#### Transcript Slide 1

```Sampling distributions
BPS chapter 11
© 2006 W. H. Freeman and Company
Objectives (BPS chapter 11)
Sampling distributions

Parameter versus statistic

The law of large numbers

What is a sampling distribution?

The sampling distribution of

The central limit theorem
x
Reminder:


Parameter versus statistic
Population: the entire group of
individuals in which we are
interested but can’t usually
assess directly.
A parameter is a number
describing a characteristic of
the population. Parameters
are usually unknown.

Sample: the part of the
population we actually examine
and for which we do have data.

A statistic is a number
describing a characteristic of a
sample. We often use a statistic
to estimate an unknown
population parameter.
Population
Sample
The law of large numbers (page 273)
Law of large numbers: As the number of randomly-drawn observations
(n) in a sample increases,
the mean of the sample (x) gets
closer and closer to the population
mean m (quantitative variable).

ˆ
the sample proportion (p
) gets
closer and closer to the population
proportion p (categorical variable).
Problem 11.29 (page 296)



Roll 2 fair six-sided dice and consider the total number of dots on
the up-faces.
Question: If we considered all possible rolls, what would be the
average number of dots on the up-faces?
Population?


Parameter?



m – the average number of dots on the up-faces – population mean
We will sample using the Law of Large Numbers Applet to see the
Law of Large Numbers in action!
Sample?


All possible rolls
Rolls done via the Applet (each additional roll increases our sample size
by 1)
Statistic?

x
The average number of dots on the up-faces of all of the rolls in the
sample – sample mean
What is a sampling distribution? (page 276)
The
sampling distribution of a statistic is the distribution of all
possible values taken by the statistic when all possible samples of a
fixed size n are taken from the population. It is a theoretical idea—we
do not actually build it. But we will simulate it!
The Big Ideas:
Averages are less variable than individual observations.
 Averages are more normal than individual observations.

Note: When sampling randomly from a given population,

the law of large numbers describes what happens when the sample size

The sampling distribution describes what happens when we take all
possible random samples of a fixed size n.
Sampling distribution of x
(the sample mean)
We take many random samples of a given fixed size n from a
population with mean m and standard deviation s.
Some sample means will be above the population mean m and some
will be below, making up the sampling distribution.
Sampling distribution of “x bar”
Histogram
of some
sample
averages
Let’s Simulate Building a Sampling
Distribution

We will use the program Sampling Sim

Be sure you understand the difference between the population
window, the sample window, and the sampling distribution window!
For any population with mean m and standard deviation s:
x
Sampling distribution of
s/√n
m
x
In English:
1
2
1. The mean of the sample means is the population mean.
2. The standard deviation of the sample means is the population
standard deviation divided by the square root of the sample size.
What does 2. say about the variation of the sample mean versus the
variation of the original population variable??
Averages are less variable than individual measurements!

Mean of a sampling distribution of
x:
There is no tendency for a sample mean to fall systematically above or
below m, even if the distribution of the raw data is skewed. Thus, the mean
of the sampling distribution of x is an unbiased estimate of the population
mean m —it will be “correct on average” in many samples.

Standard deviation of a sampling distribution of
x:
The standard deviation of the sampling distribution measures how much the
sample statistic
x
varies from sample to sample. It is smaller than the
standard deviation of the population by a factor of √n.  Averages are less
variable than individual observations.
For normally distributed populations
When a variable in a population is normally distributed, then the
sampling distribution of x for all possible samples of size n is also
normally distributed.
Sample means
If the population is
N(m,s), then the sample
means distribution is
N(m,s/√n).
Population
The central limit theorem (page 281)
Central Limit Theorem: When randomly sampling from any population
with mean m and standard deviation s, when n is large enough, the
sampling distribution of
x
is approximately normal: N(m,s/√n).
Sampling
distribution of
x for n = 2
observations
Population with
strongly skewed
distribution
(Figure 11.5
page 283)

Sampling
distribution of
x for n = 10
observations
Sampling
distribution of
x for n = 25
observations
Averages are more normal than individual
measurements!
IQ scores: population vs. sample
In a large population of adults, IQ scores have mean 112 with standard
deviation 20. Suppose 200 adults are randomly selected for a market research
campaign.

The distribution of the sample mean IQ is
A) exactly normal, mean 112, standard deviation 20.
B) approximately normal, mean 112, standard deviation 20.
C) approximately normal, mean 112 , standard deviation 1.414.
D) approximately normal, mean 112, standard deviation 0.1.
C) approximately normal, mean 112, standard deviation 1.414.
Population distribution: N (m = 112; s = 20)
Sampling distribution for n = 200 is N (m = 112; s /√n = 1.414)
What if IQ scores are normally distributed for the population??
Application
Hypokalemia is diagnosed when blood potassium levels are low, below
3.5mEq/dl. Let’s assume that we know a patient whose measured
potassium levels vary daily according to a normal distribution N(m = 3.8,
s = 0.2).
If only one measurement is made, what's the probability that this patient
will be misdiagnosed hypokalemic?
If instead measurements are taken on four separate days, what is the
probability of such a misdiagnosis?
Practical note


Large samples are not always attainable.

Sometimes the cost, difficulty, or preciousness of what is studied limits
drastically any possible sample size.

Blood samples/biopsies: no more than a handful of repetitions
acceptable. Often we even make do with just one.

Opinion polls have a limited sample size due to time and cost of
operation. During election times, though, sample sizes are increased
for better accuracy.
Not all variables are normally distributed.

Income is typically strongly skewed for example.

Is x still a good estimator of m then?
Income distribution
Let’s consider the very large database of individual incomes from the Bureau of
Labor Statistics as our population. It is strongly right-skewed.

We take 1000 SRSs of 100 incomes, calculate the sample mean for
each, and make a histogram of these 1000 means.

We also take 1000 SRSs of 25 incomes, calculate the sample mean for
each, and make a histogram of these 1000 means.
Which histogram
corresponds to the
samples of size
100? 25?
How large a sample size?
It depends on the population distribution. More observations are
required if the population distribution is far from normal.

A sample size of 25 is generally enough to obtain a normal sampling
distribution from a strong skewness or even mild outliers.

A sample size of 40 will typically be good enough to overcome extreme
skewness and outliers.
In many cases, n = 25 isn’t a huge sample. Thus,
even for strange population distributions we can
assume a normal sampling distribution of the
mean, and work with it to solve problems.
Further properties
The Central Limit Theorem is valid as long as we are sampling many
small random events, even if the events have different distributions (as
long as no one random event has an overwhelming influence).
Why is this cool?
It explains why so many variables are normally distributed.
Example: Height seems to be determined by a large number of
genetic and environmental factors, like nutrition.
So height is very much like our sample mean x.
The “individuals” are genes and environmental
factors. Your height is a mean.
Now we have a better idea of why 
the density
curve for height has this shape.
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