sampling distribution of a sample proportion lecture 25 sections 8.1 – 8.2 fri, feb 29, 2008
TRANSCRIPT
Sampling Distribution of a Sample ProportionLecture 25
Sections 8.1 – 8.2
Fri, Feb 29, 2008
Sampling Distributions
Sampling Distribution of a Statistic
The Sample Proportion
The letter p represents the population proportion.
The symbol p^ (“p-hat”) represents the sample proportion.
p^ is a random variable. The sampling distribution of p^ is the probability
distribution of all the possible values of p^.
Example
Suppose that 2/3 of all males wash their hands after using a public restroom.
Suppose that we take a sample of 1 male. Find the sampling distribution of p^.
Example
W
N
2/3
1/3
P(W) = 2/3
P(N) = 1/3
Example
Let x be the sample number of males who wash.
The probability distribution of x is
x P(x)
0 1/3
1 2/3
Example
Let p^ be the sample proportion of males who wash. (p^ = x/n.)
The sampling distribution of p^ is
p^ P(p^)
0 1/3
1 2/3
Example
Now we take a sample of 2 males, sampling with replacement.
Find the sampling distribution of p^.
Example
W
N
W
N
W
N
2/3
1/3
2/3
1/3
2/3
1/3
P(WW) = 4/9
P(WN) = 2/9
P(NW) = 2/9
P(NN) = 1/9
Example
Let x be the sample number of males who wash.
The probability distribution of x is
x P(x)
0 1/9
1 4/9
2 4/9
Example
Let p^ be the sample proportion of males who wash. (p^ = x/n.)
The sampling distribution of p^ is
p^ P(p^)
0 1/9
1/2 4/9
1 4/9
Samples of Size n = 3
If we sample 3 males, then the sample proportion of males who wash has the following distribution.
p^ P(p^)
0 1/27 = .03
1/3 6/27 = .22
2/3 12/27 = .44
1 8/27 = .30
Samples of Size n = 4
If we sample 4 males, then the sample proportion of males who wash has the following distribution.
p^ P(p^)
0 1/81 = .01
1/4 8/81 = .10
2/4 24/81 = .30
3/4 32/81 = .40
1 16/81 = .20
Samples of Size n = 5
If we sample 5 males, then the sample proportion of males who wash has the following distribution.
p^ P(p^)
0 1/243 = .004
1/5 10/243 = .041
2/5 40/243 = .165
3/5 80/243 = .329
4/5 80/243 = .329
1 32/243 = .132
Our Experiment
In our experiment, we had 80 samples of size 5.
Based on the sampling distribution when n = 5, we would expect the following
Value of p^ 0.0 0.2 0.4 0.6 0.8 1.0
Actual
Predicted 0.3 3.3 13.2 26.3 26.3 10.5
The pdf when n = 1
0 1
The pdf when n = 2
0 11/2
The pdf when n = 3
0 11/3 2/3
The pdf when n = 4
0 11/4 2/4 3/4
The pdf when n = 5
0 11/5 2/5 3/5 4/5
18/10
The pdf when n = 10
0 2/10 4/10 6/10
Observations and Conclusions
Observation: The values of p^ are clustered around p.
Conclusion: p^ is close to p most of the time.
Observations and Conclusions
Observation: As the sample size increases, the clustering becomes tighter.
Conclusion: Larger samples give better estimates.
Conclusion: We can make the estimates of p as good as we want, provided we make the sample size large enough.
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.
n
ppp
n
ppp
pp
p
p
p
1ˆ ofdeviation Standard
1ˆ of Variance
ˆ ofMean
ˆ
2ˆ
ˆ
The Central Limit Theorem for Proportions The approximation to the normal
distribution is excellent if
.51 and 5 pnnp
Example
If we gather a sample of 100 males, how likely is it that between 60 and 70 of them, inclusive, wash their hands after using a public restroom?
This is the same as asking the likelihood that 0.60 p^ 0.70.
Example
Use p = 0.66. Check that
np = 100(0.66) = 66 > 5,n(1 – p) = 100(0.34) = 34 > 5.
Then p^ has a normal distribution with
04737.0100
)34.0)(66.0(ˆ
ˆ
p
p
Example
So
P(0.60 p^ 0.70)
= normalcdf(.60,.70,.66,.04737)
= 0.6981.
Why Surveys Work
Suppose that we are trying to estimate the proportion of the male population who wash their hands after using a public restroom.
Suppose the true proportion is 66%. 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
Now we have
.01498.01000
)34.0)(66.0(ˆ
ˆ
p
p
Why Surveys Work
Now find the probability that p^ is between 0.61 and 0.71:
normalcdf(.61, .71, .66, .01498) = 0.9992. It is virtually certain that our estimate will
be within 5% of 66%.
Why Surveys Work
What if we had decided to save money and surveyed only 100 people?
If it is important to be within 5% of the correct value, is it worth it to survey 1000 people instead of only 100 people?
Quality Control
A company will accept a shipment of components if there is no strong evidence that 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 ?