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Chapter 18
Inference about a Population Proportion
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Outline
The sample proportion The sampling distribution of Conditions for inference Large-sample confidence intervals for a
population proportion Choosing the sample size Significance tests for a proportion
p̂
p̂
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1. The Sample proportion p̂
The proportion of a population that has some outcome (“success”) is p.
The proportion of successes in a sample is measured by the sample proportion:
sample the in nsobservatio of count totalsample the in successes of countp̂
“p-hat”
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2. The sampling distribution of p̂
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3. Conditions for inference
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Standard Error of
Since the population proportion p is unknown, the standard deviation of the sample proportion will need to be estimated by substituting for p.
ˆ p
n
pps
n
p)-p(1 s.d.
ˆˆ..
1e
p̂
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4. Large-sample confidence intervals for a population proportion
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Examples
Example 18.4 Estimating risky behavior (Page 476)
Example 18.5 Are the conditions met? (Page 476)
Exercise 18.8 No confidence interval. (Page 477)
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5. Accurate C.I. for a proportion
Example 18.6 (P479) Shaq’s free shows
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6. Choosing the sample size
The margin of error in our confidence interval is
We may like to choose the sample size n to achieve a certain margin of error.
n
ppzm
)ˆ1(ˆ*
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The sample size
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Guess the sample proportion:
Since we don’t know prior to sampling, we will have to use a guess p* for . There are two ways to do this:
– Use a guess p* based on a pilot study or on past experience.
– Use p*=0.50 as the guess. This guess is conservative, as it gives a margin of error bigger than the true margin of error. (Conservative)
p̂p̂
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Example
Example 18.7 Planning a poll
(Page 482)
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7. Significance tests for a proportion
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Examples
Example 18.8 Is this coin fair?
(Page 484)
Example 18.9 Estimating the chance of head (Page 485)