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Hypothesis Testing

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  1. Hypothesis Testing Variance known?

  2. Sampling Distribution • Over-the-counter stock selling prices • calculate average price of all stocks listed [] • take a sample of 25 stocks and record price • calculate average price of the 25 stocks [x-bar] • take all possible samples of size 25 • would all x-bars be equal? • average all the possible x-bars …equals 

  3. Sampling Distribution 20 H0 Levine, Prentice-Hall

  4. Sampling Distribution It is unlikely that we would get a sample mean of this value ... 20 H0 Levine, Prentice-Hall

  5. Sampling Distribution It is unlikely that we would get a sample mean of this value ... ... if in fact this were the population mean 20 H0 Levine, Prentice-Hall

  6. Sampling Distribution It is unlikely that we would get a sample mean of this value ... ... therefore, we reject the hypothesis that = 50. ... if in fact this were the population mean 20 H0 Levine, Prentice-Hall

  7. Null Hypothesis • What is tested • Always has equality sign: , or  • Designated H0 • Example ………... H0:  3 Levine, Prentice-Hall

  8. Alternative Hypothesis • Opposite of null hypothesis • Always has inequality sign: ,, or • Designated H1 • Example • H1:  < 3 Levine, Prentice-Hall

  9. Decision • Reject null hypothesis • Retain, or, fail to reject, null hypothesis • Do not use the term “accept” Levine, Prentice-Hall

  10. p-value • Probability of obtaining a test statistic more extreme (or than actual sample value given H0 is true • Called observed level of significance • Smallest value of  H0 can be rejected • Used to make rejection decision • If p-value , reject H0 Levine, Prentice-Hall

  11. Level of Significance • Defines unlikely values of sample statistic if null hypothesis is true • Called rejection region of sampling distribution • Designated (alpha) • Typical values are .01, .05, .10 • Selected by researcher at start Levine, Prentice-Hall

  12. Rejection Region (one-tail test) Sampling Distribution Level of Confidence 1 -  Levine, Prentice-Hall

  13. Rejection Region (one-tail test) Sampling Distribution Level of Confidence 1 -  Observed sample statistic Levine, Prentice-Hall

  14. Rejection Region (one-tail test) Sampling Distribution Level of Confidence 1 -  Observed sample statistic Levine, Prentice-Hall

  15. Rejection Regions(two-tailed test) Sampling Distribution Level of Confidence 1 -  Levine, Prentice-Hall

  16. Rejection Regions(two-tailed test) Sampling Distribution Level of Confidence 1 -  Observed sample statistic Levine, Prentice-Hall

  17. Rejection Regions(two-tailed test) Sampling Distribution Level of Confidence 1 -  Observed sample statistic Levine, Prentice-Hall

  18. Rejection Regions(two-tailed test) Sampling Distribution Level of Confidence 1 -  Observed sample statistic Levine, Prentice-Hall

  19. Risk of Errors in Making Decision • Type I error • Reject true null hypothesis • Has serious consequences • Probability of Type I error is alpha [ ] • Called level of significance • Type II error • Do not reject false null hypothesis • Probability of Type II error is beta [  ] Levine, Prentice-Hall

  20. Decision Results H0: Innocent Levine, Prentice-Hall

  21. Hypothesis Testing • State H0 • State H1 • Choose  • Choose n • Choose test Levine, Prentice-Hall

  22. Set up critical values Collect data Compute test statistic Make statistical decision Express decision Hypothesis Testing • State H0 • State H1 • Choose  • Choose n • Choose test Levine, Prentice-Hall

  23. Two-tailed z-test Does an average box of cereal contain 368 grams of cereal? A random sample of 25 boxes has an average weight = 372.5 grams. The company has specified to be 15 grams. Test at the .05 level. 368 gm. Levine, Prentice-Hall

  24. Two-tailed z-test Test Statistic: Decision: Conclusion: H0: H1:  n Critical Value(s): Levine, Prentice-Hall

  25. Two-tailed z-test Test Statistic: Decision: Conclusion: H0:  = 368 H1:  368  n Critical Value(s): Levine, Prentice-Hall

  26. Two-tailed z-test Test Statistic: Decision: Conclusion: H0:  = 368 H1:  368 .05 n25 Critical Value(s): Levine, Prentice-Hall

  27. Two-tailed z-test Test Statistic: Decision: Conclusion: H0:  = 368 H1:  368 .05 n25 Critical Value(s): Levine, Prentice-Hall

  28. Two-tailed z-test Test Statistic: Decision: Conclusion: H0:  = 368 H1:  368 .05 n25 Critical Value(s): Levine, Prentice-Hall

  29. Two-tailed z-test Test Statistic: Decision: Conclusion: H0:  = 368 H1:  368 .05 n25 Critical Value(s): Do not reject at  = .05 Levine, Prentice-Hall

  30. Two-tailed z-test Test Statistic: Decision: Conclusion: H0:  = 368 H1:  368 .05 n25 Critical Value(s): Do not reject at  = .05 No evidence average is not 368 Levine, Prentice-Hall

  31. Two-tailed z-test [p-value]] Z value of sample statistic  Levine, Prentice-Hall

  32. Two-tailed z-test [p-value] p-value is P(z  -1.50 or z  1.50) Z value of sample statistic  Levine, Prentice-Hall

  33. Two-tailed z-test [p-value] p-value is P(z  -1.50 or z  1.50) Z value of sample statistic  Levine, Prentice-Hall

  34. Two-tailed z-test [p-value] p-value is P(z  -1.50 or z  1.50) .4332 From Z table: lookup 1.50 Z value of sample statistic   Levine, Prentice-Hall

  35. Two-tailed z-test [p-value] p-value is P(z  -1.50 or z  1.50)  .5000- .4332 .0668 .4332 From Z table: lookup 1.50 Z value of sample statistic   Levine, Prentice-Hall

  36. Two-tailed z-test [p-value] p-value is P(z  -1.50 or z  1.50) = .1336  .5000- .4332 .0668 .4332 From Z table: lookup 1.50 Z value of sample statistic   Levine, Prentice-Hall

  37. Two-tailed z-test [p-value] 1/2 p-value = .0668 1/2 p-value = .0668 1/2  = .025 1/2  = .025 Levine, Prentice-Hall

  38. Two-tailed z-test [p-value] (p-Value = .1336)  ( = .05) Do not reject. 1/2 p-Value = .0668 1/2 p-Value = .0668 1/2  = .025 1/2  = .025 Test statistic is in ‘Do not reject’ region Levine, Prentice-Hall

  39. Two-tailed z-test( known)challenge You are a Q/C inspector. You want to find out if a new machine is making electrical cords to customer specification: average breaking strength of 70 lb. with = 3.5 lb. You take a sample of 36cords & compute a sample mean of 69.7lb. At the .05level, is there evidence that the machine is not meeting the average breaking strength? Levine, Prentice-Hall

  40. Two-tailed z-test ( known) Test Statistic: Decision: Conclusion: H0: H1:  = n = Critical Value(s): Levine, Prentice-Hall

  41. Two-tailed z-test ( known) Test Statistic: Decision: Conclusion: H0:  = 70 H1:  70  = n = Critical Value(s): Levine, Prentice-Hall

  42. Two-tailed z-test ( known) Test Statistic: Decision: Conclusion: H0:  = 70 H1:  70  = .05 n = 36 Critical Value(s): Levine, Prentice-Hall

  43. Two-tailed z-test ( known) Test Statistic: Decision: Conclusion: H0:  = 70 H1:  70  = .05 n = 36 Critical Value(s): Levine, Prentice-Hall

  44. Two-tailed z-test ( known) Test Statistic: Decision: Conclusion: H0:  = 70 H1:  70  = .05 n = 36 Critical Value(s): Levine, Prentice-Hall

  45. Two-tailed z-test ( known) Test Statistic: Decision: Conclusion: H0:  = 70 H1:  70  = .05 n = 36 Critical Value(s): Do not reject at  = .05 Levine, Prentice-Hall

  46. Two-tailed z-test ( known) Test Statistic: Decision: Conclusion: H0:  = 70 H1:  70  = .05 n = 36 Critical Value(s): Do not reject at  = .05 No evidence average is not 70 Levine, Prentice-Hall

  47. One-tailed z-test ( known) • Assumptions • Population is normally distributed • If not normal, can be approximated by normal distribution for large samples Levine, Prentice-Hall

  48. One-tailed z-test ( known) • Assumptions • Population is normally distributed • If not normal, can be approximated by normal distribution for large samples • Null hypothesis has  or  sign only Levine, Prentice-Hall

  49. One-tailed z-test ( known) • Assumptions • Population is normally distributed • If not normal, can be approximated by normal distribution for large samples • Null hypothesis has  or  sign only • Z-test statistic Levine, Prentice-Hall

  50. One-tailed z-test ( known) H0:0 H1: < 0 Must be significantly below  Levine, Prentice-Hall