Chap08 fundamentals of hypothesis

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Chap08 fundamentals of hypothesis

  1. 1. Statistics for Managers Using Microsoft® Excel 4th Edition Chapter 8 Fundamentals of Hypothesis Testing: One-Sample Tests Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-1
  2. 2. Chapter Goals After completing this chapter, you should be able to:  Formulate null and alternative hypotheses for applications involving a single population mean or proportion  Formulate a decision rule for testing a hypothesis  Know how to use the critical value and p-value approaches to test the null hypothesis (for both mean and proportion problems)  Know what Type I and Statistics for Managers Using Type II errors are Microsoft Excel, 4e © 2004 Chap 8-2 Prentice-Hall, Inc.
  3. 3. What is a Hypothesis?  A hypothesis is a claim (assumption) about a population parameter:  population mean Example: The mean monthly cell phone bill of this city is μ = $42  population proportion Example: The proportion of adults in this Statistics for Managers Using city with cell phones is p = .68 Microsoft Excel, 4e © 2004 Chap 8-3 Prentice-Hall, Inc.
  4. 4. The Null Hypothesis, H0  States the assumption (numerical) to be tested Example: The average number of TV sets in U.S. Homes is equal to three ( H0 : μ = 3 )  Is always about a population parameter, not about a sample statistic H0 : μ = 3 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. H0 : X = 3 Chap 8-4
  5. 5. The Null Hypothesis, H0 (continued)     Begin with the assumption that the null hypothesis is true  Similar to the notion of innocent until proven guilty Refers to the status quo Always contains “=” , “≤” or “≥” sign May or may not be rejected Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-5
  6. 6. The Alternative Hypothesis, H1  Is the opposite of the null hypothesis  e.g., The average number of TV sets in U.S. homes is not equal to 3 ( H1: μ ≠ 3 )  Challenges the status quo  Never contains the “=” , “≤” or “≥” sign May or may not be proven Is generally the hypothesis that the researcher is trying to prove   Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-6
  7. 7. Hypothesis Testing Process Claim: the population mean age is 50. (Null Hypothesis: H0: μ = 50 ) Population Is X= 20 likely if μ = 50? Suppose the sample REJECT Statistics for Managers Using mean age Null Hypothesis is 20: X = 20 Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Now select a random sample If not likely, Sample
  8. 8. Reason for Rejecting H0 Sampling Distribution of X 20 μ = 50 If H0 is true If it is unlikely that we would get a ... if in fact this were Statistics for Managers Using sample mean of the Microsoft Excel, 4e © 2004population mean… this value ... Prentice-Hall, Inc. X ... then we reject the null hypothesis that μ = 50. Chap 8-8
  9. 9. Level of Significance, α  Defines the unlikely values of the sample statistic if the null hypothesis is true   Is designated by α , (level of significance)   Defines rejection region of the sampling distribution Typical values are .01, .05, or .10 Is selected by the researcher at the beginning  Provides the critical Statistics for Managers Using value(s) of the test Microsoft Excel, 4e © 2004 Chap 8-9 Prentice-Hall, Inc.
  10. 10. Level of Significance and the Rejection Region Level of significance = H0: μ = 3 H1: μ ≠ 3 H0: μ ≤ 3 H1: μ > 3 α α/2 Two-tail test α/2 Represents critical value Rejection region is shaded 0 α Upper-tail test H0: μ ≥ 3 α H1: for 3 Statisticsμ < Managers Using Microsoft Excel, Lower-tail test 4e © 2004 Prentice-Hall, Inc. 0 0 Chap 8-10
  11. 11. Errors in Making Decisions  Type I Error  Reject a true null hypothesis  Considered a serious type of error The probability of Type I Error is α   Called level of significance of the test Set by researcher in advance Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-11
  12. 12. Errors in Making Decisions (continued)  Type II Error  Fail to reject a false null hypothesis The probability of Type II Error is β Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-12
  13. 13. Outcomes and Probabilities Possible Hypothesis Test Outcomes Decision Key: Outcome (Probability) Actual Situation H0 True H0 False Do Not Reject H0 No error (1 - α ) Type II Error (β) Reject H0 Type I Error (α) No Error (1-β) Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-13
  14. 14. Type I & II Error Relationship  Type I and Type II errors can not happen at the same time  Type I error can only occur if H0 is true  Type II error can only occur if H0 is false If Type I error probability ( α ) Statistics for Managers Using Type II error probability ( β ) Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. , then Chap 8-14
  15. 15. Factors Affecting Type II Error  All else equal,  β when the difference between hypothesized parameter and its true value  β when α  β when σ when n Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.  β Chap 8-15
  16. 16. Hypothesis Tests for the Mean Hypothesis Tests for µ σ Known Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. σ Unknown Chap 8-16
  17. 17. Z Test of Hypothesis for the Mean (σ Known)  Convert sample statistic ( X ) to a Z test statistic Hypothesis Tests for µ σ Known σ Unknown The test statistic is: X −μ Z = σ Statistics for Managers Using n Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-17
  18. 18. Critical Value Approach to Testing  For two tailed test for the mean, σ known:  Convert sample statistic ( X ) to test statistic (Z statistic )  Determine the critical Z values for a specified level of significance α from a table or computer  Decision Rule: If the test statistic falls in the rejection region, reject H0 ; otherwise do not Statistics for Managers Using reject H Microsoft Excel, 04e © 2004 Prentice-Hall, Inc. Chap 8-18
  19. 19. Two-Tail Tests  There are two cutoff values (critical values), defining the regions of rejection H0: μ = 3 H1 : μ ≠ 3 α/2 α/2 X 3 Reject H0 -Z Lower Statistics for Managers Using critical Microsoft Excel, 4e © 2004 value Prentice-Hall, Inc. Do not reject H0 0 Reject H0 +Z Upper critical value Chap 8-19 Z
  20. 20. Review: 10 Steps in Hypothesis Testing  1. State the null hypothesis, H0  2. State the alternative hypotheses, H1  3. Choose the level of significance, α  4. Choose the sample size, n  5. Determine the appropriate statistical technique and the test statistic to use 6. Find the critical values and determine the rejection region(s) Statistics for Managers Using  Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-20
  21. 21. Review: 10 Steps in Hypothesis Testing  7. Collect data and compute the test statistic from the sample result  8. Compare the test statistic to the critical value to determine whether the test statistics falls in the region of rejection  9. Make the statistical decision: Reject H0 if the test statistic falls in the rejection region 10. Express the decision in the context of the problem Statistics for Managers Using  Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-21
  22. 22. Hypothesis Testing Example Test the claim that the true mean # of TV sets in US homes is equal to 3. (Assume σ = 0.8)  1-2. State the appropriate null and alternative hypotheses  H: μ = 3 H1: μ ≠ 3 (This is a two tailed test) 0 Specify the desired level of significance  Suppose that α = .05 is chosen for this test  4. Choose a sample size  StatisticsSuppose a sample of size n = 100 is selected for Managers Using Microsoft Excel, 4e © 2004 Chap 8-22 Prentice-Hall, Inc.  3.
  23. 23. Hypothesis Testing Example (continued)    5. Determine the appropriate technique  σ is known so this is a Z test 6. Set up the critical values  For α = .05 the critical Z values are ±1.96 7. Collect the data and compute the test statistic  Suppose the sample results are n = 100, X = 2.84 (σ = 0.8 is assumed known) So the test statistic is: X −μ 2.84 − 3 −.16 = = =− 2.0 σ 0.8 .08 Statistics for Managers Using n 100 Microsoft Excel, 4e © 2004 Z = Prentice-Hall, Inc. Chap 8-23
  24. 24. Hypothesis Testing Example (continued)  8. Is the test statistic in the rejection region? α = .05/2 α = .05/2 Reject H Do not reject H Reject H Reject H0 if 0 -Z= -1.96 +Z= +1.96 Z < -1.96 or Z > 1.96; otherwise Here, Z = -2.0 < -1.96, so the do not test statistic Statistics for Managers Using is in the rejection reject H0 Microsoft Excel, 4e region © 2004 Chap 8-24 Prentice-Hall, Inc. 0 0 0
  25. 25. Hypothesis Testing Example (continued)  9-10. Reach a decision and interpret the result α = .05/2 Reject H0 -Z= -1.96 α = .05/2 Do not reject H0 0 Reject H0 +Z= +1.96 -2.0 Since Z = -2.0 < -1.96, we reject the null hypothesis and conclude that there is sufficient evidence that the Statistics for Managers Using mean number of TVs in US homes is not equal to 3 Microsoft Excel, 4e © 2004 Chap 8-25 Prentice-Hall, Inc.
  26. 26. p-Value Approach to Testing  p-value: Probability of obtaining a test statistic more extreme ( ≤ or ≥ ) than the observed sample value given H0 is true  Also called observed level of significance Smallest value of α for which H0 can be Statistics for Managers Using rejected  Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-26
  27. 27. p-Value Approach to Testing (continued)  Convert Sample Statistic (e.g., X ) to Test Statistic (e.g., Z statistic )  Obtain the p-value from a table or computer  Compare the p-value with α  If p-value < α , reject H0  If p-value ≥ α , do not reject H0 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-27
  28. 28. p-Value Example  Example: How likely is it to see a sample mean of 2.84 (or something further from the mean, in either direction) if the true mean is µ = 3.0? X = 2.84 is translated to a Z score of Z = -2.0 P(Z < −2.0) = .0228 P(Z > 2.0) = .0228 α/2 = .025 α/2 = .025 .0228 .0228 p-value =.0228 for Managers Statistics + .0228 = .0456 Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. -1.96 -2.0 0 1.96 2.0 Chap 8-28 Z
  29. 29. p-Value Example  Compare the p-value with α  If p-value < α , reject H0  (continued) If p-value ≥ α , do not reject H0 Here: p-value = .0456 α = .05 α/2 = .025 α/2 = .025 .0228 Since .0456 < .05, we reject the null hypothesis Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. .0228 -1.96 -2.0 0 1.96 2.0 Chap 8-29 Z
  30. 30. Connection to Confidence Intervals  For X = 2.84, σ = 0.8 and n = 100, the 95% confidence interval is: 0.8 2.84 - (1.96) to 100 0.8 2.84 + (1.96) 100 2.6832 ≤ μ ≤ 2.9968 Since this interval does not contain the hypothesized mean (3.0), we reject the Statistics for Managers Using null hypothesis at α = .05  Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
  31. 31. One-Tail Tests  In many cases, the alternative hypothesis focuses on a particular direction H0: μ ≥ 3 This is a lower-tail test since the alternative hypothesis is focused on the lower tail below the mean of 3 H1: μ < 3 This is an upper-tail test since the alternative hypothesis is focused on H1: μ > 3 the upper tail above the mean of 3 Statistics for Managers Using Microsoft Excel, 4e © 2004 Chap 8-31 Prentice-Hall, Inc. H0: μ ≤ 3
  32. 32. Lower-Tail Tests H0: μ ≥ 3  There is only one critical value, since the rejection area is in only one tail H1: μ < 3 α Reject H0 -Z Do not reject H0 0 μ Critical value Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Z X Chap 8-32
  33. 33. Upper-Tail Tests  There is only one critical value, since the rejection area is in only one tail H0: μ ≤ 3 H1: μ > 3 α Z Do not reject H0 X μ Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. 0 Zα Reject H0 Critical value Chap 8-33
  34. 34. Example: Upper-Tail Z Test for Mean (σ Known) A phone industry manager thinks that customer monthly cell phone bill have increased, and now average over $52 per month. The company wishes to test this claim. (Assume σ = 10 is known) Form hypothesis test: H0: μ ≤ 52 the average is not over $52 per month H1: μ > 52 the average is greater than $52 per month (i.e., sufficient evidence exists to support the Managers Using manager’s claim) Statistics for Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-34
  35. 35. Example: Find Rejection Region (continued)  Suppose that α = .10 is chosen for this test Find the rejection region: Reject H0 α = .10 Do not reject H0 0 1.28 Reject H0 Statistics for Managers Using Reject H0 if Z > 1.28 Microsoft Excel, 4e © 2004 Chap 8-35 Prentice-Hall, Inc.
  36. 36. Review: One-Tail Critical Value What is Z given α = 0.10? .90 .10 α = .10 .90 z Standard Normal Distribution Table (Portion) 0 1.28 Statistics for Critical Value Managers Using = 1.28 Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Z .07 .08 .09 1.1 .8790 .8810 .8830 1.2 .8980 .8997 .9015 1.3 .9147 .9162 .9177 Chap 8-36
  37. 37. Example: Test Statistic (continued) Obtain sample and compute the test statistic Suppose a sample is taken with the following results: n = 64, X = 53.1 (σ=10 was assumed known)  Then the test statistic is: X−μ 53.1 − 52 Z = = = 0.88 σ 10 Statistics for Managers Using n 64 Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-37
  38. 38. Example: Decision (continued) Reach a decision and interpret the result: Reject H0 α = .10 Do not reject H0 1.28 0 Z = .88 Reject H0 Do not reject H0 since Z = 0.88 ≤ 1.28 Statistics for Managers Using i.e.: there is not sufficient evidence that the Microsoft Excel,mean 2004 over $52 4e © bill is Chap 8-38 Prentice-Hall, Inc.
  39. 39. p -Value Solution Calculate the p-value and compare to α (continued) (assuming that μ = 52.0) p-value = .1894 Reject H0 α = .10 0 Do not reject H0 1.28 Z = .88 Reject H0 P( X ≥ 53.1) 53.1 − 52.0   = P Z ≥  10/ 64   = P(Z ≥ 0.88) = 1 − .8106 = .1894 Statistics for Managers Using Do Excel, 4e © since Microsoftnot reject H02004 p-value = .1894 > α = .10 Chap 8-39 Prentice-Hall, Inc.
  40. 40. Z Test of Hypothesis for the Mean (σ Known)  Convert sample statistic ( X ) to a t test statistic Hypothesis Tests for µ σ Known σ Unknown The test statistic is: t n-1 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. X−μ = S n Chap 8-40
  41. 41. Example: Two-Tail Test (σ Unknown) The average cost of a hotel room in New York is said to be $168 per night. A random sample of 25 hotels resulted in X = $172.50 and S = $15.40. Test at the α = 0.05 level. Statistics for Managers Usingnormal) (Assume the population distribution is Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. H0: μ = 168 H1: μ ≠ 168 Chap 8-41
  42. 42. Example Solution: Two-Tail Test     H0: μ = 168 H1: μ ≠ 1680.05 α = α/2=.025 Reject H0 -t n-1,α/2 -2.0639 n = 25 σ is unknown, so use a t statistic Critical Value: t n−1 = α/2=.025 Do not reject H0 0 1.46 Reject H0 t n-1,α/2 2.0639 X −μ 172.50 −168 = = 1.46 S 15.40 n 25 Do not t24 = ± for Managers Using reject H0: not sufficient evidence that Statistics 2.0639 Microsoft Excel, 4e © 2004 true mean cost is different than $168 Chap 8-42 Prentice-Hall, Inc.
  43. 43. Connection to Confidence Intervals  For X = 172.5, S = 15.40 and n = 25, the 95% confidence interval is: 172.5 - (2.0639) 15.4/ 25 to 172.5 + (2.0639) 15.4/ 25 166.14 ≤ μ ≤ 178.86 Since this interval contains the Hypothesized mean (168), we do not reject the null hypothesis at α = .05 Statistics for Managers Using  Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
  44. 44. Hypothesis Tests for Proportions  Involves categorical variables  Two possible outcomes    “Success” (possesses a certain characteristic) “Failure” (does not possesses that characteristic) Fraction or proportion of the population in the “success” category is denoted by p Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-44
  45. 45. Proportions (continued)  Sample proportion in the success category is denoted by ps  X number of successes in sample ps = = n sample size When both np and n(1-p) are at least 5, ps can be approximated by a normal distribution with mean and standard deviation  μps = p Using σ ps = p(1 − p) Statistics for Managers n Microsoft Excel, 4e © 2004  Prentice-Hall, Inc. Chap 8-45
  46. 46. Hypothesis Tests for Proportions  The sampling distribution of ps is approximately normal, so the test statistic is a Z value: Hypothesis Tests for p np ≥ 5 and n(1-p) ≥ 5 ps − p Z= p(1 − p) n Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. np < 5 or n(1-p) < 5 Not discussed in this chapter Chap 8-46
  47. 47. Z Test for Proportion in Terms of Number of Successes  An equivalent form to the last slide, but in terms of the number of successes, X: X − np Z= np(1 − p) Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Hypothesis Tests for X X≥5 and n-X ≥ 5 X<5 or n-X < 5 Not discussed in this chapter Chap 8-47
  48. 48. Example: Z Test for Proportion A marketing company claims that it receives 8% responses from its mailing. To test this claim, a random sample of 500 were surveyed with 25 responses. Test at the α = .05 significance level. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Check: n p = (500)(.08) = 40 n(1-p) = (500)(.92) = 460 Chap 8-48 
  49. 49. Z Test for Proportion: Solution Test Statistic: H0: p = .08 H1: p ≠ . α = .05 08 Z= n = 500, ps = .05 Critical Values: ± 1.96 Reject .025 Reject .025 z -1.96 0 1.96Using for Managers Statistics -2.47 Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. ps − p = p(1 − p) n .05 − .08 = −2.47 .08(1 − .08) 500 Decision: Reject H0 at α = .05 Conclusion: There is sufficient evidence to reject the company’s claim of 8% response rate. Chap 8-49
  50. 50. p-Value Solution (continued) Calculate the p-value and compare to α (For a two sided test the p-value is always two sided) Reject H0 Do not reject H0 α/2 = .025 Reject H0 p-value = .0136: α/2 = .025 P(Z ≤ −2.47) + P(Z ≥ 2.47) .0068 .0068 -1.96 Z = -2.47 0 = 2(.0068) = 0.0136 1.96 Z = 2.47 Statistics for Managers Using Reject H0 since p-value = .0136 < α = .05 Microsoft Excel, 4e © 2004 Chap 8-50 Prentice-Hall, Inc.
  51. 51. Using PHStat Options Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-51
  52. 52. Sample PHStat Output Input Output Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-52
  53. 53. Potential Pitfalls and Ethical Considerations  Use randomly collected data to reduce selection biases  Do not use human subjects without informed consent  Choose the level of significance, α, before data collection  Do not employ “data snooping” to choose between onetail and two-tail test, or to determine the level of significance  Do not practice “data cleansing” to hide observations that do not support a stated hypothesis Report all pertinent findings Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.  Chap 8-53
  54. 54. Chapter Summary  Addressed hypothesis testing methodology  Performed Z Test for the mean (σ known)  Discussed critical value and p–value approaches to hypothesis testing  Performed one-tail and two-tail tests Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-54
  55. 55. Chapter Summary (continued)  Performed t test for the mean (σ unknown)  Performed Z test for the proportion  Discussed pitfalls and ethical issues Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-55

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