Download Sampling Distribution

Topics for Today
Sampling Distribution
The Central Limit Theorem
Stat203
Fall 2011 – Week 5 Lecture 2
Page 1 of 28
Law of Large Numbers
Draw ___________ at ______ from any
population with mean μ.
As the number of individuals _________, the
mean, x , of the sample gets _______to the
mean μ of the population.
Stat203
Fall 2011 – Week 5 Lecture 2
Page 2 of 28
LLN is the foundations of businesses such as
casinos and insurance companies.
Winnings or losses of a gambler on a few
plays are uncertain, which is why gambling is
exciting. It is only in the long run that the mean
is predictable. The house plays tens of
thousands of times so, unlike the individual
gambler, they can count on the long run
regularity described by the law of large
numbers.
Stat203
Fall 2011 – Week 5 Lecture 2
Page 3 of 28
Fallacy of the LLN
Large means large.
Gamblers often succumb to believing the LLN
will help them to predict the next occurrence of
an event (eg: red, blackjack, etc).
This is not the case. Consecutive runs are
independent.
Google this for more: Monte Carlo 1913
Another poor example:
A mathematician always takes a bomb on board
an airplane reasoning that the odds of a bomb on
a plane are small … so the odds of two bombs on
a plane are virtually zero.
Stat203
Fall 2011 – Week 5 Lecture 2
Page 4 of 28
Sampling Distributions
The law of large numbers _______ us that if
we measure enough ___________, the
_________ x will eventually get very close to
μ.
What more can we say about the
____________ of x ?
Suppose we took a ______ of 10 individuals
from a population and calculate x . What does
the ____________ of x look like?
Stat203
Fall 2011 – Week 5 Lecture 2
Page 5 of 28
Question: What would happen if we took
____________ of 10 individuals from a
population?
 Take a large number of samples of size 10
from the population
 Calculate x for each sample
 Make a histogram of these values of x and
examine it.
http://www.stat.tamu.edu/~west/ph/sampledist.html
Stat203
Fall 2011 – Week 5 Lecture 2
Page 6 of 28
Example: Constructing a
Sampling Distribution
Extensive studies have found that the odor
threshold of adults follows roughly a ______
distribution with ____ μ = 25 micrograms/litre
and a __________________ σ = 7
micrograms/litre.
With this information, we can simulate many
runs of our study with _________ individuals
drawn at ______ from the population.
The next figure illustrates the process: the
authors took ____________ of
______________, found the mean odor
threshold x and made a histogram of these
1000 x ’s.
Stat203
Fall 2011 – Week 5 Lecture 2
Page 7 of 28
Stat203
Fall 2011 – Week 5 Lecture 2
Page 8 of 28
What can we say about the _____, ______,
and ______ of this distribution?
The histogram shows how x would behave if
we drew many samples; the
_____________________ of the statistic x .
The figure on the next page compares the
mean odor threshold
_____________________ to the
______________________ distribution of
odor thresholds for a single adult.
Stat203
Fall 2011 – Week 5 Lecture 2
Page 9 of 28
Stat203
Fall 2011 – Week 5 Lecture 2
Page 10 of 28
The _____________________ is the
_____________ that would emerge if we
look at ____________ samples of 10
individuals from our population.
So, connecting to some earlier ideas:
The Probability Distribution is the underlying
theoretical distribution of
____________________ for an entire
population.
The Sampling Distribution is the underlying
theoretical distribution of the
___________________ (like the mean) for
all possible samples from a population.
Stat203
Fall 2011 – Week 5 Lecture 2
Page 11 of 28
Mean and Standard Deviation of x
Suppose that x is the mean of a ______ of
size n drawn from a large population with
mean μ and standard deviation σ.
Then the ____ of the
_____________________ of x is μ and its
standard deviation is  n .
… this is true __________ of the underlying
________________________!
Stat203
Fall 2011 – Week 5 Lecture 2
Page 12 of 28
Mean of the Sampling Distribution
 The ____ of x ’s sampling distribution is
always the same as the mean, μ, of the
population.
 The sampling distribution of x is
________ at μ.
 In repeated sampling, x will sometimes
fall above the true value of the population
parameter μ and sometimes below, but
there is _____________ tendency to
____________ or _____________.
We say x is an ________ estimator of μ.
Stat203
Fall 2011 – Week 5 Lecture 2
Page 13 of 28
Standard Deviation of the
Sampling Distribution
 How close is x to μ?
 Averages are _____________ than
individual observations.
 The standard deviation of x is _______
than the standard deviation of the
individuals.
 Standard deviation of x =

n
.
 The results of _____________ are ____
variable than the results of small samples.
Stat203
Fall 2011 – Week 5 Lecture 2
Page 14 of 28
 If __________, the standard deviation of
x will be _____, and almost all samples
will give values of x that are ______to μ.
 Note that to cut the standard deviation of
x in half we must take four times as many
observations, not just twice as many.
Stat203
Fall 2011 – Week 5 Lecture 2
Page 15 of 28
Normal Distributions
If a variable measured on a population has a
normal distribution, then the distribution of the
sample means x generated by a random
sample ____ has a normal distribution.
If X is ____________________ with mean μ
and standard deviation σ, then the distribution
of the sample mean x of a _____________ of
n observations has a ___________________
with mean μ and standard deviation  n .
Stat203
Fall 2011 – Week 5 Lecture 2
Page 16 of 28
What happens when the population is ___
______?
As the sample size _________, the distribution
of x _______ shape, it looks less like that of
the population and more like a ______
distribution.
This is true no matter what shape the
population distribution has.
This famous fact is called:
The Central Limit Theorem
Stat203
Fall 2011 – Week 5 Lecture 2
Page 17 of 28
The Central Limit Theorem
Draw a _____________ of size n from any
population with finite mean μ and standard
deviation σ. When n is _____, the ________
____________ of the sample mean is
____________________.
x is approximately normal with mean μ and
standard deviation

n
.
How large a sample?
More ___________ are required if the shape
of the population distribution is ___ from
normal.
Stat203
Fall 2011 – Week 5 Lecture 2
Page 18 of 28
Stat203
Fall 2011 – Week 5 Lecture 2
Page 19 of 28
The Central Limit Theorem in Action
The above figure shows how the central limit
theorem works for a fairly non-normal
population.
The first figure a) displays the probability
distribution of a single individual, that is, of the
entire population.
 The distribution is __________ skewed
with the most probable outcomes near 0.
 The mean μ of this distribution is 1, and its
standard deviation σ is also 1.
 This distribution is called an ___________
distribution.
Stat203
Fall 2011 – Week 5 Lecture 2
Page 20 of 28
Exponential distributions are used as models
for the lifetime in service of electronic
components as well as the time required to
serve a customer or repair a machine.
Stat203
Fall 2011 – Week 5 Lecture 2
Page 21 of 28
The next three figures are the density curves
of the sample means of size _, __, and __
observations from this population.
As the sample size n _________, the shape of
the distributions becomes ___________. The
mean μ = 1, and the standard deviation
decreases taking the values 1 n .
The density curve for 10 observations is
________ positively skewed but is close to
resembling the normal distribution with mean μ
= 1 and standard deviation σ = 1 10 = .32.
The density curve for a sample of 25
observations is even closer to the ______
distribution.
We can clearly see the contrast in shapes
between the population distribution and the
distributions of the means.
Stat203
Fall 2011 – Week 5 Lecture 2
Page 22 of 28
Try some others:
http://www.stat.tamu.edu/~west/ph/sampledist.html
Stat203
Fall 2011 – Week 5 Lecture 2
Page 23 of 28
Example: Maintaining air conditioners
The time (X) that a worker requires to perform
preventative maintenance on an air
conditioning unit is governed by the
exponential distribution.
The mean repair time is μ =1 hour and the
standard deviation σ = 1 hour. If your company
operates 70 of these units, what is the
probability that their average maintenance
time exceeds 50 minutes?
Stat203
Fall 2011 – Week 5 Lecture 2
Page 24 of 28
Example: Flaws in carpets
The number of defects per square meter in a
type of carpet material varies with mean μ =
1.6 flaws/m2 and standard deviation σ = 1.2
flaws/ m2. The population distribution is not
normal since the number of flaws is a count.
An inspector looks at 200 square meters of the
material and records the number of flaws per
square meter and calculates the sample mean
x.
Use the central limit theorem to find the
probability that the mean number of flaws is
greater than 2 per square meter.
Stat203
Fall 2011 – Week 5 Lecture 2
Page 25 of 28
Stat203
Fall 2011 – Week 5 Lecture 2
Page 26 of 28
Today’s Topics
Sampling Distribution of the mean
- has the same mean as the population
- has standard deviation is the population
standard deviation divided by the squareroot of n
Central Limit Theorem
- Says that no matter what the underlying
probability distribution, the sampling
distribution of the mean will be Normal
Stat203
Fall 2011 – Week 5 Lecture 2
Page 27 of 28
Reading for next lecture
Chapter 6 – Confidence Intervals
Stat203
Fall 2011 – Week 5 Lecture 2
Page 28 of 28