Transcript 9.1: Sampling Distributions

9.1: Sampling
Distributions
Parameter vs. Statistic
Remember!
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PPSS!
Population – Parameter
that
describes
the
Sample
– Statistic
Parameter: a number
population
A parameter is an actual number, but we
don’t know what it is because we cannot
reach/interview the entire population.
Statistic: a number that describes a sample
The value of a statistic can change from
sample to sample.
A statistic is used to estimate an unknown
parameter.
Ex 1: Making Money
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The mean income of the sample of
households contacted by the Current
Population Survey was x = $60,528.
What is the statistic?
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$60,528, because it describes this one Current
Population Survey Sample
What is the parameter?
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The mean income of 113 million U.S. households.
Population vs. Sample
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Use μ for the population mean.
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Use x for the sample mean.
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The sample mean x is an estimate of the actual
mean μ of the population.
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The fact that the value of a statistic varies in
repeated random sampling is called sampling
variability.
Ex 2: Do you believe in ghosts?
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The Gallup Poll asked a random sample of
515 U.S. adults whether they believe in
ghosts. Of the respondents, 160 said “Yes.”
What is the proportion of the sample who say
they believe in ghosts?
160
pˆ 
 0.31
515
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In this example, what is the statistic?
0.31
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Could we
conclude that the
proportion of all
U.S. adults who
believe in ghosts
is 0.31?
What is the parameter?
The proportion of all U.S. adults who believe in ghosts.
No! A different
sample would
probably yield a
different p̂ .
Sampling Variability and Distribution
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Take a large number of samples from the
same population.
Calculate the sample mean x or sample
proportion p̂ for each sample.
Make a histogram of the values of x or p̂.
Examine the distribution displayed in a
histogram for shape, center, and spread, as
well as outliers or other deviations.
Ex 3: Baggage Check!
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Instead of systematically searching
passengers, Customs agents at Guadalajara
airport have passengers push a button that is
connected to a modified stoplight.
If the stoplight turns green, passengers may
proceed to their flight. If the stoplight turns
red, the passenger is searched.
The stoplight is programmed to turn green
with a probability of 0.70.
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Let the number 1 – 7 represent passengers
who get the green light. Number 8 – 10
represent passengers who get the red light.
Use your calculator to generate 100 random
numbers into L1. (Randint(1,10,100))
Sort L1 in ascending order.
Compute the proportion of “green lights”.
The Sampling Distribution of
Guadalajara Airport Simulation
Sampling Distribution
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The sampling distribution of a statistic is
the distribution of values taken by the statistic
in all possible samples of the same size from
the same population.
This is an ideal definition
in that it would be difficult
to actually consider all the
possible samples of a
large population.
Ex 4: Are you a Survivor fan?
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According to 2005 Nielson ratings, Survivor:
Guatemala was one of the most-watched
television shows during every week that it
aired.
Suppose that the true proportion of U.S.
adults who watched Survivor: Guatemala is
p = 0.37.
Distribution of 1000 SRSs of size
n = 100 from the population.
Describe
the SOCS!
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The overall shape of the distribution is
roughly symmetric and approximately normal.
The center of the distribution is very close to
the true value p = 0.37.
The values of p̂ have a large spread. They
range from 0.22 to 0.535.
There are no outliers.
Approximate sampling distributions for
two different sample sizes from a
population with p = 0.37
Both statistics are unbiased
because the means of their
distributions equal the true
population value p = 0.37.
Sample Size 100
Sample Size 1000
Unbiased Statistic
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A statistic used to estimate a parameter is
unbiased if the mean of its sampling
distribution is equal to the true value of the
parameter being estimated.
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Note that an individual statistic does not need to
have the exact same mean as the population to
be unbiased.
The idea is that over many SRSs, the mean of the
sampling distribution will approach the mean of
the actual population.
The Variability of a Statistic
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The sample proportion from a random
sample of any size is an unbiased estimate of
the population parameter.
Larger samples are more likely to produce an
estimate close to the true value of the
parameter because there is much less
variability among large samples than among
small samples.
Ex 5: Sample Size and Variability
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Using an SRS size of 100 yields a distribution
with mean 0.37 and standard deviationWhich
0.05.
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Using the empirical rule, 95% of the
will fall between which two values?
27% and 47%
sample size
hasof
lessp̂
values
variability?
Using an SRS size of 1000 yields a distribution
with mean 0.37 and standard deviation 0.01.
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Using the empirical rule, 95% of the values of p̂
will fall between which two values?
35% and 39%
Variability of a Statistic
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The variability of a statistic is described by
the spread of its sampling distribution. This
spread is determined by the sampling design
and the size of the sample. Larger samples
give smaller spread.
As long as the population is much larger than
the sample, the spread of a sampling
distribution will depend only on sample size
and not on the size of the population.
Think of it like a bull's-eye…
This bull's-eye is our goal!