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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. = )