Business Statistics, 3eby Ken Black

Chapter 16Chi-Square andOther NonparametricStatistics

Discrete Distributions

Learning ObjectivesRecognize the advantages and disadvantages of nonparametric statistics.Understand the 2 goodness-of-fit test and how to use it.Analyze data using the 2 test of independence.Understand how to use the runs test to test for randomness.Know when and how to use the Mann-Whitney U test, the Wilcoxon matched-pairs signed rank test, the Kruskal-Wallis test, and the Friedman test.Learn when and how to measure correlation using Spearmans rank correlation measurement.

Parametric vs Nonparametric StatisticsParametric Statistics are statistical techniques based on assumptions about the population from which the sample data are collected.Assumption that data being analyzed are randomly selected from a normally distributed population. Requires quantitative measurement that yield interval or ratio level data.

Nonparametric Statistics are based on fewer assumptions about the population and the parameters. Sometimes called distribution-free statistics.A variety of nonparametric statistics are available for use with nominal or ordinal data.

Advantages of Nonparametric TechniquesSometimes there is no parametric alternative to the use of nonparametric statistics.Certain nonparametric test can be used to analyze nominal data.Certain nonparametric test can be used to analyze ordinal data.The computations on nonparametric statistics are usually less complicated than those for parametric statistics, particularly for small samples.Probability statements obtained from most nonparametric tests are exact probabilities.

Disadvantages of Nonparametric StatisticsNonparametric tests can be wasteful of data if parametric tests are available for use with the data.Nonparametric tests are usually not as widely available and well know as parametric tests.For large samples, the calculations for many nonparametric statistics can be tedious.

2 Goodness-of-Fit TestThe 2 goodness-of-fit test compares expected (theoretical) frequencies of categories from a population distribution to the observed (actual) frequencies from a distribution to determine whether there is a difference between what was expected and what was observed.

2 Goodness-of-Fit Test

Milk Sales Data for Demonstration Problem 16.1

Hypotheses and Decision Rules for Demonstration Problem 16.1

Calculations for Demonstration Problem 16.1

Demonstration Problem 16.1: Conclusion

Bank Customer Arrival Data for Demonstration Problem 16.2

Hypotheses and Decision Rules for Demonstration Problem 16.2

Calculations for Demonstration Problem 16.2: Estimating the Mean Arrival Rate

Calculations for Demonstration Problem 16.2: Poisson Probabilities for = 2.3PoissonProbabilities for = 2.3

2 Calculations for Demonstration Problem 16.2

Demonstration Problem 16.2: Conclusion

Using a 2 Goodness-of-Fit Test to Test a Population Proportion

Using a 2 Goodness-of-Fit Test to Test a Population Proportion: Calculations

Using a 2 Goodness-of-Fit Test to Test a Population Proportion: Conclusion

2 Test of IndependenceUsed to analyze the frequencies of two variables with multiple categories to determine whether the two variables are independent.

2 Test of Independence: Investment ExampleIn which region of the country do you reside?A. NortheastB. MidwestC. SouthD. WestWhich type of financial investment are you most likely to make today?E. StocksF. BondsG. Treasury bills

2 Test of Independence: Investment Example

2 Test of Independence: Formulas

2 Test of Independence: Gasoline Preference Versus Income Category

Gasoline Preference Versus Income Category: Observed Frequencies

Gasoline Preference Versus Income Category: Expected Frequencies

Gasoline Preference Versus Income Category: 2 Calculation

Gasoline Preference Versus Income Category: Conclusion

Runs TestTest for randomness - is the order or sequence of observations in a sample random or notEach sample item possesses one of two possible characteristicsRun - a succession of observations which possess the same characteristicExample with two runs: F, F, F, F, F, F, F, F, M, M, M, M, M, M, MExample with fifteen runs: F, M, F, M, F, M, F, M, F, M, F, M, F, M, F

Runs Test: Sample Size ConsiderationSample size: nNumber of sample member possessing the first characteristic: n1Number of sample members possessing the second characteristic: n2n = n1 + n2If both n1 and n2 are 20, the small sample runs test is appropriate.

Runs Test: Small Sample Example H0: The observations in the sample are randomly generated.Ha: The observations in the sample are not randomly generated.

= .05

n1 = 18n2 = 8

If 7 R 17, do not reject H0Otherwise, reject H0.

1 2 3 4 5 6 7 8 9 10 11 12D CCCCC D CC D CCCC D C D CCC DDD CCC

R = 12Since 7 R = 12 17, do not reject H0

Runs Test: Large SampleIf either n1 or n2 is > 20, the sampling distribution of R is approximately normal.

Runs Test: Large Sample ExampleH0: The observations in the sample are randomly generated.Ha: The observations in the sample are not randomly generated.

= .05

n1 = 40n2 = 10

If -1.96 Z 1.96, do not reject H0Otherwise, reject H0. 1 1 2 3 4 5 6 7 8 9 0 11NNN F NNNNNNN F NN FF NNNNNN F NNNN F NNNNN

12 13FFFF NNNNNNNNNNNN R = 13

Runs Test: Large Sample Example-1.96 Z = -1.81 1.96,do not reject H0

Mann-Whitney U TestNonparametric counterpart of the t test for independent samplesDoes not require normally distributed populationsMay be applied to ordinal dataAssumptionsIndependent SamplesAt Least Ordinal Data

Mann-Whitney U Test: Sample Size ConsiderationSize of sample one: n1Size of sample two: n2If both n1 and n2 are 10, the small sample procedure is appropriate.If either n1 or n2 is greater than 10, the large sample procedure is appropriate.

Mann-Whitney U Test: Small Sample ExampleH0: The health service population is identical to the educational service population on employee compensationHa: The health service population is not identical to the educational service population on employee compensation

Mann-Whitney U Test: Small Sample Example = .05

If the final p-value < .05, reject H0.

W1 = 1 + 2 + 3 + 4 + 6 + 7 + 8= 31

W2 = 5 + 9 + 10 + 11 + 12 + 13 + 14 + 15= 89

Mann-Whitney U Test: Small Sample Example

Mann-Whitney U Test: Formulas for Large Sample Case

Incomes of PBS and Non-PBS ViewersHo: The incomes for PBS viewers and non-PBS viewers are identicalHa: The incomes for PBS viewers and non-PBS viewers are not identical

Ranks of Income from Combined Groups of PBS and Non-PBS Viewers

PBS and Non-PBS Viewers: Calculation of U

PBS and Non-PBS Viewers: Conclusion

Wilcoxon Matched-PairsSigned Rank TestA nonparametric alternative to the t test for related samplesBefore and After studiesStudies in which measures are taken on the same person or object under different conditionsStudies or twins or other relatives

Wilcoxon Matched-PairsSigned Rank TestDifferences of the scores of the two matched samplesDifferences are ranked, ignoring the signRanks are given the sign of the differencePositive ranks are summedNegative ranks are summedT is the smaller sum of ranks

Wilcoxon Matched-Pairs Signed Rank Test: Sample Size Considerationn is the number of matched pairsIf n > 15, T is approximately normally distributed, and a Z test is used.If n 15, a special small sample procedure is followed.The paired data are randomly selected.The underlying distributions are symmetrical.

Wilcoxon Matched-Pairs Signed Rank Test: Small Sample ExampleH0: Md = 0Ha: Md 0

n = 6

=0.05

If Tobserved 1, reject H0.

Wilcoxon Matched-Pairs Signed Rank Test: Small Sample Example

Wilcoxon Matched-Pairs Signed Rank Test: Large Sample Formulas

Airline Cost Data for 17 Cities, 1997 and 1999H0: Md = 0Ha: Md 0

Airline Cost: T Calculation

Airline Cost: Conclusion

Kruskal-Wallis TestA nonparametric alternative to one-way analysis of varianceMay used to analyze ordinal dataNo assumed population shapeAssumes that the C groups are independentAssumes random selection of individual items

Kruskal-Wallis K Statistic

Number of Patients per Day per Physician in Three Organizational Categories Ho: The three populations are identicalHa: At least one of the three populations is different

Patients per Day Data: Kruskal-Wallis Preliminary Calculations

Patients per Day Data: Kruskal-Wallis Calculations and Conclusion

Friedman TestA nonparametric alternative to the randomized block designAssumptionsThe blocks are independent.There is no interaction between blocks and treatments.Observations within each block can be ranked.Hypotheses Ho: The treatment populations are equal Ha:At least one treatment population yields larger values than at least one other treatment population

Friedman Test

Friedman Test: Tensile Strength of Plastic HousingsHo:The supplier populations are equalHa:At least one supplier population yields larger values than at least one other supplier population

Friedman Test: Tensile Strength of Plastic Housings

Friedman Test: Tensile Strength of Plastic Housings

Friedman Test: Tensile Strength of Plastic Housings

Spearmans Rank CorrelationAnalyze the degree of association of two variablesApplicable to ordinal level data (ranks)

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