Inference
So far we have been looking at how different kinds of casino-like data
generating processes can produce data.
Statistical inference is the art of looking at the data and making guesses about
the underlying population or data generating process.
Figure 6.1 The process of inference involves picking a box…
Population or
Data
generation
process 1
Population or
Data
generation
process 2
?
Observations or data
(sample)
?
?
?
Population or
Data
generation
process 3
Population or
Data
generation
process 4…
Two key aspects of statistical inference are estimation and hypothesis testing.
Before we get on to those we have to step back a little.
Populations
A rough definition of a population: “Set of all things under consideration” e.g.
However, the term is used in two slightly different senses – a) the set of
individual items in which we are interested and b) one or more measured
attributes of that set of items.
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Table 6.1: Examples of populations
Group
People entitled to vote in the next election
All past and future students of City University
Potential applicants to course X
Grains of sand in a bucket
Fish in the North Sea
Owners of Porsche 911s
Present and future members of the human species
Outcomes of random number generating process
Quantitative Techniques
Attribute e.g.
Height; voting intention
Ownership of laptops
Current country of residence
Mass; hardness
Mercury levels
IQ score
Length of life; ability to jump
Value
The attribute can be any kind of data discussed in Lecture 1. The population
can be finite or infinite, as can the set of possible outcomes.
A “population” in the normal sense can be regarded as a sample in the
statistical sense. The people currently living on Orkney might be taken as a
sample of all people living north of Edinburgh.
Samples
The main purpose of discussing populations is to distinguish them from
samples. Normally samples are all we observe, and an important job of
statistics is to make inferences about attribute of the population as a whole
from measured characteristics of the sample. This is one definition of
statistical inference.
Sampling can be systematic or random. With simple random sampling every
member of the population has the same chance of being chosen. We shall
focus on the characteristics of simple random sampling.
There are several other approaches to sampling. For example, stratified
sampling can, if done properly, give more accurate results than simple
random sampling.
The opposite of random sampling is selective sampling. For example, one’s
sample might be based only on people passing Piccadilly Circus between 3
and 4 pm on Friday 13th. Or people may select themselves for interview etc.
Selective sampling potentially gives rise to selection bias.
Sometimes quota sampling is used as a means of both stratifying a sample
and reducing the selection bias that arises because random sampling may be
difficult to achieve in practice.
Survey specialists need to understand these biases and design their surveys
accordingly. We shall ignore (“abstract from”) these problems for the time
being.
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Sampling distributions
Things like the means and standard deviations of samples have the properties
of probability distributions and density functions. These are called sampling
distributions.
Suppose we take a sample of size 1 from a population. How will it be
distributed?
Exactly like the original population.
Now suppose we take a sample of size 2. Each member will be distributed
like the population. But how will the sample mean be distributed?
How will the mean be distributed as the sample size increases?
The Central Limit Theorem says that, as the sample size n increases these
means will be distributed:
a) approximately normally
b) with a mean equal to the population mean
c) with a variance equal to the population variance/n.
Lab 5 and Coursework 1 ask you to investigate the Central Limit theorem
empirically using the skills you have built up so far in the labs.
We can use this knowledge to work out confidence intervals.
e.g. Suppose our sample estimate of the IQ level of a group of 16 students in
a single year of the course is 120 and we know that the standard deviation is
10 we can get an estimate of the 95% confidence interval for the population
from which the students are drawn.
This is the process: the explanation will follow.
1. Take the sample mean Xbar.
2. Get the population variance and divide by n. Take the square root to get
the standard deviation sxbar.
3. Find the critical values z of the Normal distribution from the Tables e.g.
Appendix B1 in Ashenfelter et all. In this case 1.96.
4. Then the 95% confidence interval goes
from Xbar - z.sxbar
to Xbar + z.sxbar
In the present case
Xbar =120
z =1.96
sxbar = 10/√16
= 2.5
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z.sxbar = 1.96 x 2.5 = 4.9
So the 95% confidence interval runs from 115.1 to 124.9
Supplementary question: what is “the population from which the students are
drawn”.
Likelihoods and probabilities
Although this looks straightforward enough, strictly speaking we are talking
about likelihoods rather than probabilities here.
Econometricians make more of a distinction here than normal people.
Roughly speaking, a probability refers to an event which has not happened
and which therefore can be influenced by random factors. A likelihood refers
to something that has happened or is not subject to future random factors, but
about which we have incomplete knowledge.
Example: Consider the table you may have produced in lab 4 showing the
probabilities of different numbers of heads in five tosses depending on the
bias in the coin.
Table 6.2 The binomial table as a function g(h,p)
Likelihood
↓
p (h)
0
0.1
0.2
0.3
0
1
0.59049
0.32768
0.16807
1
0
0.32805
0.4096
0.36015
0.4
0.5
0.6
0.7
0.8
0.9
1
0.07776
0.03125
0.01024
0.00243
0.00032
1E-05
0
0.2592
0.15625
0.0768
0.02835
0.0064
0.00045
0
Number of heads h
2
3
0
0
0.0729
0.0081
0.2048
0.0512
0.3087
0.1323
0.3456
0.3125
0.2304
0.1323
0.0512
0.0081
0
0.2304
0.3125
0.3456
0.3087
0.2048
0.0729
0
4
0
0.00045
0.0064
0.02835
5
0
0.00001
0.00032
0.00243
0.0768
0.15625
0.2592
0.36015
0.4096
0.32805
0
0.01024
0.03125
0.07776
0.16807
0.32768
0.59049
1
Probabilities →
We can view Table 6.2 in two ways:
a) going along the rows, for a given value of p, what is the probability
associated with a given number of heads out of five i.e. g(h|p) ;
b) going down the columns, for a given number of heads, what is the
likelihood that it was generated by a coin with a particular characteristic
p ? i.e. g(p|h). The latter is sometimes written as L(p|h). (L for
likelihood).
Question: if I observe two heads out of five, what value of p maximises the
likelihood function?
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Question: For the above example of the confidence interval, identify and draw
a) a probability distribution when  = 115.1 b) a relevant likelihood function
when Xbar =120.
Reading
Ashenfelter et. al. Chapter 6 . Salvatore and Reagle Chapter 4.
Exercises
Ashenfelter 6.11 numbers 1, 2, 3,4, 5, 6, 8.
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