Download Sampling Distribution Proportion

Sampling
Distribution of a
Sample Proportion
Lecture 28
Sections 8.1 – 8.2
Wed, Mar 7, 2007
Sampling Distributions

Sampling Distribution of a Statistic
The Sample Proportion

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Let p be the population proportion.
Then p is a fixed value (for a given population).
Let p^ (“p-hat”) be the sample proportion.
Then p^ is a random variable; it takes on a new
value every time a sample is collected.
The sampling distribution of p^ is the probability
distribution of all the possible values of p^.
Example
Suppose that this class is 3/4 freshmen.
 Suppose that we take a sample of 1
student.
 Find the sampling distribution of p^.

Example
3/4
F
P(F) = 3/4
N
P(N) = 1/4
1/4
Example
Let X be the number of freshmen in the
sample.
 The probability distribution of X is

x
0
1
P(x)
1/4
3/4
Example
Let p^ be the proportion of freshmen in the
sample. (p^ = X/n.)
 The sampling distribution of p^ is

x
0
1
P(p^ = x)
1/4
3/4
Example
Now we take a sample of 2 student,
sampling with replacement.
 Find the sampling distribution of p^.

Example
3/4
3/4
F
P(FF) = 9/16
N
P(FN) = 3/16
F
P(NF) = 3/16
N
P(NN) = 1/16
1/4
1/4
3/4
N
F
1/4
Example
Let X be the number of freshmen in the
sample.
 The probability distribution of X is

x
0
1
2
P(x)
1/16
6/16
9/16
Example
Let p^ be the proportion of freshmen in the
sample. (p^ = X/n.)
 The sampling distribution of p^ is

x
0
1/2
1
P(p^ = x)
1/16
6/16
9/16
Samples of Size n = 3

If we sample 3 people (with replacement)
from a population that is 3/4 freshmen,
then the proportion of freshmen in the
sample has the following distribution.
x
0
P(p^ = x)
1/64 = .02
1/3
2/3
1
9/64 = .14
27/64 = .42
27/64 = .42
Samples of Size n = 4

If we sample 4 people (with replacement)
from a population that is 3/4 freshmen,
then the proportion of freshmen in the
sample has the following distribution.
x
P(p^ = x)
0
1/256 = .004
1/4
12/256 = .05
2/4
54/256 = .21
3/4
108/256 = .42
1
81/256 = .32
The pdf when n = 1
0
1
The pdf when n = 2
0
1/2
1
The pdf when n = 3
0
1
The pdf when n = 4
0
1/4
2/4
3/4
1
The pdf when n = 8
0
1/4
2/4
3/4
1
The pdf when n = 16
0
1/4
2/4
3/4
1
The pdf when n = 48
0
1/4
2/4
3/4
1
Observations and Conclusions
Observation: The values of p^ are
clustered around p.
 Conclusion: p^ is probably close to p.
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Observations and Conclusions
Observation: As the sample size
increases, the clustering becomes tighter.
 Conclusion: Larger samples give more
reliable estimates.
 Conclusion: For sample sizes that are
large enough, we can make very good
estimates of the value of p.

Observations and Conclusions
Observation: The distribution of p^ appears
to be approximately normal.
 Conclusion: We can use the normal
distribution to calculate just how close to p
we can expect p^ to be.

One More Observation
However, we must know the values of 
and  for the distribution of p^.
 That is, we have to quantify the sampling
distribution of p^.

The Central Limit Theorem for
Proportions

It turns out that the sampling distribution of
p^ is approximately normal with the
following parameters.
Mean of pˆ  p
p 1  p 
Variance of pˆ 
n
Standard deviation of pˆ 
p 1  p 
n
The Central Limit Theorem for
Proportions
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The approximation to the normal
distribution is excellent if
np  5 and n1  p   5.
Why Surveys Work
Suppose that we are trying to estimate the
proportion of the population who own a
cell phone.
 Suppose the true proportion is 75%.
 If we survey a random sample of 1000
people, how likely is it that our error will be
no greater than 5%?

Why Surveys Work
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First, describe the sampling distribution of
p^ if the sample size is n = 1000 and p =
0.75.
np = 750  5 and n(1 – p) = 250  5,
so p^ is approximately normal.
 Check:
Why Surveys Work

Then find the parameters p^ and p^.
 pˆ  p  0.75.
 pˆ 
p1  p 

n
0.750.25  0.01369.
1000
Why Surveys Work
Now find the probability that p^ is between
0.70 and 0.80.
 normalcdf(.70, .80, .75, .01369) = 0.9997.
 It is virtually certain that our estimate with
be within 5% of 75%.
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Why Surveys Work
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What if we had surveyed only 200 people?
Surveys
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What range of percentages contains 95%
of the sample proportions?
Surveys
Suppose that Candidate X has 48% of the
vote and Candidate Y has 52% of the vote.
 What is the probability that a survey of 100
people will indicate that Candidate X is
ahead?
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Surveys
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What is the probability that a survey of
2000 people will indicate that Candidate X
is ahead?
Quality Control
A company will accept a shipment of
components if they are convinced that no
more than 5% of them are defective.
 H0: 5% of the parts are defective.
 H1: More than 5% of the parts are
defective.

Quality Control
They will take a random sample of 100
parts and test them. If no more than 10 of
them are defective, they will accept the
shipment.
 What is ?
 What is ?
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