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STA291

STA291. Statistical Methods Lecture 17. Bias versus Efficiency. A. B. C. D. Last time:. Thanks to the CLT … We know is approximately normal with mean and standard deviation ___ and ______, respectively. Time to use this fact to do some inference about … ?.

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STA291

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  1. STA291 Statistical Methods Lecture 17

  2. Bias versus Efficiency A B C D

  3. Last time: Thanks to the CLT … We know is approximately normal with mean and standard deviation ___ and ______, respectively. Time to use this fact to do some inference about … ?

  4. Estimation of a Proportion The sample proportion is an unbiased and efficient point estimator of the population proportion p

  5. Confidence Interval for a Proportion • A large sample confidence interval for the population proportion p has the form • where is the sample proportion

  6. Example • In a recent telephone survey (conducted in mid- October), people were asked whether they have seen a ghost or felt its presence. • Of 1013 adults interviewed, 230 answered yes, and 783 answered no. • Find the point estimate of the population proportion of adults who would answer yes • Construct and interpret a 95% confidence interval for the population proportion. • Can you conclude that fewer than half of the population has seen a ghost or felt its presence?

  7. Conditions & Interpretation • Conditions: • Random sample • “Infinite population”, or population size at least 10 times that of the sample • p not too near 0, 1 (usually checked by verifying that & .) • Interpretation “We found the interval [ __ , __ ] using a method that, if done with many randomly drawn samples, would result in intervals that would include the true population proportion (of _____) ___ % of the time.”

  8. Sample size considerations • Suppose you’re working for a candidate and you return with a 95% CI for the proportion of likely voters who replied in the affirmative when asked if they were going to vote for your candidate; that interval is [0.45,0.57]. • Cause for celebration? or consternation? • Difficulty: while the point estimate (0.51) is on the “right side” of 50%, the margin of error is so large, we aren’t confident we’re going to win. • Solution?

  9. Sample size calculation • Suppose we’re given a target bound on our margin of error, ME • This can be solved for the sample size, n: • But wait, we don’t know p…

  10. Looking back • Estimation of proportions • Point estimate (little used) • Confidence interval estimate • Assumptions • Interpretation (!) • Sample size calculation • Cases

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