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Estimation
1. Appreciate the importance of random sampling
2. Understand the concept of estimation from samples
3. Understand the Central Limit Theorem
4. Be able to determine unbiased estimates of the
variance
What is so important about these words?
• Population
– A Population is the set of all elements of interest for a
particular study. Quantities such as the population mean μ are
know as population parameters
– You can’t normally get information on every element in a
population
• Sample
– A subset of the population. The sample mean
, sample
statistics & are estimates of the population parameters.
– Inferences need to be made from a sample
– Vital any sample is representative of the population
If everyone was the same their would be no
need for statistics or statisticians!
How will ‘first time’ voters vote?
If I take a survey how will I know they are representative?
As the sample gets
bigger the results should
get closer to the
population true mean
Getting estimates of the mean
We have come across estimating means before:
Example:
Mean height
f
mid=x f x
160 ≤ x < 165
1
162.5
162.5
165 ≤ x < 170
5
167.5
837.5
170 ≤ x < 175
13
172.5
2242.5
175 ≤ x < 180
10
177.5
1775
180 ≤ x < 185
1
182.5
182.5
Total
30
5200
Estimated mean is 5200 ÷ 30 = 173.3
Or when all values in a set of data are known
then the mean of a set is got by
X
n
X
i 1
n
i
How will they differ from older voters
If I take a survey how will I know they are representative?
X
Is an UNBIASED
ESTIMATOR of the
population mean μ
What inferences can be made from samples and can
they be trusted – that is the subject of this lesson.
Which issues concern voters most?
If I take a survey how will I know they are representative?
In this chapter we learn
some statistical magic.
The distribution of all sample means is
normal, as long as the sample size are large
enough. And it doesn't matter what the
distribution of the original population looks
like. (CLT)
What this means is that we can get an
unbiased estimate of the population mean
without doing a census
The mean of the distribution of sample means is
approximately equal to the population mean.
X
Is an UNBIASED
ESTIMATOR of the
population mean μ
And the distribution of the means is normal
That means that the normal distribution tables can be used to solve
problems.
But first lets
Similarities and differences of between population and sample distribution
Mean
Population
Number of Samples
μ
X Mean of means. Used for population
mean estimate – unbiased estimator
Distribution
σ distribution is
measured in standard
deviations from the
mean
distribution is measured in standard
2
errors which is the
n
square root of the (variance ÷ sample size)
Variance
σ²
This is the unbiased estimator
of population variance
S2 
from sample
Normal Tables
X ̴ N (μ,
σ2)
IF X can be modelled by a
Normal distribution then :
X
  xi  x 
2
n 1
 2 
N  , 
n 

Sample means have standard errors in the
same way as population means have standard
deviations from the mean.
Standard Error of sample means
2
n
=

n
n
n
n
n
n
n
Standard deviation from population mean
Task
EXERCISE B
Page 94
Answers page 171
Just as we standardised for a population
when the when the mean was not zero and
the standard deviation was not one.
z
x

We standardise samples too but this time we
divide by the standard error
z
X 

n
standard error 

2
n


n
X
IF random samples of size n are taken from a population which can be
modelled by a normal distribution N(μ,σ²) then we can assume 3 facts
about the sampling distribution
X
1. The mean of a sample
is μ
X
2

2. The variance of the sampling distribution of X is
n
X
3. The sampling distribution of
These can be summarised as:
X ̴ N(μ,σ²) then X
̴ N(μ, 
2
n
)
is a normal distribution.
Task
Follow example 2 & 3 page 96.
These give you practice working with samples and the normal
distribution and percentage points tables.
Then do Exercise C page 97
Always do a sketch of the problem.
What you don’t finish in class do for homework.
The Central Limit Theorem (CLT)
(Statistical Magic)
The CLT says that if a random sample of size n is
taken from any distribution (it does not need to be
normal) with a mean μ and a variance σ² then,
provided n is large enough:
X
 2 
N   ,  approximately
n 

This is important as in real life we don’t always
know the distribution of the population.
Example 4 (page 100)
A packaging machine produces packs of butter with a μ weight of 250g and a σ of 5g.
Random samples of 10 packs are taken regularly from the production line and the
mean weight of the sample is found. Use the CLT to find the approximate proportion
of samples that should have a mean weight > 253g.
The distribution of the population is not known but it is implied that n is big enough to
use CLT
From CLT
X

52 
N  250, 
10 

so sample means follow a normal distribution where
the mean is 250 and the standard error is
25
253  250
 2.5  1.58 so z 
 1.90
10
1.58
from the tables Φ(1.90)=0.9713
and since we want the probability of packets being above that we take it away form 1
Task
• Exercise D page 100
• Questions 1 to 3 and do the rest for homework
Estimating the Variance
All the questions that you have done so far the variance
or standard deviation has been know. In most real
cases however it wont be known and you will have to
use the samples to work out an unbiased estimator of
the variance.
The unbiased estimator (S2) of the population variance
(σ²) from a sample of size n is given by:
n
S2 
2
(
X

X
)
 i
i 1
n 1
Many people get confused when to use n and
when to use n-1 as the denominator.
•If the data entered is for the whole
population use n
•If the data entered is for a sample
use
n-1
Task
Exercise E page 104
Warning!!!
IF the population distribution is not known and
the sample size is less than 30 the normal
distribution tables can’t be used.
The normal distribution tables can be used if the sample size
is grater than 30 (from CLT)
Summary
Similarities and differences of between population and sample distribution
Population
Number of Samples
Mean
μ
Mean of means. Used for population
mean estimate – unbiased estimator
Distribution
σ distribution is
measured in standard
deviations from the
mean
distribution is measured in standard
2
errors which is the
n
square root of the (variance ÷ sample size)
Variance
σ²
This is the unbiased estimator
2
x

x


of population variance
 i
S2 
n 1
from sample
Normal Tables
X ̴ N (μ,
X
σ2)
IF X can be modelled by a
Normal distribution then :
X
 2 
N  , 
n 

Estimation
1. Appreciate the importance of random sampling
2. Understand the concept of estimation from samples
3. Understand the Central Limit Theorem
4. Be able to determine unbiased estimates of the
variance
Task and homework
• Exercise F page 106
• Homework mixed questions and test yourself
pages 107 & 108