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Last Time. Central Limit Theorem Illustrations How large n? Normal Approximation to Binomial Statistical Inference Estimate unknown parameters Unbiasedness (centered correctly) Standard error (measures spread). Administrative Matters. Midterm II, coming Tuesday, April 6.

malorie-scholz
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Last Time

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  1. Last Time • Central Limit Theorem • Illustrations • How large n? • Normal Approximation to Binomial • Statistical Inference • Estimate unknown parameters • Unbiasedness (centered correctly) • Standard error (measures spread)

  2. Administrative Matters Midterm II, coming Tuesday, April 6

  3. Administrative Matters Midterm II, coming Tuesday, April 6 • Numerical answers: • No computers, no calculators

  4. Administrative Matters Midterm II, coming Tuesday, April 6 • Numerical answers: • No computers, no calculators • Handwrite Excel formulas (e.g. =9+4^2) • Don’t do arithmetic (e.g. use such formulas)

  5. Administrative Matters Midterm II, coming Tuesday, April 6 • Numerical answers: • No computers, no calculators • Handwrite Excel formulas (e.g. =9+4^2) • Don’t do arithmetic (e.g. use such formulas) • Bring with you: • One 8.5 x 11 inch sheet of paper

  6. Administrative Matters Midterm II, coming Tuesday, April 6 • Numerical answers: • No computers, no calculators • Handwrite Excel formulas (e.g. =9+4^2) • Don’t do arithmetic (e.g. use such formulas) • Bring with you: • One 8.5 x 11 inch sheet of paper • With your favorite info (formulas, Excel, etc.)

  7. Administrative Matters Midterm II, coming Tuesday, April 6 • Numerical answers: • No computers, no calculators • Handwrite Excel formulas (e.g. =9+4^2) • Don’t do arithmetic (e.g. use such formulas) • Bring with you: • One 8.5 x 11 inch sheet of paper • With your favorite info (formulas, Excel, etc.) • Course in Concepts, not Memorization

  8. Administrative Matters Midterm II, coming Tuesday, April 6 • Material Covered: HW 6 – HW 10

  9. Administrative Matters Midterm II, coming Tuesday, April 6 • Material Covered: HW 6 – HW 10 • Note: due Thursday, April 2

  10. Administrative Matters Midterm II, coming Tuesday, April 6 • Material Covered: HW 6 – HW 10 • Note: due Thursday, April 2 • Will ask grader to return Mon. April 5 • Can pickup in my office (Hanes 352)

  11. Administrative Matters Midterm II, coming Tuesday, April 6 • Material Covered: HW 6 – HW 10 • Note: due Thursday, April 2 • Will ask grader to return Mon. April 5 • Can pickup in my office (Hanes 352) • So today’s HW not included

  12. Administrative Matters Extra Office Hours before Midterm II Monday, Apr. 23 8:00 – 10:00 Monday, Apr. 23 11:00 – 2:00 Tuesday, Apr. 24 8:00 – 10:00 Tuesday, Apr. 24 1:00 – 2:00 (usual office hours)

  13. Study Suggestions • Work an Old Exam • On Blackboard • Course Information Section

  14. Study Suggestions • Work an Old Exam • On Blackboard • Course Information Section • Afterwards, check against given solutions

  15. Study Suggestions • Work an Old Exam • On Blackboard • Course Information Section • Afterwards, check against given solutions • Rework HW problems

  16. Study Suggestions • Work an Old Exam • On Blackboard • Course Information Section • Afterwards, check against given solutions • Rework HW problems • Print Assignment sheets • Choose problems in “random” order

  17. Study Suggestions • Work an Old Exam • On Blackboard • Course Information Section • Afterwards, check against given solutions • Rework HW problems • Print Assignment sheets • Choose problems in “random” order • Rework (don’t just “look over”)

  18. Reading In Textbook Approximate Reading for Today’s Material: Pages 356-369, 487-497 Approximate Reading for Next Class: Pages 498-501, 418-422, 372-390

  19. Law of Averages Case 2: any random sample CAN SHOW, for n “large” is “roughly” Terminology: • “Law of Averages, Part 2” • “Central Limit Theorem” (widely used name)

  20. Central Limit Theorem Illustration: Rice Univ. Applet http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.html Starting Distribut’n user input (very non-Normal) Dist’n of average of n = 25 (seems very mound shaped?)

  21. Extreme Case of CLT Consequences: roughly roughly Terminology: Called The Normal Approximation to the Binomial

  22. Normal Approx. to Binomial How large n? • Bigger is better • Could use “n ≥ 30” rule from above Law of Averages • But clearly depends on p • Textbook Rule: OK when {np ≥ 10 & n(1-p) ≥ 10}

  23. Statistical Inference Idea: Develop formal framework for handling unknowns p & μ e.g. 1: Political Polls e.g. 2a: Population Modeling e.g. 2b: Measurement Error

  24. Statistical Inference A parameter is a numerical feature of population, not sample An estimate of a parameter is some function of data (hopefully close to parameter)

  25. Statistical Inference Standard Error: for an unbiased estimator, standard error is standard deviation Notes: • For SE of , since don’t know p, use sensible estimate • For SE of , use sensible estimate

  26. Statistical Inference Another view: Form conclusions by

  27. Statistical Inference Another view: Form conclusions by quantifying uncertainty

  28. Statistical Inference Another view: Form conclusions by quantifying uncertainty (will study several approaches, first is…)

  29. Confidence Intervals Background:

  30. Confidence Intervals Background: The sample mean, , is an “estimate” of the population mean,

  31. Confidence Intervals Background: The sample mean, , is an “estimate” of the population mean, How accurate?

  32. Confidence Intervals Background: The sample mean, , is an “estimate” of the population mean, How accurate? (there is “variability”, how much?)

  33. Confidence Intervals Idea: Since a point estimate (e.g. or )

  34. Confidence Intervals Idea: Since a point estimate is never exactly right (in particular )

  35. Confidence Intervals Idea: Since a point estimate is never exactly right give a reasonable range of likely values (range also gives feeling for accuracy of estimation)

  36. Confidence Intervals Idea: Since a point estimate is never exactly right give a reasonable range of likely values (range also gives feeling for accuracy of estimation)

  37. Confidence Intervals E.g.

  38. Confidence Intervals E.g. with σ known

  39. Confidence Intervals E.g. with σ known Think: measurement error

  40. Confidence Intervals E.g. with σ known Think: measurement error Each measurement is Normal

  41. Confidence Intervals E.g. with σ known Think: measurement error Each measurement is Normal Known accuracy (maybe)

  42. Confidence Intervals E.g. with σ known Think: population modeling

  43. Confidence Intervals E.g. with σ known Think: population modeling Normal population

  44. Confidence Intervals E.g. with σ known Think: population modeling Normal population Known s.d. (a stretch, really need to improve)

  45. Confidence Intervals E.g. with σ known Recall the Sampling Distribution:

  46. Confidence Intervals E.g. with σ known Recall the Sampling Distribution: (recall have this even when data not normal, by Central Limit Theorem)

  47. Confidence Intervals E.g. with σ known Recall the Sampling Distribution: Use to analyze variation

  48. Confidence Intervals Understand error as: (normal density quantifies randomness in )

  49. Confidence Intervals Understand error as: (distribution centered at μ)

  50. Confidence Intervals Understand error as: (spread: s.d. = )

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