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Chapter 4
Probability and
Sampling Distributions
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Random Variable
Definition: A random variable is a variable whose value is a numerical outcome of a random phenomenon.
The statistic calculated from a randomly chosen sample is an example of a random variable.
A statistic from a random sample will take different values if we take more samples from the same population
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Section 4.
The Sampling
Distribution of a
Sample Mean
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Sampling Distributions
In this class we will focus on three
sampling distributions.
The sampling distribution of the sample mean.
The sampling distribution of the sample proportion.
The sampling distribution of b 1 the sample slope.
We start with the sampling distribution of
x
p ˆ
x
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Sampling Distribution of
To get the sampling distribution of we
must investigate three things
The center of the distribution
The spread of the distribution
The shape of the distribution
We start with the center
x
x
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Example
Suppose that we are interested in the workout times of ISU students at the Recreation center. Let’s assume that μ is the average workout time of all ISU students To estimate μ lets take a simple random sample of 100 students at ISU
- We will record each students work out time (xi)
- Then we find the average workout time for the 100 students
- The population mean μ is the parameter of interest.
- The sample mean, , is the statistic (which is a random variable).
- Use to estimate μ (This seems like a sensible thing to do).
x
x x
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Example
A SRS should be a fairly good representation of the population so the x-bar should be somewhere near μ.
x-bar from a SRS is an unbiased estimate of μ
We don’t expect x-bar to be exactly equal to μ
There is variability in x-bar from sample to sample
If we take another simple random sample (SRS) of 100 students, then the x-bar will probably be different.
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If x-bar is rarely exactly right and varies from
sample to sample, why is it a reasonable estimate of the population mean μ?
answer: if we keep on taking larger and larger samples, the statistic x-bar is guaranteed to get closer and closer to the parameter μ
We have the comfort of knowing that if we can afford to keep on measuring more subjects, eventually we will estimate the mean workout time for all ISU students very accurately…
Statistical Estimation
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The Law of Large Numbers
Law of Large Numbers (LLN) : draw independent observations at random from any population with finite mean μ as the number of observations drawn increases, the mean x-bar of the observed values gets closer and closer to the mean μ of the population If n is the sample size as n gets large
The Law of Large Numbers holds for any population, not just for special classes such as Normal distributions
x !μ
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Example
Suppose we have a bowl with 21 small pieces of paper inside. Each paper is labeled with a number 0-20. We will draw several random samples out of the bowl of size n and record the sample means, x-bar, for each sample. What is the population? Since we know the values for each individual in the population (i.e. for each paper in the bowl), we can actually calculate the value of μ, the true population mean. μ= 10 Draw a random sample of size n = 1. Calculate x-bar for this sample.
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Example
Draw a second random sample of size n = 5. Calculate for this sample.
Draw a third random sample of size n = 10. Calculate for this sample.
Draw a fourth random sample of size n = 15. Calculate for this sample.
Draw a fifth random sample of size n = 20. Calculate for this sample.
What can we conclude about the value of as the sample size increases?
THIS IS CALLED THE LAW OF LARGE NUMBERS.
x
x
x
x
x
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Another Example
Example: If we were to roll a pair of dice and sum of the number of dots showing the average would be 7.
Go to applet
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How Large is a Large
Number?
The Law of Large Numbers says that the actual mean outcome of many trials gets close to the distribution mean μ as more trials are made
It doesn’t say how many trials are needed to guarantee a mean outcome close to μ that depends on the variability of the random outcomes
The more variable the outcomes, the more trials are needed to ensure that the mean outcome x- bar is close to the distribution μ 20
More Laws of Large Numbers
The Law of Large Numbers is one of the
central facts about probability
it assures us that statistical estimation will be accurate if we can afford enough observations
The basic Law of Large Numbers applies to
independent observations that all have the
same distribution
mathematicians have extended the law to many more general settings
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Sample Statistic Facts.
Some sample statistic facts:
A statistic varies from sample to sample.
A statistic almost always differs from a parameter. A statistic approaches parameter as sample size increases.
Question:
How do we investigate the behavior of the statistic?
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Questions about
How well does (statistic) estimate μ (parameter)? Does vary about μ?
How much could differ from μ?
How much could vary from sample to sample?
Can we compute probabilities on The sampling distribution of will answer all these questions
x
x
x
x
x
x
x
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Recall:
Theoretical sampling distribution of :
- The distribution of all -values from all possible samples of the same size from the same population. Can we take all possible samples?
- Maybe, if population size = 100 and n = 10 then there are over 17,300,000,000,000 possible samples. What do we do then
- Take many, many SRS’s.
- Compute for each.
- Approximate the theoretical sampling distribution of
Sampling Distribution
x
x
x
x x
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The idea of a sampling distribution is
the foundation of statistical inference
the laws of probability can tell us about sampling distributions without the need to actually choose or simulate a large number of samples
Sampling Distributions
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Mean and Standard Deviation of a
Sample Mean
Suppose that x-bar is the mean of a SRS of size n drawn from a large population with mean μ and standard deviation σ The mean of the sampling distribution of x-bar is μ The standard deviation is of the sampling distribution of x-bar is
In short: where n is the sample size
- Notice: averages are less variable than individual observations!
n
!
n
x x
μ !" μ
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The mean of the statistic x-bar is always the same
as the mean μ of the population the sampling distribution of x-bar is centered at μ in repeated sampling, x-bar will sometimes fall above the true value of the parameter μ and sometimes below, but there is no systematic tendency to overestimate or underestimate the parameter because the mean of x-bar is equal to μ, we say that the statistic x-bar is an unbiased estimator of the parameter μ
Mean and Standard Deviation of a
Sample Mean
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An unbiased estimator is “correct on the
average” in many samples how close the estimator falls to the parameter in most samples is determined by the spread of the sampling distribution if individual observations have standard deviation σ, then sample means x-bar from samples of size n have standard deviation
Again, notice that averages are less variable than individual observations
n
!
Mean and Standard Deviation of a
Sample Mean
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Not only is the standard deviation of the
distribution of x-bar smaller than the
standard deviation of individual
observations, but it gets smaller as we take
larger samples
the results of large samples are less variable than the results of small samples when dealing with sample means
Mean and Standard Deviation of a
Sample Mean
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If n is large, the standard deviation of x-bar
is small and almost all samples will give
values of x-bar that lie very close to the true
parameter μ
the sample mean from a large sample can be trusted to estimate the population mean accurately
Mean and Standard Deviation of a
Sample Mean
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Example
Suppose we take samples of size 15
from a distribution with mean 25 and
standard deviation 7
the distribution of x-bar has:
- A mean of :
- And a standard deviation of :
μ = 25
! n
=
7 15
" 1.
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How Large a Sample is Needed?
It depends on how close to Normal the
population distribution is
more observations are required if the shape of the population distribution is far from Normal
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Example
The time X that a technician requires to perform preventive maintenance on an air-conditioning unit is governed by a right skewed distribution (see figure 4.17 (a)) with mean time μ = 1 hour and standard deviation σ = 1 hour
Your company operates 70 of these units
The distribution of the mean time your company spends on preventative maintenance is approx.:
( 1 , 0. 12 )
N 1 , != N
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What is the probability that
your company’s units
average maintenance time
exceeds 50 minutes?
- 50/60 = 0.83 hour
- So we want to know P( x-bar > 0.83)
- Use Normal distribution calculations we learned in Chp 2!
( )
( )
( )
1 0. 0778 0. 9222
P z
P z
n
x P
P x
μ
Example
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4.86 ACT scores
The scores of students on the ACT
college entrance examination in a recent
year had the Normal distribution with
mean μ = 18.6 and standard deviation σ =
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What is the probability that a single
student randomly chosen from all those
taking the test scores 21 or higher?
4.86 ACT scores
1 0. 6591 0. 3409
( 0. 4068 ) 1 ( 0. 41 )
- 9
21 18. 6
( 21 )
=! =
= " =! <
$
% & '
(! "
!
"
Pz P z
x P
P x
)
μ
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About 34% of students (from this
population) scored a 21 or higher on the
ACT
The probability that a single student
randomly chosen from this population
would have a score of 21 or higher is 0.
4.86 ACT scores
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Now take a SRS of 50 students who took
the test. What are the mean and
standard deviation of the sample mean
score x-bar of these 50 students?
Mean = 18.6 [same as μ]
Standard Deviation = 0.8344 [sigma/sqrt(50)]
4.86 ACT scores
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What is the probability that the mean
score x-bar of these students is 21 or
higher?
4.86 ACT scores
1 0. 9980 0. 002
( 2. 8778 ) 1 ( 2. 88 )
- 834
21 18. 6
( 21 )
=! =
= " =! <
$
%
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&
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&
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( ! "
$
% & '
(
!
"
Pz P z
n
x P
P x
)
μ
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About 0.2 % of all random samples of
size 50 (from this population) would have
a mean score x-bar of 21 or higher.
The probability of having a mean score x-
bar of 21 or higher from a sample of 50
students (from this population) is 0.002.
4.86 ACT scores
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Section 4.4 Summary
When we want information about the
population mean μ for some variable, we
often take a SRS and use the sample
mean x-bar to estimate the unknown
parameter μ.
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The Law of Large Numbers states that
the actually observed mean outcome x-
bar must approach the mean μ of the
population as the number of observations
increases.
Section 4.4 Summary
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The sampling distribution of x-bar
describes how the statistic x-bar varies in
all possible samples of the same size
from the same population.
Section 4.4 Summary
