Lecture09

# Lecture09 - Assignment After this lecture start the...

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Assignment After this lecture, start the Pre-Midterm Extra Problems. It’s in the course binder.

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Lecture 9 Sampling distribution and confidence intervals for proportions
Overview We know about sampling distribution standard error confidence interval for a mean Today we’ll learn the same for a proportion Connection: A proportion is the mean of a dummy variable.

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1. Sampling distribution of a proportion
Sampling: 1936 election Literary Digest poll unrepresentative sample of 10 million voters wrong: called election for Landon Gallup poll “quota sample” of 50,000 voters right: called election for Roosevelt Sociology 549 poll simple random sample of 100 voters right or wrong?

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Review: Sampling distribution of the mean Across all possible samples has a normal distribution with mean and standard error Y Y Y μ μ= N Y Y / σ =
Review: Dummy variables Let Y be a dummy variable (1=Roosevelt, 0=not). In a sample, where p is the proportion of the sample with Y =1 In the population , where π is the proportion of the population with Y =1 ) 1 ( p p s p Y Y - = = ) 1 ( π σ μ - = = Y Y

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Result: Sampling distribution of the proportion Across all possible samples has a normal distribution with mean and standard error of the sample proportions Y Y Y μ μ= N Y Y / σ = p p = π N p / ) 1 ( π - =
Example 1: Standard error of a sample proportion π =.61, or 61%, of voters will vote for Roosevelt But we don’t know this. We sample N =100 voters In sample, p will vote for Roosevelt Standard error of p σ p =[ π(1-π29 / Ν ] 1/2 =[ .61(1-.6129 / 100 ] 1/2 = .05 , or 5%

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Example 1: Margin of error Poll’s “margin of error” is typically ~2 SE’s Using normal distribution, In 95% of all samples the sample proportion p are within 1.96 standard errors of population proportion Here, in 95% of samples, we’ll get .61 +/- 1.96(.05) = .51 to .71, or 51% to 71% voting for Roosevelt
.50

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Lecture09 - Assignment After this lecture start the...

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