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Lecture 19

Lecture 19. Parameters and statistics. Example: A random sample of 1014 voters are asked if they think the President is too liberal, too conservative, or about right. 514 (51%) say ‘too liberal’. The observed percentage in the sample , 51%, is a statistic .

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Lecture 19

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  1. Lecture 19

  2. Parameters and statistics • Example: A random sample of 1014 voters are asked if they think the President is too liberal, too conservative, or about right. • 514 (51%) say ‘too liberal’. • The observed percentage in the sample, 51%, is a statistic. • The unknown percent of the population who would say yes is a parameter. • Presumably it’s close to 51%, but we will never know. • Question we’d like to ask: “How likely is the statistic to be how close to the parameter?”

  3. Sampling experiment • http://demonstrations.wolfram.com/UrnProblem/

  4. Experiment • Population (urn) – our class (22 people) • Question: Dogs vs Cats • We will sample 5 • Assign numbers • Use R to sample • Repeat 5 times • ascertain truth and run R

  5. US Polls • 2012 estimate of the number of eligible voters is 206,072,000. • We sampled 1014 people at random and got 514 yes.

  6. Main idea • We learned that given a model we can check if the data is consistent with it • Idea: Find models that are consistent with the data.

  7. US Polls • 2012 estimate of the number of eligible voters is 206,072,000. • We sampled 1014 people at random and got 514 yes. • We will consider various models: • True proportion is p = 0.05, 0.10, 0.15, … • Which of the models is the data in agreement with?

  8. Results

  9. “P-value” • Consider proportion of fake samples < 514 • Values close to 0 or 1 are not consistent with the model

  10. Cutoff • Using better resolution of models • A usual cutoff 0.05 split between both sides (0.975 and .025) • Models selected: [.477, .538]

  11. Fake data • 3 sub-groups • 42,343,562 (R) • 59,280,986 (D) • 104,447,452 (I) • We sample roughly in proportion: • Sampled 220, 201 yes (R) • Sampled 284, 31 yes (D) • Sampled 510, 282 yes (I)

  12. Issue • Too many levers to “fiddle” (three different Ks for each group) • Cannot simply look how well the data fits. • Needs more sophisticated statistics

  13. Naïve parametric bootstrap • Compute a number/numbers estimated from the data (point estimate). • Use this number to simulate a lot of fake data and see how the fake data can vary. • Use this variability to estimate the uncertainty in our estimator

  14. Original problem • Estimated K=206,072,000 * (514/1014)=104,458,588 • Estimated p in (.476,.537)

  15. Stratified problem • Estimate K for each group – combine to get joint p estimate • pcombined=.20*.91 +.29*.11 +.51*.55=.50 • This is not that much different from ignoring stratification • There is a (small) gain in uncertainty • Bootstrap interval (.474,.525)

  16. Bootstrap Stratified

  17. Bootstrap SRS

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