Sampling Distribution of a Sample Proportion

1
Sampling Distribution of a Sample Proportion

• Lecture 26
• Sections 8.1 8.2
• Wed, Mar 8, 2006

2
Preview of the Central Limit Theorem

• We looked at the distribution of the sum of 1, 2,
  and 3 uniform random variables U(0, 1).
• We saw that the shapes of their distributions was
  moving towards the shape of the normal
  distribution.
• If we replace sum with average, we will
  obtain the same phenomenon, but on the scale from
  0 to 1 each time.

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Preview of the Central Limit Theorem
2
1
0
1
4
Preview of the Central Limit Theorem
2
1
0
1
5
Preview of the Central Limit Theorem
2
1
0
1
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Preview of the Central Limit Theorem

• Some observations
• Each distribution is centered at the same place,
  ½.
• The distributions are being drawn in towards
  the center.
• That means that their standard deviation is
  decreasing.
• Can we quantify this?

7
Preview of the Central Limit Theorem
2

• ½
• ?2 1/12

1
0
1
8
Preview of the Central Limit Theorem
2

• ½
• ?2 1/24

1
0
1
9
Preview of the Central Limit Theorem
2

• ½
• ?2 1/36

1
0
1
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Preview of the Central Limit Theorem

• This tells us that a mean based on three
  observations is much more likely to be close to
  the population mean than is a mean based on only
  one or two observations.

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Parameters and Statistics

• THE PURPOSE OF A STATISTIC IS TO ESTIMATE A
  POPULATION PARAMETER.
• A sample mean is used to estimate the population
  mean.
• A sample proportion is used to estimate the
  population proportion.
• Sample statistics, by their very nature, are
  variable.
• Population parameters are fixed.

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Some Questions

• We hope that the sample proportion is close to
  the population proportion.
• How close can we expect it to be?
• Would it be worth it to collect a larger sample?
• If the sample were larger, would we expect the
  sample proportion to be closer to the population
  proportion?
• How much closer?

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The Sampling Distribution of a Statistic

• Sampling Distribution of a Statistic The
  distribution of values of the statistic over all
  possible samples of size n from that population.

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The Sample Proportion

• 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.

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Example

• Suppose that this class is 3/4 freshmen.
• Suppose that we take a sample of 2 students,
  selected with replacement.
• Find the sampling distribution of p.

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Example
17
Example

• Let X be the number of freshmen in the sample.
• The probability distribution of X is

x P(x)
0 1/16
1 6/16
2 9/16
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Example

• Let p be the proportion of freshmen in the
  sample. (p X/n.)
• The sampling distribution of p is

x P(p x)
0 1/16
1/2 6/16
1 9/16
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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 P(p x)
0 1/64 .02
1/3 9/64 .14
2/3 27/64 .42
1 27/64 .42
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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
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The Parameters of the Sampling Distributions

• When n 1, the sampling distribution is
• The mean and standard deviation are
• ? 3/4 0.75
• ?2 3/16 0.1875

p P(p)
0 1/4
1 3/4
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The Parameters of the Sampling Distributions

• When n 2, the sampling distribution is
• The mean and standard deviation are
• ? 3/4 0.75
• ?2 3/32 0.09375

p P(p)
0 1/16
1/2 6/16
1 9/16
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The Parameters of the Sampling Distributions

• When n 3, the sampling distribution is
• The mean and standard deviation are
• ? 3/4 0.75
• ?2 3/48 0.0625

p P(p)
0 1/64 .02
1/3 9/64 .14
2/3 27/64 .42
1 27/64 .42
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The Parameters of the Sampling Distributions

• When n 4, the sampling distribution is
• The mean and standard deviation are
• ? 3/4 0.75
• ?2 3/64 0.046875

p P(p)
0 1/256 .004
1/4 12/256 .05
2/4 54/256 .21
3/4 108/256 .42
1 81/256 .32
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Sampling Distributions

• Run the program
• Central Limit Theorem for Proportions.exe.
• Use n 30 and p 0.75 generate 100 samples.

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100 Samples of Size n 30
?? 0.75
?? 0.079
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Observations and Conclusions

• Observation 1 The values of p are clustered
  around p.
• Conclusion 1 p is probably close to p.

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Larger Sample Size

• Now we will select 100 samples of size 120
  instead of size 30.
• Run the program
• Central Limit Theorem for Proportions.exe.
• Pay attention to the spread (standard deviation)
  of the distribution.

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100 Samples of Size n 120
?? 0.75
?? 0.0395
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Observations and Conclusions

• Observation 2 As the sample size increases, the
  clustering is tighter.
• Conclusion 2A Larger samples give more reliable
  estimates.
• Conclusion 2B For sample sizes that are large
  enough, we can make very good estimates of the
  value of p.

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Larger Sample Size

• Now we will select 10000 samples of size 120
  instead of only 100 samples.
• Run the program
• Central Limit Theorem for Proportions.exe.
• Pay attention to the shape of the distribution.

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10,000 Samples of Size n 120
?? 0.75
?? 0.0395
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10,000 Samples of Size n 126
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More Observations and Conclusions

• Observation 3 The distribution of p appears to
  be approximately normal.

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One More Conclusion

• Conclusion 3 We can use the normal distribution
  to calculate just how close to p we can expect p
  to be.
• However, we must know the values of ? and ? for
  the distribution of p.
• That is, we have to quantify the sampling
  distribution of p.

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The Sampling Distribution of p

• It turns out that the sampling distribution of p
  is approximately normal with the following
  parameters.
• This is the Central Limit Theorem for
  Proportions, summarized on page 519.

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The Sampling Distribution of p

• The approximation to the normal distribution is
  excellent if

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Why Surveys Work

• Suppose 51 of the population plan to vote for
  candidate X, i.e., p 0.51.
• What is the probability that an exit survey of
  1000 people would show candidate X with less than
  45 support, i.e., p lt .45?

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Why Surveys Work

• First, describe the sampling distribution of p
  if the sample size is n 1000 and p 0.51.
• Check np 510 ? 5 and n(1 p) 490 ? 5.
• p is approximately normal.

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Why Surveys Work

• The z-score of 0.45 is z (0.45 0.51)/.01581
  -3.795.
• P(p lt 0.45) P(Z lt -3.795)
• 0.00007385 (not likely!)
• Or use normalcdf(-E99, 0.45, 0.51, 0.01581).

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Why Surveys Work

• Perform the same calculation, but with a smaller
  sample size, say n 50.
• The probability turns out to be 0.1980, nearly a
  20 chance.
• By symmetry, there is also a 20 chance that the
  sample proportion is greater than 57.
• Thus, there is a 40 chance that the sample
  proportion is off by at least 6 percentage points.