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¾Section 6.4
z
How Likely Are the Possible
Values of a Statistic?
z
The Sampling Distribution
Agresti/Franklin Statistics, 1e, 1 of 21
Parameter and Statistic
z
z
z
Parameter: A numerical summary of a
population, such as a population
proportion (p) or a population mean (μ).
Statistic: A numerical summary of sample
data, such as a sample proportion or a
sample mean.
Statistic estimates Parameter.
Agresti/Franklin Statistics, 1e, 2 of 21
Example: 2003 California Recall
Election
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Prior to counting the votes, the proportion
(p) in favor of recalling Governor Gray
Davis was an unknown parameter.
z
An exit poll of 3160 voters reported that the
sample proportion in favor of a recall was
0.54.
•
•
That is x=1706 voters in favor of a recall.
The sample proportion=x/n=1706/3160=0.54.
Agresti/Franklin Statistics, 1e, 3 of 21
Example: 2003 California Recall
Election
z
z
z
If a different random sample of 3160 voters
were selected resulting 1590 in favor or a
(different) sample proportion
1675/3160=0.53, which is different from
0.54.
Imagine all the distinct samples of 3160
voters you could possibly get.
Then under re-sampling sense, the sample
proportion is a random variable.
Agresti/Franklin Statistics, 1e, 4 of 21
Sampling Distribution
z
z
Question: How do we know that a
sample statistic is a good estimate of
a population parameter?
Answer: The sampling distribution.
• The sampling distribution of a statistic is
the probability distribution that specifies
probabilities for the possible values the
statistic can take.
Agresti/Franklin Statistics, 1e, 5 of 21
Example: Sampling Distribution
z
Which Brand of Pizza Do You Prefer?
• Two Choices: A or D.
• Assume that half of the population prefers
A and half prefers D.
• Parameter of interest: p=population proportion.
That is p=0.5.
• Take a random sample of n = 3 tasters.
Agresti/Franklin Statistics, 1e, 6 of 21
Example: Sampling Distribution
Sample of
size 3=n
No. Prefer
Pizza A (x)
Proportion
(x/n)
(A,A,A)
3
1
(A,A,D)
2
2/3
(A,D,A)
2
2/3
(D,A,A)
2
2/3
(A,D,D)
1
1/3
(D,A,D)
1
1/3
(D,D,A)
1
1/3
(D,D,D)
0
0
Agresti/Franklin Statistics, 1e, 7 of 21
Example: Sampling Distribution
Sample
Proportion
Probability
0
1/8
1/3
3/8
2/3
3/8
1
1/8
Agresti/Franklin Statistics, 1e, 8 of 21
P ( x )= n C x p x (1 − p ) n − x
Example: Sampling Distribution
Use binomial distribution:
P ( x)= n C x p x (1 − p ) n − x
Agresti/Franklin Statistics, 1e, 9 of 21
Mean and Standard Deviation of the
Sampling Distribution of the Sample
Proportion
z
For a binomial random variable with n trials and
probability p of success for each, the sampling
distribution of the proportion of successes has:
Mean = p and standard deviation =
z
p(1 - p)
=standard error
n
To obtain these value, take the mean np and
standard deviation np (1 − p ) for the binomial
distribution of the number of successes and divide
by n.
Agresti/Franklin Statistics, 1e, 10 of 21
Example: 2003 California Recall
Election
z
z
Sample: Exit poll of 3160 voters.
•
n=3160
Suppose that exactly 50% of the
population of all voters voted in favor
of the recall.
Agresti/Franklin Statistics, 1e, 11 of 21
Example: 2003 California Recall
Election
z
Describe the mean and standard deviation of
the sampling distribution of the number in the
sample who voted in favor of the recall.
•
n=3160, p=0.50.
• µ = np = 3160(0.50) = 1580
σ = np(1 - p) = 3160 (0.50 )( 0.50 ) = 28 .1
•
Agresti/Franklin Statistics, 1e, 12 of 21
Example: 2003 California Recall
Election
z
Describe the mean and standard deviation of the
sampling distribution of the proportion in the
sample who voted in favor of the recall.
Mean = p = 0.50
Standard Deviation =
p (1 − p )
(0.50)(0.50)
=
= 0.000079 = 0.0089
n
3160
Agresti/Franklin Statistics, 1e, 13 of 21
Example: 2003 California Recall
Election
z
z
If the population proportion supporting
recall was 0.50, would it have been
unlikely to observe the exit-poll sample
proportion of 0.54?
Based on your answer, would you be
willing to predict that Davis would be
recalled from office?
Agresti/Franklin Statistics, 1e, 14 of 21
Example: 2003 California Recall
Election
z
Convert the sample proportion value of
0.54 to a z-score:
(0.54 - 0.50)
z=
= 4.5
0.0089
zThe
sample proportion of 0.54 is more than four standard errors
from the expected value of 0.50.
zThe sample proportion of 0.54 voting for recall would be very
unlikely if the population support were p = 0.50.
Agresti/Franklin Statistics, 1e, 15 of 21
Example: 2003 California Recall
Election
z
z
z
A sample proportion of 0.54 would be
even more unlikely if the population
support were less than 0.50.
We have strong evidence that the actually
p was large than 0.50.
The exit poll gives strong evidence that
Governor Davis would be recalled.
Agresti/Franklin Statistics, 1e, 16 of 21
Example: 2003 California Recall
Election
z
Fact: The sampling
distribution of the
sample proportion
has a bell-shape with
a mean µ = 0.50 and
a standard deviation
σ = 0.0089 if np≥15.
n(1-p) ≥15.
Agresti/Franklin Statistics, 1e, 17 of 21
Recap: Summary of the Sampling
Distribution of a Proportion (p)
z
For a random sample of size n from a population
with proportion p, the sampling distribution of the
sample proportion has
p(1 - p)
Mean = p and standard error =
n
z
If n is sufficiently large such that the expected
numbers of outcomes of the two types, np and n(1p), are both at least 15, then this sampling
distribution has a bell-shape.
Agresti/Franklin Statistics, 1e, 18 of 21
¾Section 6.5
How Close Are Sample Means to
Population Means?
Agresti/Franklin Statistics, 1e, 19 of 21
The Sampling Distribution of the
Sample Mean
z
z
z
The sample mean, x, is a random
variable.
The sample mean varies from sample
to sample.
By contrast, the population mean, µ,
is a single fixed number.
Agresti/Franklin Statistics, 1e, 20 of 21
Mean and Standard Error of the
Sampling Distribution of the Sample
Mean
z
For a random sample of size n from a population
having mean µ and standard deviation σ, the
sampling distribution of the sample mean has:
•
Center described by the mean µ (the same as the
mean of the population).
•
Spread described by the standard error, which
equals the population standard deviation divided by
the square root of the sample size: σ
n
Agresti/Franklin Statistics, 1e, 21 of 21
Example: How Much Do Mean
Sales Vary From Week to Week?
z
Daily sales at a pizza restaurant vary
from day to day.
z
The sales figures fluctuate around a
mean µ = $900 with a standard
deviation σ = $300.
Agresti/Franklin Statistics, 1e, 22 of 21
Example: How Much Do Mean
Sales Vary From Week to Week?
z
z
z
The mean sales for the seven days in a
week are computed each week.
The weekly means are plotted over time.
These weekly means form a sampling
distribution.
Agresti/Franklin Statistics, 1e, 23 of 21
Example: How Much Do Mean
Sales Vary From Week to Week?
z
What are the center and spread of the
sampling distribution?
μ = $900
300
σ=
= 113
7
Agresti/Franklin Statistics, 1e, 24 of 21
Sampling Distribution vs.
Population Distribution
Agresti/Franklin Statistics, 1e, 25 of 21
Standard Error
z
Knowing how to find a standard error
gives us a mechanism for
understanding how much variability
to expect in sample statistics “just by
chance.”
Agresti/Franklin Statistics, 1e, 26 of 21
Standard Error
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The standard error of the sample mean:
σ
n
z
z
As the sample size n increases, the denominator
increase, so the standard error decreases.
With larger samples, the sample mean is more
likely to fall close to the population mean.
Agresti/Franklin Statistics, 1e, 27 of 21
Central Limit Theorem
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Question: How does the sampling
distribution of the sample mean relate
with respect to shape, center, and
spread to the probability distribution
from which the samples were taken?
Agresti/Franklin Statistics, 1e, 28 of 21
Central Limit Theorem
z
z
For random sampling with a large
sample size n, the sampling
distribution of the sample mean is
approximately a normal distribution.
This result applies no matter what the
shape of the probability distribution
from which the samples are taken.
Agresti/Franklin Statistics, 1e, 29 of 21
Central Limit Theorem:
How Large a Sample?
z
The sampling distribution of the sample
mean takes more of a bell shape as the
random sample size n increases. The more
skewed the population distribution, the
larger n must be before the shape of the
sampling distribution is close to normal. In
practice, the sampling distribution is
usually close to normal when the sample
size n is at least about 30.
Agresti/Franklin Statistics, 1e, 30 of 21
A Normal Population Distribution
and the Sampling Distribution
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If the population distribution is
approximately normal, then the
sampling distribution is
approximately normal for all sample
sizes.
Agresti/Franklin Statistics, 1e, 31 of 21
How Does the Central Limit Theorem
Help Us Make Inferences
z
z
For large n, the sampling distribution
is approximately normal even if the
population distribution is not.
This enables us to make inferences
about population means regardless of
the shape of the population
distribution.
Agresti/Franklin Statistics, 1e, 32 of 21