Chapter 11
Sampling Distributions
BPS - 5th Ed.
Chapter 11
1
Sampling Terminology
X
Parameter
– fixed, unknown number that describes the population
X
Statistic
– known value calculated from a sample
– a statistic is often used to estimate a parameter
X
Variability
– different samples from the same population may yield
different values of the sample statistic
X
Sampling Distribution
– tells what values a statistic takes and how often it
takes those values in repeated sampling
BPS - 5th Ed.
Chapter 11
2
The Law of Large Numbers
Consider sampling at random from a
population with true mean µ. As the
number of (independent) observations
sampled increases, the mean of the
sample gets closer and closer to the
true mean of the population.
( x gets closer to µ )
BPS - 5th Ed.
Chapter 11
3
Sampling Distribution
X
The sampling distribution of a statistic
is the distribution of values taken by the
statistic in all possible samples of the
same size (n) from the same population
– to describe a distribution we need to specify
the shape, center, and spread
– we will discuss the distribution of the sample
mean (x-bar) in this chapter
BPS - 5th Ed.
Chapter 11
4
Mean and Standard Deviation of
Sample Means
If numerous samples of size n are taken from
a population with mean μ and standard
deviation σ , then the mean of the sampling
distribution of X is μ (the population mean)
and the standard deviation is: σ
n
(σ is the population s.d.)
BPS - 5th Ed.
Chapter 11
5
Mean and Standard Deviation of
Sample Means
X Since
the mean of X is μ, we say that X is
an unbiased estimator of μ
X Individual
observations have standard
deviation σ, but sample means X from
samples of size n have standard deviation
σ
n . Averages are less variable than
individual observations.
BPS - 5th Ed.
Chapter 11
6
Sampling Distribution of
Sample Means
If individual observations have the N(µ, σ)
distribution, then the sample mean X of n
independent observations has the N(µ, σ/ n )
distribution.
“If measurements in the population follow a
Normal distribution, then so does the sample
mean.”
BPS - 5th Ed.
Chapter 11
7
Central Limit Theorem
If a random sample of size n is selected from
ANY population with mean μ and standard
deviation σ , then when n is large the
sampling distribution of the sample mean X
is approximately Normal:
X is approximately N(µ, σ/ n )
“No matter what distribution the population
values follow, the sample mean will follow a
Normal distribution if the sample size is large.”
BPS - 5th Ed.
Chapter 11
8
Central Limit Theorem:
Sample Size
X
How large must n be for the CLT to hold?
– depends on how far the population
distribution is from Normal
Y the
further from Normal, the larger the sample
size needed
Y a sample size of 30 is typically large enough for
any population distribution encountered in
practice
Y recall: if the population is Normal, any sample
size will work (n≥1)
BPS - 5th Ed.
Chapter 11
9
Detecting gypsy moths. The gypsy moth is a serious threat to oak and
aspen trees. A state agriculture department places traps throughout the
state to detect the moths. When traps are checked periodically, the mean
number of moths trapped is only 0.5, but some traps have several moths.
The distribution of moth counts is discrete and stongly skewed, with
standard deviation 0.7.
Detecting gypsy moths. The gypsy moth is a serious threat to oak and
aspen trees. A state agriculture department places traps throughout the
state to detect the moths. When traps are checked periodically, the mean
number of moths trapped is only 0.5, but some traps have several moths.
The distribution of moth counts is discrete and stongly skewed, with
standard deviation 0.7.
a. What are the mean and standard deviation of the average number
of moths x̄ of 50 traps?
Detecting gypsy moths. The gypsy moth is a serious threat to oak and
aspen trees. A state agriculture department places traps throughout the
state to detect the moths. When traps are checked periodically, the mean
number of moths trapped is only 0.5, but some traps have several moths.
The distribution of moth counts is discrete and stongly skewed, with
standard deviation 0.7.
a. What are the mean and standard deviation of the average number
of moths x̄ of 50 traps?
b. Use the central limit theorem to find the probability that the
average number of moths in 50 traps is greater than 0.6.
Insurance. An insurance company knows that in the entire population of
millions of homeowners, the mean annual loss from fire is µ = $250 and
the standard deviation of the loss is σ = $1, 000. The distribution of
losses is strongly right-skewed: most policies have $0 loss, but a few have
large losses. If the company sells 10,000 policies, can it safely base its
rates on the assumption that its average loss will be no greater than
$275?