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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)