Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our User Agreement and Privacy Policy.

Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our Privacy Policy and User Agreement for details.

Like this presentation? Why not share!

- Chap06 normal distributions & conti... by Uni Azza Aunillah 2925 views
- Chap09 2 sample test by Uni Azza Aunillah 2109 views
- Chap04 basic probability by Uni Azza Aunillah 1396 views
- Chap07 interval estimation by Uni Azza Aunillah 1987 views
- Chap16 decision making by Uni Azza Aunillah 967 views
- Chap17 statistical applications on ... by Uni Azza Aunillah 1085 views

1,697 views

1,463 views

1,463 views

Published on

Modul Statistik Bisnis II

No Downloads

Total views

1,697

On SlideShare

0

From Embeds

0

Number of Embeds

4

Shares

0

Downloads

148

Comments

0

Likes

2

No embeds

No notes for slide

- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Using PHStat Options Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-51
- 52. Sample PHStat Output Input Output Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-52
- 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. 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. 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

No public clipboards found for this slide

Be the first to comment