Statistics for Managers Using Microsoft Excel Chapter 9 Two Sample Tests With Numerical Data

Transcription

1 Statistics for Managers Using Microsoft Excel Chapter 9 Two Sample Tests With Numerical Data 999 Prentice-Hall, Inc. Chap. 9 -

2 Chapter Topics Comparing Two Independent Samples: Z Test for the Difference in Two Means t Test for Difference in Two Means F Test for Difference in two Variances Comparing Two Related Samples: t Tests for the Mean Difference Wilcoxon Rank-Sum Test: Difference in Two Medians 999 Prentice-Hall, Inc. Chap. 9 -

3 Independent Samples Different Data Sources: Unrelated Independent Sample selected from one population has no effect or bearing on the sample selected from the other population. Use Difference Between the Sample Means Use Pooled Variance t Test 999 Prentice-Hall, Inc. Chap. 9-3

4 Z Test for Differences in Two Means (Variances Known Assumptions: Samples are Randomly and Independently drawn Data Collected are Numerical Population Variances Are Known Samples drawn are Large ( X Z Test Statistic: X n ( n 999 Prentice-Hall, Inc. Chap. 9-4

5 t Test for Differences in Two Means (Variances Unknown Assumptions: Both Populations Are Normally Distributed Or, If Not Normal, Can Be Approximated by Normal Distribution Samples are Randomly and Independently drawn Population Variances Are Unknown But Assumed Equal 999 Prentice-Hall, Inc. Chap. 9-5

6 Developing the Pooled-Variance t Test (Part Setting Up the Hypothesis: H 0 : = H : OR H 0 : - = 0 H : - 0 Two Tail H 0 : H : > OR H 0 : - 0 H : - > 0 Right Tail H 0 : H : < OR H 0 : - H : - < 0 Left Tail 999 Prentice-Hall, Inc. Chap. 9-6

7 Developing the Pooled-Variance t Test (Part Calculate the Pooled Sample Variances as an Estimate of the Common Populations Variance: ( n S ( n S p ( n ( n S S p = Pooled-Variance n = Size of Sample S S = Variance of Sample = Variance of sample n = Size of Sample 999 Prentice-Hall, Inc. Chap. 9-7

8 Developing the Pooled-Variance t Test (Part 3 Compute the Test Statistic: t ( _ X X ( S p _ n n Hypothesized Difference df n n S ( ( n S n S P ( n ( n 999 Prentice-Hall, Inc. Chap. 9-8

9 Pooled-Variance t Test: Example You re a financial analyst for Charles Schwab. Is there a difference in dividend yield between stocks listed on the NYSE & NASDAQ? You collect the following data: NYSE NASDAQ Number 5 Mean Std Dev.30.6 Assuming equal variances, is there a difference in average yield (a = 0.05? T/Maker Co. 999 Prentice-Hall, Inc. Chap. 9-9

10 Calculating the Test Statistic: t ( X X S P n ( n ( S n S n S P n n ( ( ( ( ( ( ( ( Prentice-Hall, Inc. Chap. 9-0

11 Solution H 0 : - = 0 ( = H : - 0 ( a = 0.05 df = = 44 Critical Value(s: Reject H 0 Reject H t t Test Statistic: Decision: Reject at a = 0.05 Conclusion: There is evidence of a difference in means Prentice-Hall, Inc. Chap. 9 -

12 The F test Statistic: F Test for Differences in Two Variances F = S S S = Variance of Sample n - = degrees of freedom S = Variance of Sample n - = degrees of freedom 0 F 999 Prentice-Hall, Inc. Chap. 9 -

13 F Test for the Difference in Two Population Variances Tests for Differences in Independent Population Variances Parametric Test Procedure Assumptions Both Populations Are Normally Distributed Test Is Not Robust to Violations 999 Prentice-Hall, Inc. Chap. 9-3

14 F Test for the Difference in Two Population Variances Hypotheses Reject H 0 H 0 : = H : Test Statistic F = S /S Two Sets of Degrees of Freedom df = n - ; df = n - a/ Do Not Reject 0 F L F U Reject H 0 a/ F Critical Values: F L( and F U( n -, n - n -, n - F L = /F U * (*degrees of freedom switched 999 Prentice-Hall, Inc. Chap. 9-4

15 F Test: An Example Assume you are a financial analyst for Charles Schwab. You want to compare dividend yields between stocks listed on the NYSE & NASDAQ. You collect the following data: NYSE NASDAQ Number 5 Mean Std Dev.30.6 Is there a difference in the variances between the NYSE & NASDAQ at the 0.05 level? T/Maker Co. 999 Prentice-Hall, Inc. Chap. 9-5

16 F Test: Example Solution H 0 : = H : a.05 df 0 df 4 Critical Value(s: Reject.05 Reject F Test Statistic: S. 30 F S Decision: Do not reject at a = 0.05 Conclusion: There is no evidence of a difference in variances. 999 Prentice-Hall, Inc. Chap. 9-6

17 F Test: One Tail Reject a.05 H 0 : H : < F L = FU or a =.05 (n -, n - H 0 : H : > Degrees of freedom switched Reject a.05 0 F 0 F F L F U 999 Prentice-Hall, Inc. Chap. 9-7

18 Comparing Two Related Samples: t Test for Mean Difference Tests Means of Related Populations Paired or Matched Repeated Measures (Before/After Use Difference Between Pairs D n = X n - X n Eliminates Variation Among Subjects Assumptions Both Population Are Normally Distributed Or, if Not Normal, use large samples 999 Prentice-Hall, Inc. Chap. 9-8

19 Paired Sample t Test: Example Assume you work in the finance department. Is the new financial package faster (0.05 level? You collect the following data entry times: User Current Leader ( New Software ( Difference D i C.B Seconds 9.88 Seconds.0 T.F M.H R.K M.O D.S S.S C.T K.T S.Z D = S D i =.084 n S D ( Di n D 999 Prentice-Hall, Inc. Chap. 9-9

20 Paired Sample t Test: Example Solution Is the new financial package faster (0.05 level? t D S D H 0 : D H : D > a.5 / D n D =.084 Critical Value=.833 df = n - = 9 Test Statistic / Reject a.5 Decision: Reject H 0 t Stat. in the rejection zone. Conclusion: The new software package is faster. 999 Prentice-Hall, Inc. Chap. 9-0

21 Wilcoxon Rank Sum Test for Differences in Medians Tests Two Independent Population Medians Populations Need Not be Normal Distribution Free Procedure Only Rank of Data Obtained Can Use Normal Approximation If n i > Prentice-Hall, Inc. Chap. 9 -

22 Wilcoxon Rank Sum Test: Procedure Assign Ranks, R i, to the n + n Sample Observations If Unequal Sample Sizes, Let n Refer to Smaller- Sized Sample Smallest Value = Average Ties Sum the Ranks, T i, for Each Sample Obtain Test Statistic, T (Smallest Sample 999 Prentice-Hall, Inc. Chap. 9 -

23 Wilcoxon Rank Sum Test: Setting of Hypothesis Two -Tail Test Left-Tail Test Right -Tail Test H 0 : M = M H : M M H 0 : M M H : M < M H 0 : M M H : M > M M = median of population M = median of population 999 Prentice-Hall, Inc. Chap. 9-3

24 Wilcoxon Rank Sum Test: Assume you re a production planner. You want to see if the median operating rates for the factories is the same. For factory, the rates (% of capacity are 7, 8, 77, 9, 88. For factory, the rates are 85, 8, 94 & 97. Do the factories have the same median rates at the 0.0 level. Example 999 Prentice-Hall, Inc. Chap. 9-4

25 Wilcoxon Rank Sum Test: Computation Table Factory Factory Rate Rank Rate Rank Tie Tie Rank Sum Prentice-Hall, Inc. Chap. 9-5

26 Wilcoxon Rank Sum Test: Solution H 0 : M = M H : M M a =.0 n = 4 n = 5 Critical Value(s: Reject Do Not Reject Reject 8 S Ranks Test Statistic: T = = 5.5 (Smallest Sample Decision: Do not reject at a = 0.0 Conclusion: There is no evidence medians are not equal. 999 Prentice-Hall, Inc. Chap. 9-6

27 Wilcoxon Rank Sum Test: Large Sample For Large Sample Size: the test statistic T is approximately normal with mean standard deviation. Computing the Z value: T T Z T T and T Where T n ( n n n, n = n + n and T nn (n 999 Prentice-Hall, Inc. Chap. 9-7

28 Chapter Summary Compared Two Independent Samples: Performed Z Test for the Differences in Two Means Performed t Test for Differences in Two Means Addressed F Test for Difference in two Variances Compared Two Related Samples: Performed t Tests for the Mean Difference Addressed Wilcoxon Rank Sum Test: Performed Tests on Differences in Two Medians 999 Prentice-Hall, Inc. Chap. 9-8