Lecture Business statistics in practice (7/e): Chapter 13 - Bowerman, O'Connell, Murphree

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Chapter 13 - Chi-square tests. After mastering the material in this chapter, you will be able to: Test hypotheses about multinomial probabilities by using a chi-square goodness-of-fit test, perform a goodness-of-fit test for normality, decide whether two qualitative variables are independent by using a chi-square test for independence.

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Nội dung Text: Lecture Business statistics in practice (7/e): Chapter 13 - Bowerman, O'Connell, Murphree

Chapter 13 ChiSquare Tests McGrawHill/Irwin Copyright © 2014 by The McGrawHill Companies, Inc. All rights reserved.
ChiSquare Tests 13.1 ChiSquare GoodnessofFit Tests 13.2 A ChiSquare Test for Independence 132
LO13-1: Test hypotheses about multinomial probabilities by using a chi-square goodness-of-fit test. 13.1 ChiSquare GoodnessofFit Tests Collect count data to study how counts are distributed among cells Often use categorical data for statistical inference May use a multinomial experiment ◦Similar to a binomial experiment only more than two outcomes are possible 133
LO13-1 The Multinomial Experiment 1. Carry out n identical trials with k possible outcomes of each trial 2. Probabilities are denoted p1, p2, … , pk where p1 + p2 + … + pk = 1 3. The trials are independent 4. The results are observed frequencies of the number of trials that result in each of k possible outcomes, denoted f1, f2, …, fk 134
LO13-1 ChiSquare Goodness of Fit Tests Consider the outcome of a multinomial experiment where each of n randomly selected items is classified into one of k groups Let fi = number of items classified into group i (ith observed frequency) Ei = npi = expected number in ith group if pi is probability of being in group i (ith expected frequency) 135
LO13-1 A Goodness of Fit Test for Multinomial Probabilities  H0: multinomial probabilities are p1, p2, … , pk  Ha: at least one of the probabilities differs from p1, p2, … , pk 2 2 k ( fi Ei ) =  Test statistic: i 1 Ei  Reject H0 if ◦ 2 > or pvalue
LO13-2: Perform a goodness of fit test for normality. Normal Distribution Have seen many statistical methods based on the assumption of a normal distribution Can check the validity of this assumption using frequency distributions, stemandleaf displays, histograms, and normal plots Another approach is to use a chisquare goodness of fit test 137
LO13-2 A Goodness of Fit Test for a Normal Distribution 1. Test the following null and alternative hypotheses: H0: the population has a normal distribution Ha: population does not have normal distribution 2. Select random sample and compute sample mean and standard deviation 3. Define k intervals for the test 4. Record observed frequency (fi) for each interval 5. Calculate expected frequency (Ei) 2 6. Calculate the chisquare statistic 2 k fi Ei 7. Make a decision i 1 Ei 138
LO13-3: Decide whether two qualitative variables are independent by using a chi-square test for independence. 13.2 A ChiSquare Test for Independence Each of n randomly selected items is classified on two dimensions into a contingency table with r rows an c columns and let ◦fij = observed cell frequency for ith row and jth column ◦ri = ith row total cj = jth column total Expected cell frequency for ith row and jth column under independence ri c j Eˆ ij n 139
LO13-3 A ChiSquare Test for Independence Continued  H0: the two classifications are statistically independent  Ha: the two classifications are statistically dependent  Test statistic 2 ( f ij Eˆ ij ) 2 = all cells Eˆ ij  Reject H0 if > 2 or if pvalue