Lecture Business statistics in practice (7/e): Chapter 18 - Bowerman, O'Connell, Murphree

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Chapter 18 - Nonparametric methods. After mastering the material in this chapter, you will be able to: Use the sign test to test a hypothesis about a population median, compare the locations of two distributions using a rank sum test for independent samples, compare the locations of two distributions using a signed ranks test for paired samples,...

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Nội dung Text: Lecture Business statistics in practice (7/e): Chapter 18 - Bowerman, O'Connell, Murphree

Chapter 18 Nonparametric Methods McGrawHill/Irwin Copyright © 2014 by The McGrawHill Companies, Inc. All rights reserved.
Nonparametric Methods 18.1 The Sign Test: A Hypothesis Test about the Median 18.2 The Wilcoxon Rank Sum Test 18.3 The Wilcoxon Signed Ranks Test 18.4 Comparing Several Populations Using the KruskalWallis H Test 18.5 Spearman’s Rank Correlation Coefficient 182
LO18-1: Use the sign test to test a hypothesis about a population median. 18.1 Sign Test: A Hypothesis Test about the Median Define… ◦S = the number of sample measurements (less/greater) than M0 ◦x to be a binomial random variable with p = 0.5 We can reject H0: Md = M0 at the level of significance (probability of Type I error equal to ) by using the appropriate pvalue 183
LO18-1 Sign Test: A Hypothesis Test about the Median Continued Alternative Test Statistic pValue Ha: Md > Mo S=number of The probability that x measurements is greater than or equal greater than Mo to S Ha: Md
LO18-2: Compare the locations of two distributions using a 18.2 The Wilcoxon Rank Sum Test rank sum test for independent samples.  Given two independent samples of sizes n1 and n2 from populations 1 and 2 with distributions D1 and D2  Rank the (n1+ n2) observations from smallest to largest (average ranks for ties) ◦ T1 = sum of ranks, sample 1 ◦ T2 = sum of ranks, sample 2 ◦ T = T1 if n1 n2 and T = T2 if n1> n2  We can reject H0: D1 and D2 are identical probability distributions at the level of significance if and only if the test statistic T satisfies the appropriate rejection condition 185
LO18-2 The Wilcoxon Rank Sum Test Continued Alternative Reject H0 if Ha: D1 is shifted right of D2 T ≥ Tu if n1 ≤ n2 T ≤ Tu if n1 > n2 Ha: D1 is shifted left of D2 T ≤ TL if n1 ≤ n2 T ≥ TL if n1 > n2 Ha: D1 is shifted right or left of D2 T ≤ Tu or T ≥ Tu 186
LO18-3: Compare the locations of two distributions using a signed ranks test for paired samples. 18.3 The Wilcoxon Signed Rank Test  Given two matched pairs of n observations, selected at random from populations 1 and 2 with distributions D1 and D2 compute the n differences (D1 – D2)  Rank the absolute value of the differences from smallest to largest ◦Drop zero differences from sample ◦Assign average ranks for ties ◦T = sum of ranks, negative differences ◦T+ = sum of ranks, positive differences  We can reject H0: D1 and D2 are identical probability distributions at the level of significance if and only if the appropriate test statistic satisfies the corresponding rejection point condition 187
LO18-3 The Wilcoxon Signed Rank Test Continued Alternative Test Statistic Reject H0 if Ha: D1 is shifted right of T T ≤ T0 D2 Ha: D1 is shifted left of T+ T+ ≤ T0 D2 Ha: D1 is shifted right or T=smaller of T T ≤ T0 left of D2 or T+ 188
LO18-4: Compare the locations of three or more distributions using a Kruskal–Wallis test for independent samples. 18.4 Comparing Several Populations Using The KruskalWallis H Test Given p independent samples (n1, …, np 5) from p populations Rank the (n1+ … + np) observations from smallest to largest (average ranks for ties) Let T1 equal sum of ranks, sample 1, continuing until Tp equals sum of ranks, sample p 189
LO18-4 The KruskalWallis H Test Continued To test… H0: The p populations are identical Ha: At least two of the populations differ in location Test statistic p 2 12 Ti H= 3(n 1 ) n(n 1 ) i 1 ni  Reject H 0 if H > 2 or if pvalue
LO18-5: Measure and test the association between two variables by using Spearman’s rank correlation 18.5 Spearman’s Rank Correlation coefficient. Coefficient  Given n pairs of measurements on two variables, rank each separately, assigning average ranks for ties  The Spearman rank correlation coefficient, rs is given by rs = corr[xrank, yrank] is the standard Pearson correlation coefficient  If there are no ties, the Spearman correlation coefficient can be calculated as 6 di2 rs=1 n(n 2 1 )  Where di is the difference between the xrank and the yrank for the ith observation 1811
LO18-5 Spearman’s Rank Correlation Test Critical Value Rule Alternative Hypothesis Reject H0 if Ha: ps > 0 rs > r Ha: Ha: ps