Slide 1 Shakeel Nouman M.Phil Statistics Nonparametric Methods and Chi-Square Tests (1) Nonparametric Methods and Chi-Square Tests (1) By Shakeel Nouman.

Shakeel Nouman M.Phil Statistics Nonparametric Methods and Chi-Square Tests (1) Nonparametric Methods and Chi-Square Tests (1) By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Using Statistics The Sign Test The Runs Test - A Test for Randomness The Mann-Whitney U Test The Wilcoxon Signed-Rank Test The Kruskal-Wallis Test - A Nonparametric Alternative to One-Way ANOVA Nonparametric Methods and Chi-Square Tests (1)  Nonparametric Methods and Chi-Square Tests (1) By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

The Friedman Test for a Randomized Block Design The Spearman Rank Correlation Coefficient A Chi-Square Test for Goodness of Fit Contingency Table Analysis - A Chi- Square Test for Independence A Chi-Square Test for Equality of Proportions Summary and Review of Terms Nonparametric Methods and Chi-Square Tests (2)  Nonparametric Methods and Chi-Square Tests (1) By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Parametric Methods Inferences based on assumptions about the nature of the population distribution » Usually: population is normal Types of tests » z-test or t-test » Comparing two population means or proportions » Testing value of population mean or proportion » ANOVA » Testing equality of several population means Parametric Methods Inferences based on assumptions about the nature of the population distribution » Usually: population is normal Types of tests » z-test or t-test » Comparing two population means or proportions » Testing value of population mean or proportion » ANOVA » Testing equality of several population means 14-1 Using Statistics (Parametric Tests) Nonparametric Methods and Chi-Square Tests (1) By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Nonparametric Tests Distribution-free methods making no assumptions about the population distribution Types of tests » Sign tests » Sign Test: Comparing paired observations » McNemar Test: Comparing qualitative variables » Cox and Stuart Test: Detecting trend » Runs tests » Runs Test: Detecting randomness » Wald-Wolfowitz Test: Comparing two distributions Nonparametric Tests Distribution-free methods making no assumptions about the population distribution Types of tests » Sign tests » Sign Test: Comparing paired observations » McNemar Test: Comparing qualitative variables » Cox and Stuart Test: Detecting trend » Runs tests » Runs Test: Detecting randomness » Wald-Wolfowitz Test: Comparing two distributions Nonparametric Tests Nonparametric Methods and Chi-Square Tests (1) By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Nonparametric Tests Ranks tests Mann-Whitney U Test: Comparing two populations Wilcoxon Signed-Rank Test: Paired comparisons Comparing several populations: ANOVA with ranks – Kruskal-Wallis Test – Friedman Test: Repeated measures Spearman Rank Correlation Coefficient Chi-Square Tests Goodness of Fit Testing for independence: Contingency Table Analysis Equality of Proportions Nonparametric Tests Ranks tests Mann-Whitney U Test: Comparing two populations Wilcoxon Signed-Rank Test: Paired comparisons Comparing several populations: ANOVA with ranks – Kruskal-Wallis Test – Friedman Test: Repeated measures Spearman Rank Correlation Coefficient Chi-Square Tests Goodness of Fit Testing for independence: Contingency Table Analysis Equality of Proportions Nonparametric Tests (Continued) Nonparametric Methods and Chi-Square Tests (1) By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Deal with enumerative (frequency counts) data. Do not deal with specific population parameters, such as the mean or standard deviation. Do not require assumptions about specific population distributions (in particular, the normality assumption). Deal with enumerative (frequency counts) data. Do not deal with specific population parameters, such as the mean or standard deviation. Do not require assumptions about specific population distributions (in particular, the normality assumption). Nonparametric Tests (Continued) Nonparametric Methods and Chi-Square Tests (1) By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Comparing paired observations Paired observations: X and Y p = P(X>Y) » Two-tailed test H 0 : p = 0.50 H 1 : p  0.50 » Right-tailed testH 0 : p  0.50 H 1 : p  0.50 » Left-tailed testH 0 : p  0.50 H 1 : p  0.50 » Test statistic: T = Number of + signs Comparing paired observations Paired observations: X and Y p = P(X>Y) » Two-tailed test H 0 : p = 0.50 H 1 : p  0.50 » Right-tailed testH 0 : p  0.50 H 1 : p  0.50 » Left-tailed testH 0 : p  0.50 H 1 : p  0.50 » Test statistic: T = Number of + signs 14-2 Sign Test Nonparametric Methods and Chi-Square Tests (1) By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Small Sample: Binomial Test For a two-tailed test, find a critical point corresponding as closely as possible to  /2 (C 1 ) and define C 2 as n-C 1. Reject null hypothesis if T  C 1 or T  C 2. For a right-tailed test, reject H 0 if T  C, where C is the value of the binomial distribution with parameters n and p = 0.50 such that the sum of the probabilities of all values less than or equal to C is as close as possible to the chosen level of significance, . For a left-tailed test, reject H 0 if T  C, where C is defined as above. Small Sample: Binomial Test For a two-tailed test, find a critical point corresponding as closely as possible to  /2 (C 1 ) and define C 2 as n-C 1. Reject null hypothesis if T  C 1 or T  C 2. For a right-tailed test, reject H 0 if T  C, where C is the value of the binomial distribution with parameters n and p = 0.50 such that the sum of the probabilities of all values less than or equal to C is as close as possible to the chosen level of significance, . For a left-tailed test, reject H 0 if T  C, where C is defined as above. Sign Test Decision Rule Nonparametric Methods and Chi-Square Tests (1) By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Cumulative Binomial Probabilities (n=15, p=0.5) x F(x) 00.00003 10.00049 20.00369 30.01758 40.05923 50.15088 60.30362 70.50000 80.69638 90.84912 100.94077 110.98242 120.99631 130.99951 140.99997 151.00000 CEO Before After Sign 1 34 1+ 2 55 0 3 2 3 1+ 4 24 1+ 5 44 0 6 23 1+ 7 12 1+ 8 54 -1- 9 45 1+ 10 54 -1- 11 34 1+ 12 25 1+ 13 25 1+ 14 23 1+ 15 1 2 1+ 16 32 -1- 17 45 1+ CEO Before After Sign 1 34 1+ 2 55 0 3 2 3 1+ 4 24 1+ 5 44 0 6 23 1+ 7 12 1+ 8 54 -1- 9 45 1+ 10 54 -1- 11 34 1+ 12 25 1+ 13 25 1+ 14 23 1+ 15 1 2 1+ 16 32 -1- 17 45 1+ n = 15 T = 12  0.025 C1=3 C2 = 15-3 = 12 H 0 rejected, since T  C2 C1 Example 14-1 Nonparametric Methods and Chi-Square Tests (1) By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Example 14-1- Using the Template H 0 : p = 0.5 H 1 : p  0.5 Test Statistic: T = 12 p-value = 0.0352. For a = 0.05, the null hypothesis is rejected since 0.0352 < 0.05. Thus one can conclude that there is a change in attitude toward a CEO following the award of an MBA degree. Nonparametric Methods and Chi-Square Tests (1) By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

A run is a sequence of like elements that are preceded and followed by different elements or no element at all. Case 1: S|E|S|E|S|E|S|E|S|E|S|E|S|E|S|E|S|E|S|E : R = 20 Apparently nonrandom Case 2: SSSSSSSSSS|EEEEEEEEEE : R = 2 Apparently nonrandom Case 3: S|EE|SS|EEE|S|E|SS|E|S|EE|SSS|E : R = 12 Perhaps random Case 1: S|E|S|E|S|E|S|E|S|E|S|E|S|E|S|E|S|E|S|E : R = 20 Apparently nonrandom Case 2: SSSSSSSSSS|EEEEEEEEEE : R = 2 Apparently nonrandom Case 3: S|EE|SS|EEE|S|E|SS|E|S|EE|SSS|E : R = 12 Perhaps random A two-tailed hypothesis test for randomness: H 0 : Observations are generated randomly H 1 : Observations are not generated randomly Test Statistic: R=Number of Runs Reject H 0 at level  if R  C1 or R  C2, as given in Table 8, with total tail probability P(R  C 1 ) + P(R  C 2 ) =   A two-tailed hypothesis test for randomness: H 0 : Observations are generated randomly H 1 : Observations are not generated randomly Test Statistic: R=Number of Runs Reject H 0 at level  if R  C1 or R  C2, as given in Table 8, with total tail probability P(R  C 1 ) + P(R  C 2 ) =   14-3 The Runs Test - A Test for Randomness Nonparametric Methods and Chi-Square Tests (1) By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Table 8: Number of Runs (r) (n 1,n 2 )11121314151617181920. (10,10)0.5860.7580.8720.9490.9810.9960.9991.0001.0001.000 Table 8: Number of Runs (r) (n 1,n 2 )11121314151617181920. (10,10)0.5860.7580.8720.9490.9810.9960.9991.0001.0001.000 Case 1: n 1 = 10 n 2 = 10 R= 20 p-value  0 Case 2: n 1 = 10 n 2 = 10 R = 2 p-value  0 Case 3: n 1 = 10 n 2 = 10 R= 12 p-value    PR          F(11)] = (2)(1-0.586) = (2)(0.414) = 0.828 H 0 not rejected Case 1: n 1 = 10 n 2 = 10 R= 20 p-value  0 Case 2: n 1 = 10 n 2 = 10 R = 2 p-value  0 Case 3: n 1 = 10 n 2 = 10 R= 12 p-value    PR          F(11)] = (2)(1-0.586) = (2)(0.414) = 0.828 H 0 not rejected Runs Test: Examples Nonparametric Methods and Chi-Square Tests (1) By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Large-Sample Runs Test: Using the Normal Approximation Nonparametric Methods and Chi-Square Tests (1) By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Example 14-2: n 1 = 27 n 2 = 26 R = 15 H 0 should be rejected at any common level of significance. Large-Sample Runs Test: Example 14-2 Nonparametric Methods and Chi-Square Tests (1) By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Large-Sample Runs Test: Example 14-2 – Using the Template Note:The computed p-value using the template is 0.0005 as compared to the manually computed value of 0.0006. The value of 0.0005 is more accurate. Note: The computed p-value using the template is 0.0005 as compared to the manually computed value of 0.0006. The value of 0.0005 is more accurate. Reject the null hypothesis that the residuals are random. Nonparametric Methods and Chi-Square Tests (1) By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

The null and alternative hypotheses for the Wald-Wolfowitz test: H 0 : The two populations have the same distribution H 1 : The two populations have different distributions The test statistic: R = Number of Runs in the sequence of samples, when the data from both samples have been sorted The null and alternative hypotheses for the Wald-Wolfowitz test: H 0 : The two populations have the same distribution H 1 : The two populations have different distributions The test statistic: R = Number of Runs in the sequence of samples, when the data from both samples have been sorted Salesperson A:35443950 48296075 4966 Salesperson B:17231324 33211816 32 Salesperson A:35443950 48296075 4966 Salesperson B:17231324 33211816 32 Using the Runs Test to Compare Two Population Distributions (Means): the Wald-Wolfowitz Test Example 14-3: Nonparametric Methods and Chi-Square Tests (1) By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Table Number of Runs (r) (n 1,n 2 )2345. (9,10) 0.0000.0000.0020.004... Table Number of Runs (r) (n 1,n 2 )2345. (9,10) 0.0000.0000.0020.004... Sales SalesSalesPerson SalesPerson(Sorted)(Sorted)Runs 35A13 B 44A16 B 39A17 B 48A21 B 60A24 B 1 75A29 A 2 49A32 B 66A33 B 3 17B35 A 23B39 A 13B44 A 24B48 A 33B49 A 21B50 A 18B60 A 16B66 A 32B75 A 4 Sales SalesSalesPerson SalesPerson(Sorted)(Sorted)Runs 35A13 B 44A16 B 39A17 B 48A21 B 60A24 B 1 75A29 A 2 49A32 B 66A33 B 3 17B35 A 23B39 A 13B44 A 24B48 A 33B49 A 21B50 A 18B60 A 16B66 A 32B75 A 4 n 1 = 10 n 2 = 9 R= 4 p-value  P   H 0 may be rejected n 1 = 10 n 2 = 9 R= 4 p-value  P   H 0 may be rejected The Wald-Wolfowitz Test: Example 14-3 Nonparametric Methods and Chi-Square Tests (1) By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Ranks tests Mann-Whitney U Test: Comparing two populations Wilcoxon Signed-Rank Test: Paired comparisons Comparing several populations: ANOVA with ranks Kruskal-Wallis Test Friedman Test: Repeated measures Ranks tests Mann-Whitney U Test: Comparing two populations Wilcoxon Signed-Rank Test: Paired comparisons Comparing several populations: ANOVA with ranks Kruskal-Wallis Test Friedman Test: Repeated measures Ranks Tests Nonparametric Methods and Chi-Square Tests (1) By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

The null and alternative hypotheses: H 0 : The distributions of two populations are identical H 1 : The two population distributions are not identical The Mann-Whitney U statistic: where n 1 is the sample size from population 1 and n 2 is the sample size from population 2. The null and alternative hypotheses: H 0 : The distributions of two populations are identical H 1 : The two population distributions are not identical The Mann-Whitney U statistic: where n 1 is the sample size from population 1 and n 2 is the sample size from population 2. 14-4 The Mann-Whitney U Test (Comparing Two Populations) Nonparametric Methods and Chi-Square Tests (1) By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Cumulative Distribution Function of the Mann-Whitney U Statistic n 2 =6 n 1 =6 u. 40.0130 50.0206 60.0325. Rank ModelTimeRankSum A35 5 A38 8 A4010 A4212 A4111 A36 652 B29 2 B27 1 B30 3 B33 4 B39 9 B37 726 Rank ModelTimeRankSum A35 5 A38 8 A4010 A4212 A4111 A36 652 B29 2 B27 1 B30 3 B33 4 B39 9 B37 726 P( u  5) The Mann-Whitney U Test: Example 14-4 Nonparametric Methods and Chi-Square Tests (1) By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Example 14-5: Large-Sample Mann-Whitney U Test Score Rank Score Program Rank Sum 85120.020.0 87121.041.0 92127.068.0 98130.098.0 90126.0124.0 88123.0147.0 75117.0164.0 72113.5177.5 6016.5184.0 93128.0212.0 88123.0235.0 89125.0260.0 96129.0289.0 73115.0304.0 6218.5312.5 Score Rank Score Program Rank Sum 85120.020.0 87121.041.0 92127.068.0 98130.098.0 90126.0124.0 88123.0147.0 75117.0164.0 72113.5177.5 6016.5184.0 93128.0212.0 88123.0235.0 89125.0260.0 96129.0289.0 73115.0304.0 6218.5312.5 Score Rank Score Program Rank Sum 65210.0 10.0 5724.0 14.0 74216.0 30.0 4322.0 32.0 3921.0 33.0 88223.0 56.0 6228.5 64.5 69211.0 75.5 70212.0 87.5 72213.5101.0 5925.0106.0 6026.5112.5 80218.0130.5 83219.0149.5 5023.0152.5 Score Rank Score Program Rank Sum 65210.0 10.0 5724.0 14.0 74216.0 30.0 4322.0 32.0 3921.0 33.0 88223.0 56.0 6228.5 64.5 69211.0 75.5 70212.0 87.5 72213.5101.0 5925.0106.0 6026.5112.5 80218.0130.5 83219.0149.5 5023.0152.5 Since the test statistic is z = -3.32, the p-value  0.0005, and H 0 is rejected. Since the test statistic is z = -3.32, the p-value  0.0005, and H 0 is rejected. Nonparametric Methods and Chi-Square Tests (1) By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Example 14-5: Large-Sample Mann-Whitney U Test – Using the Template Since the test statistic is z = - 3.32, the p-value  0.0005, and H 0 is rejected. That is, the LC (Learning Curve) program is more effective. Nonparametric Methods and Chi-Square Tests (1) By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

The null and alternative hypotheses: H 0 : The median difference between populations are 1 and 2 is zero H 1 : The median difference between populations are 1 and 2 is not zero Find the difference between the ranks for each pair, D = x 1 -x 2, and then rank the absolute values of the differences. The Wilcoxon T statistic is the smaller of the sums of the positive ranks and the sum of the negative ranks: For small samples, a left-tailed test is used, using the values in Appendix C, Table 10. The large-sample test statistic: The null and alternative hypotheses: H 0 : The median difference between populations are 1 and 2 is zero H 1 : The median difference between populations are 1 and 2 is not zero Find the difference between the ranks for each pair, D = x 1 -x 2, and then rank the absolute values of the differences. The Wilcoxon T statistic is the smaller of the sums of the positive ranks and the sum of the negative ranks: For small samples, a left-tailed test is used, using the values in Appendix C, Table 10. The large-sample test statistic: 14-5 The Wilcoxon Signed- Ranks Test (Paired Ranks) Nonparametric Methods and Chi-Square Tests (1) By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Sold Sold Rank Rank Rank (1) (2) D=x 1 -x 2 ABS(D) ABS(D)(D>0) (D<0) 564016169.09.00 4870-222212.00.012 10060404015.015.00 857015158.08.00 22814147.07.00 4440442.02.00 3545-10106.00.06 287212111.011.00 5260-885.00.05 7770773.53.50 8990-111.00.01 10100**** 6585-202010.00.010 9061292913.013.00 7040303014.014.00 3326773.53.50 Sum:8634 Sold Sold Rank Rank Rank (1) (2) D=x 1 -x 2 ABS(D) ABS(D)(D>0) (D<0) 564016169.09.00 4870-222212.00.012 10060404015.015.00 857015158.08.00 22814147.07.00 4440442.02.00 3545-10106.00.06 287212111.011.00 5260-885.00.05 7770773.53.50 8990-111.00.01 10100**** 6585-202010.00.010 9061292913.013.00 7040303014.014.00 3326773.53.50 Sum:8634 T=34 n=15 P=0.0530 P=0.02525 P=0.0120 P=0.00516 H 0 is not rejected (Note the arithmetic error in the text for store 13) T=34 n=15 P=0.0530 P=0.02525 P=0.0120 P=0.00516 H 0 is not rejected (Note the arithmetic error in the text for store 13) Example 14-6 Nonparametric Methods and Chi-Square Tests (1) By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Hourly Rank Rank Rank Messages Md 0 D=x 1 -x 2 ABS(D) ABS(D) (D>0) (D<0) 151149221.01.00.0 144149-552.00.02.0 123149-262613.00.013.0 178149292915.015.00.0 105149-444423.00.023.0 112149-373720.00.020.0 140149-994.00.04.0 167149181810.010.00.0 177149282814.014.00.0 185149363619.019.00.0 129149-202011.00.011.0 16014911116.06.00.0 110149-393921.00.021.0 170149212112.012.00.0 198149494925.025.00.0 16514916168.08.00.0 109149-404022.00.022.0 118149-313116.50.016.5 155149663.03.00.0 102149-474724.00.024.0 16414915157.07.00.0 180149313116.516.50.0 139149-10105.00.05.0 16614917179.09.00.0 82149333318.018.00.0 Sum:163.5161.5 Hourly Rank Rank Rank Messages Md 0 D=x 1 -x 2 ABS(D) ABS(D) (D>0) (D<0) 151149221.01.00.0 144149-552.00.02.0 123149-262613.00.013.0 178149292915.015.00.0 105149-444423.00.023.0 112149-373720.00.020.0 140149-994.00.04.0 167149181810.010.00.0 177149282814.014.00.0 185149363619.019.00.0 129149-202011.00.011.0 16014911116.06.00.0 110149-393921.00.021.0 170149212112.012.00.0 198149494925.025.00.0 16514916168.08.00.0 109149-404022.00.022.0 118149-313116.50.016.5 155149663.03.00.0 102149-474724.00.024.0 16414915157.07.00.0 180149313116.516.50.0 139149-10105.00.05.0 16614917179.09.00.0 82149333318.018.00.0 Sum:163.5161.5 Example 14-7 Nonparametric Methods and Chi-Square Tests (1) By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Example 14-7 using the Template Note 1: Note 1: You should enter the claimed value of the mean (median) in every used row of the second column of data. In this case it is 149. Note 2: Note 2: In order for the large sample approximations to be computed you will need to change n > 25 to n >= 25 in cells M13 and M14. Nonparametric Methods and Chi-Square Tests (1) By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

The Kruskal-Wallis hypothesis test: H 0 : All k populations have the same distribution H 1 : Not all k populations have the same distribution The Kruskal-Wallis test statistic: If each n j > 5, then H is approximately distributed as a  2. The Kruskal-Wallis hypothesis test: H 0 : All k populations have the same distribution H 1 : Not all k populations have the same distribution The Kruskal-Wallis test statistic: If each n j > 5, then H is approximately distributed as a  2. 14-6 The Kruskal-Wallis Test - A Nonparametric Alternative to One-Way ANOVA Nonparametric Methods and Chi-Square Tests (1) By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

SoftwareTimeRank Group RankSum 14514190 13810256 15616325 16017 14715 16518 2308 24011 2287 24413 2255 24212 3224 3193 3151 3319 3276 3172 SoftwareTimeRank Group RankSum 14514190 13810256 15616325 16017 14715 16518 2308 24011 2287 24413 2255 24212 3224 3193 3151 3319 3276 3172  2 (2,0.005) =10.5966, so H 0 is rejected. Example 14-8: The Kruskal- Wallis Test Nonparametric Methods and Chi-Square Tests (1) By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Example 14-8: The Kruskal-Wallis Test – Using the Template Nonparametric Methods and Chi-Square Tests (1) By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

If the null hypothesis in the Kruskal-Wallis test is rejected, then we may wish, in addition, compare each pair of populations to determine which are different and which are the same. Further Analysis (Pairwise Comparisons of Average Ranks) Nonparametric Methods and Chi-Square Tests (1) By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Pairwise Comparisons: Example 14-8 Nonparametric Methods and Chi-Square Tests (1) By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

The Friedman test is a nonparametric version of the randomized block design ANOVA. Sometimes this design is referred to as a two-way ANOVA with one item per cell because it is possible to view the blocks as one factor and the treatment levels as the other factor. The test is based on ranks. 14-7 The Friedman Test for a Randomized Block Design The Friedman hypothesis test: H 0 : The distributions of the k treatment populations are identical H 1 : Not all k distribution are identical The Friedman test statistic: The degrees of freedom for the chi-square distribution is (k – 1). The Friedman hypothesis test: H 0 : The distributions of the k treatment populations are identical H 1 : Not all k distribution are identical The Friedman test statistic: The degrees of freedom for the chi-square distribution is (k – 1). Nonparametric Methods and Chi-Square Tests (1) By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

Example 14-10 – using the Template Note: Note: The p-value is small relative to a significance level of a = 0.05, so one should conclude that there is evidence that not all three low- budget cruise lines are equally preferred by the frequent cruiser population Nonparametric Methods and Chi-Square Tests (1) By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

The Spearman Rank Correlation Coefficient is the simple correlation coefficient calculated from variables converted to ranks from their original values. 14-8 The Spearman Rank Correlation Coefficient Nonparametric Methods and Chi-Square Tests (1) By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer