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Last Time. Hypothesis Testing Yes – No Questions Assess with p-value P[what saw or m.c. | Boundary] Interpretation Small is conclusive 1-sided vs. 2-sided. Administrative Matters. Midterm I, coming Tuesday, Feb. 24. Administrative Matters. Midterm I, coming Tuesday, Feb. 24

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Last Time

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  1. Last Time • Hypothesis Testing • Yes – No Questions • Assess with p-value P[what saw or m.c. | Boundary] • Interpretation • Small is conclusive • 1-sided vs. 2-sided

  2. Administrative Matters Midterm I, coming Tuesday, Feb. 24

  3. Administrative Matters Midterm I, coming Tuesday, Feb. 24 • Numerical answers: • No computers, no calculators • Handwrite Excel formulas (e.g. =9+4^2) • Don’t do arithmetic (e.g. use such formulas)

  4. Administrative Matters Midterm I, coming Tuesday, Feb. 24 • 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: • 8.5 x 11 inch sheet of paper • With your favorite info (formulas, Excel, etc.)

  5. Administrative Matters Midterm I, coming Tuesday, Feb. 24 • 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: • 8.5 x 11 inch sheet of paper • With your favorite info (formulas, Excel, etc.) • Course in Concepts, not Memorization

  6. Administrative Matters State of BlackBoard Discussion Board • Generally happy with result

  7. Administrative Matters State of BlackBoard Discussion Board • Generally happy with result • But think carefully about “where to post” • Look at current Thread HW 4 • Note “diffusion of questions” • Hard to find what you want

  8. Administrative Matters State of BlackBoard Discussion Board • Generally happy with result • But think carefully about “where to post” • Look at current Thread HW 4 • Note “diffusion of questions” • Hard to find what you want • Suggest keep HW problems all together • i.e. One “Root node” per HW problem

  9. Administrative Matters State of BlackBoard Discussion Board • Suggest keep HW problems all together • i.e. One “Root node” per HW problem

  10. Administrative Matters State of BlackBoard Discussion Board • Suggest keep HW problems all together • i.e. One “Root node” per HW problem • Choose where to post (in tree) carefully

  11. Administrative Matters State of BlackBoard Discussion Board • Suggest keep HW problems all together • i.e. One “Root node” per HW problem • Choose where to post (in tree) carefully • Use better “Subject Lines” • Not just dumb “Replies” • You can enter anything you want • Try to make it clear to readers… • Especially when “not following current line”

  12. Reading In Textbook Approximate Reading for Today’s Material: Pages 261-262, 9-14 Approximate Reading for Next Class: 270-276, 30-34

  13. Hypothesis Testing In General: p-value = P[what was seen, or more conclusive | at boundary between H0 & H1] Caution: more conclusive requires careful interpretation

  14. Hypothesis Testing Caution: more conclusive requires careful interpretation Reason: Need to decide between 1 - sided Hypotheses, like H0 : p < vs. H1: p ≥ And 2 - sided Hypotheses, like H0 : p = vs. H1: p ≠

  15. Hypothesis Testing e.g. a slot machine bears a sign which says “Win 30% of the time” In 10 plays, I don’t win any. Can I conclude sign is false? (& thus have grounds for complaint, or is this a reasonable occurrence?)

  16. Hypothesis Testing e.g. a slot machine bears a sign which says “Win 30% of the time” In 10 plays, I don’t win any. Conclude false? Let p = P[win], let X = # wins in 10 plays Model: X ~ Bi(10, p) Test: H0: p = 0.3 vs. H1: p ≠ 0.3

  17. Hypothesis Testing Test: H0: p = 0.3 vs. H1: p ≠ 0.3 p-value = P[X = 0 or more conclusive | p = 0.3]

  18. Hypothesis Testing Test: H0: p = 0.3 vs. H1: p ≠ 0.3 p-value = P[X = 0 or more conclusive | p = 0.3] (understand this by visualizing # line)

  19. Hypothesis Testing Test: H0: p = 0.3 vs. H1: p ≠ 0.3 p-value = P[X = 0 or more conclusive | p = 0.3] 0 1 2 3 4 5 6

  20. Hypothesis Testing Test: H0: p = 0.3 vs. H1: p ≠ 0.3 p-value = P[X = 0 or more conclusive | p = 0.3] 0 1 2 3 4 5 6 30% of 10, most likely when p = 0.3 i.e. least conclusive

  21. Hypothesis Testing Test: H0: p = 0.3 vs. H1: p ≠ 0.3 p-value = P[X = 0 or more conclusive | p = 0.3] 0 1 2 3 4 5 6 so more conclusive includes

  22. Hypothesis Testing Test: H0: p = 0.3 vs. H1: p ≠ 0.3 p-value = P[X = 0 or more conclusive | p = 0.3] 0 1 2 3 4 5 6 so more conclusive includes but since 2-sided, also include

  23. Hypothesis Testing Generally how to calculate? 0 1 2 3 4 5 6

  24. Hypothesis Testing Generally how to calculate? Observed Value 0 1 2 3 4 5 6

  25. Hypothesis Testing Generally how to calculate? Observed Value Most Likely Value 0 1 2 3 4 5 6

  26. Hypothesis Testing Generally how to calculate? Observed Value Most Likely Value 0 1 2 3 4 5 6 # spaces = 3

  27. Hypothesis Testing Generally how to calculate? Observed Value Most Likely Value 0 1 2 3 4 5 6 # spaces = 3 so go 3 spaces in other direct’n

  28. Hypothesis Testing Result: More conclusive means X ≤ 0 or X ≥ 6 0 1 2 3 4 5 6

  29. Hypothesis Testing Result: More conclusive means X ≤ 0 or X ≥ 6 p-value = P[X = 0 or more conclusive | p = 0.3]

  30. Hypothesis Testing Result: More conclusive means X ≤ 0 or X ≥ 6 p-value = P[X = 0 or more conclusive | p = 0.3] = P[X ≤ 0 or X ≥ 6 | p = 0.3]

  31. Hypothesis Testing Result: More conclusive means X ≤ 0 or X ≥ 6 p-value = P[X = 0 or more conclusive | p = 0.3] = P[X ≤ 0 or X ≥ 6 | p = 0.3] = P[X ≤ 0] + (1 – P[X ≤ 5])

  32. Hypothesis Testing Result: More conclusive means X ≤ 0 or X ≥ 6 p-value = P[X = 0 or more conclusive | p = 0.3] = P[X ≤ 0 or X ≥ 6 | p = 0.3] = P[X ≤ 0] + (1 – P[X ≤ 5]) = 0.076

  33. Hypothesis Testing Result: More conclusive means X ≤ 0 or X ≥ 6 p-value = P[X = 0 or more conclusive | p = 0.3] = P[X ≤ 0 or X ≥ 6 | p = 0.3] = P[X ≤ 0] + (1 – P[X ≤ 5]) = 0.076 Excel result from: http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg4.xls

  34. Hypothesis Testing Test: H0: p = 0.3 vs. H1: p ≠ 0.3 p-value = 0.076

  35. Hypothesis Testing Test: H0: p = 0.3 vs. H1: p ≠ 0.3 p-value = 0.076 Yes-No Conclusion: 0.076 > 0.05, so not safe to conclude “P[win] = 0.3” sign is wrong, at level 0.05

  36. Hypothesis Testing Test: H0: p = 0.3 vs. H1: p ≠ 0.3 p-value = 0.076 Yes-No Conclusion: 0.076 > 0.05, so not safe to conclude “P[win] = 0.3” sign is wrong, at level 0.05 (10 straight losses is reasonably likely)

  37. Hypothesis Testing Test: H0: p = 0.3 vs. H1: p ≠ 0.3 p-value = 0.076 Yes-No Conclusion: 0.076 > 0.05, so not safe to conclude “P[win] = 0.3” sign is wrong, at level 0.05 Gray Level Conclusion: in “fuzzy zone”, some evidence, but not too strong

  38. Hypothesis Testing Alternate Question: Same setup, can we conclude: P[win] < 30% ???

  39. Hypothesis Testing Alternate Question: Same setup, can we conclude: P[win] < 30% ??? • Seems like same question?

  40. Hypothesis Testing Alternate Question: Same setup, can we conclude: P[win] < 30% ??? • Seems like same question? • Careful, “≠” became “<”

  41. Hypothesis Testing Alternate Question: Same setup, can we conclude: P[win] < 30% ??? • Seems like same question? • Careful, “≠” became “<” • I.e. 2-sided hypo became 1-sided hypo

  42. Hypothesis Testing Alternate Question: Same setup, can we conclude: P[win] < 30% ??? • Seems like same question? • Careful, “≠” became “<” • I.e. 2-sided hypo became 1-sided hypo • Difference can have major impact

  43. Hypothesis Testing Alternate Question: Same setup, can we conclude: P[win] < 30% ???

  44. Hypothesis Testing Alternate Question: Same setup, can we conclude: P[win] < 30% ??? Test: H0: p ≥ 0.3 vs. H1: p < 0.3

  45. Hypothesis Testing Alternate Question: Same setup, can we conclude: P[win] < 30% ??? Test: H0: p ≥ 0.3 vs. H1: p < 0.3 p-value = P[ X = 0 or m. c. | p = 0.3]

  46. Hypothesis Testing Alternate Question: Same setup, can we conclude: P[win] < 30% ??? Test: H0: p ≥ 0.3 vs. H1: p < 0.3 p-value = P[ X = 0 or m. c. | p = 0.3] same boundary between H0 & H1

  47. Hypothesis Testing Alternate Question: Same setup, can we conclude: P[win] < 30% ??? Test: H0: p ≥ 0.3 vs. H1: p < 0.3 p-value = P[ X = 0 or m. c. | p = 0.3]

  48. Hypothesis Testing Alternate Question: Same setup, can we conclude: P[win] < 30% ??? Test: H0: p ≥ 0.3 vs. H1: p < 0.3 p-value = P[ X = 0 or m. c. | p = 0.3] = P[ X ≤ 0 | p = 0.3]

  49. Hypothesis Testing Alternate Question: Same setup, can we conclude: P[win] < 30% ??? Test: H0: p ≥ 0.3 vs. H1: p < 0.3 p-value = P[ X = 0 or m. c. | p = 0.3] = P[ X ≤ 0 | p = 0.3] = 0.028

  50. Hypothesis Testing Alternate Question: Same setup, can we conclude: P[win] < 30% ??? Test: H0: p ≥ 0.3 vs. H1: p < 0.3 p-value = P[ X = 0 or m. c. | p = 0.3] = P[ X ≤ 0 | p = 0.3] = 0.028 Excel result from: http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg4.xls

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