Inference for Proportions

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Inference for Proportions
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Proportions

• Our earlier analysis focused on inference about
  population means.
• Now we turn our attention to inference about the
  proportion of some outcome in a population.
• We will consider a single population and then
  compare proportions from two populations or
  treatments.

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The Basics

• Population Proportion
• Sample Proportion
• is a point estimate of P

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Sampling Distribution of Sample Proportion

• Choose an SRS of size n from a population that
  contains proportion p of characteristic. Let
  the sample proportion be defined as the number in
  the sample with the characteristic divided by
  n.
• As the sample size increases, the sampling
  distribution of the sample proportion becomes
  approximately normal.
• The mean of the sampling distribution is p, the
  population proportion.
• The standard deviation of the sampling
  distribution is

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Assumptions for Inference about a Proportion

• The data are an SRS from the population of
  interest.
• The population is at least 10 times as large as
  the sample.
• Fact The normal approximation to the
  distribution of the sample proportion is most
  accurate when p 0.5

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Lets Try Some Examples
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Comparing Two Proportions

• Setting 1 Independent Samples from two
  populations
• proportion of items in population 1 with
  characteristic
• proportion of items in population 2 with
  characteristic
•  
• Setting 2 Randomly assign subjects to one of two
  treatments
• Probability of success with treatment 1
• Probability of success with treatment 2
• In both settings, we wish to compare
  .

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The Data
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For large
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Hypothesis Test for

• Ho
• Under Ho, estimate the same
  quantity.

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Test Statistic
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Set Up for Ha
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• How about another example?