10.1: Multinomial Experiments Multinomial experiment A probability experiment consisting of a fixed number of trials in which there are more than two possible.

Slides:



Advertisements
Similar presentations
Chi Squared Tests. Introduction Two statistical techniques are presented. Both are used to analyze nominal data. –A goodness-of-fit test for a multinomial.
Advertisements

CHAPTER 23: Two Categorical Variables: The Chi-Square Test
Chapter 11 Inference for Distributions of Categorical Data
Chapter 10 Chi-Square Tests and the F- Distribution 1 Larson/Farber 4th ed.
Chapter 11 Chi-Square Procedures 11.1 Chi-Square Goodness of Fit.
Introduction to Chi-Square Procedures March 11, 2010.
Chapter 16 Chi Squared Tests.
Ch 15 - Chi-square Nonparametric Methods: Chi-Square Applications
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 14 Goodness-of-Fit Tests and Categorical Data Analysis.
Hypothesis Testing for Variance and Standard Deviation
11-2 Goodness-of-Fit In this section, we consider sample data consisting of observed frequency counts arranged in a single row or column (called a one-way.
Chi-Square Test.
BCOR 1020 Business Statistics
Chi-Square Tests and the F-Distribution
Chi-square Goodness of Fit Test
Copyright © Cengage Learning. All rights reserved. 11 Applications of Chi-Square.
Chapter 13: Inference for Tables – Chi-Square Procedures
The table shows a random sample of 100 hikers and the area of hiking preferred. Are hiking area preference and gender independent? Hiking Preference Area.
Analysis of Count Data Chapter 26
EDRS 6208 Analysis and Interpretation of Data Non Parametric Tests
Copyright © 2013 Pearson Education, Inc. All rights reserved Chapter 10 Inferring Population Means.
Section 10.3 Comparing Two Variances Larson/Farber 4th ed1.
13.1 Goodness of Fit Test AP Statistics. Chi-Square Distributions The chi-square distributions are a family of distributions that take on only positive.
Section 10.1 Goodness of Fit. Section 10.1 Objectives Use the chi-square distribution to test whether a frequency distribution fits a claimed distribution.
Other Chi-Square Tests
Chapter 11: Inference for Distributions of Categorical Data.
Copyright © 2010, 2007, 2004 Pearson Education, Inc. 1.. Section 11-2 Goodness of Fit.
Chapter 16 – Categorical Data Analysis Math 22 Introductory Statistics.
Chapter 10 Chi-Square Tests and the F-Distribution
Chapter Chi-Square Tests and the F-Distribution 1 of © 2012 Pearson Education, Inc. All rights reserved.
Chi-Square Procedures Chi-Square Test for Goodness of Fit, Independence of Variables, and Homogeneity of Proportions.
Chapter 10 Chi-Square Tests and the F- Distribution 1 Larson/Farber 4th ed.
Chi-Square Test.
Chapter 10 Chi-Square Tests and the F-Distribution
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc Chapter 16 Chi-Squared Tests.
GOODNESS OF FIT Larson/Farber 4th ed 1 Section 10.1.
Chi-Square Test James A. Pershing, Ph.D. Indiana University.
AP Statistics Section 14.. The main objective of Chapter 14 is to test claims about qualitative data consisting of frequency counts for different categories.
Section 10.2 Independence. Section 10.2 Objectives Use a chi-square distribution to test whether two variables are independent Use a contingency table.
© Copyright McGraw-Hill CHAPTER 11 Other Chi-Square Tests.
1 Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.
Dan Piett STAT West Virginia University Lecture 12.
Statistics 300: Elementary Statistics Section 11-2.
381 Goodness of Fit Tests QSCI 381 – Lecture 40 (Larson and Farber, Sect 10.1)
Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Lecture Slides Elementary Statistics Twelfth Edition and the Triola Statistics Series.
Chapter 11 Chi-Square Procedures 11.1 Chi-Square Goodness of Fit.
+ Section 11.1 Chi-Square Goodness-of-Fit Tests. + Introduction In the previous chapter, we discussed inference procedures for comparing the proportion.
11.1 Chi-Square Tests for Goodness of Fit Objectives SWBAT: STATE appropriate hypotheses and COMPUTE expected counts for a chi- square test for goodness.
Chi-Två Test Kapitel 6. Introduction Two statistical techniques are presented, to analyze nominal data. –A goodness-of-fit test for the multinomial experiment.
Section 10.2 Objectives Use a contingency table to find expected frequencies Use a chi-square distribution to test whether two variables are independent.
Goodness-of-Fit and Contingency Tables Chapter 11.
Chapter 10 Chi-Square Tests and the F-Distribution.
Chi Square Analysis. What is the chi-square statistic? The chi-square (chi, the Greek letter pronounced "kye”) statistic is a nonparametric statistical.
Section 10.1 Goodness of Fit © 2012 Pearson Education, Inc. All rights reserved. 1 of 91.
Check your understanding: p. 684
Chapter 10 Chi-Square Tests and the F-Distribution
10 Chapter Chi-Square Tests and the F-Distribution Chapter 10
Chi-Squared Goodness of Fit
Chapter 8 Hypothesis Testing with Two Samples.
Chapter 10 Chi-Square Tests and the F-Distribution
Section 10-1 – Goodness of Fit
Elementary Statistics: Picturing The World
Chi-Square Test.
Chi-Square Test.
Chi-Square - Goodness of Fit
Goodness of Fit Test - Chi-Squared Distribution
Overview and Chi-Square
Chi-Square Test.
Chapter 26 Part 2 Comparing Counts.
Presentation transcript:

10.1: Multinomial Experiments Multinomial experiment A probability experiment consisting of a fixed number of trials in which there are more than two possible outcomes for each independent trial. A binomial experiment had only two possible outcomes. The probability for each outcome is fixed and each outcome is classified into categories. 1 Larson/Farber Example: A radio station claims that the distribution of music preferences for listeners in the broadcast region is as shown below. Distribution of music Preferences Classical4%Oldies2% Country36%Pop18% Gospel11%Rock29% Each outcome is classified into categories. The probability for each possible outcome is fixed.

Chi-Square Goodness-of-Fit Test Used to test whether a frequency distribution fits an expected distribution. H 0 : The frequency distribution FITS the specified distribution. H a : The frequency distribution DOES NOT FIT the specified distribution. 2 Larson/Farber Example: To test the radio station’s claim, the executive can perform a chi-square goodness-of-fit test using the following hypotheses. H 0 : The distribution of music preferences in the broadcast region is 4% classical, 36% country, 11% gospel, 2% oldies, 18% pop, and 29% rock. (claim) H a : The distribution of music preferences differs from the claimed or expected distribution.

Chi-Square Goodness-of-Fit Test Observed frequency O - frequency for the category observed in the sample. 3 Larson/Farber 4th ed Expected frequency E - calculated frequency for the category.  Expected frequencies are obtained assuming the specified (or hypothesized) distribution. The expected frequency for the i th category is: E i = np i n = number of trials (sample size) p i = assumed probability of i th category. Example: A marketing executive randomly selects 500 radio music listeners from the broadcast region and asks each whether he or she prefers classical, country, gospel, oldies, pop, or rock music. The results are shown at the right. Find the observed frequencies and the expected frequencies for each type of music. Survey results (n = 500) Classical8 Country210 Gospel72 Oldies10 Pop75 Rock125 Type of music % of listeners Observed frequency Expected frequency Classical 4%8 Country36%210 Gospel11%72 Oldies 2%10 Pop18%75 Rock29% (0.04) = (0.36) = (0.11) = (0.02) = (0.18) = (0.29) = 145 n = 500

Chi-Square Goodness-of-Fit Test 1.The observed frequencies must be obtained by using a random sample. 2.Each expected frequency must be greater than or equal to 5. If these two conditions are satisfied, then the sampling distribution for the goodness-of-fit test is approximated by a chi-square distribution with k – 1 degrees of freedom, where k is the number of categories and test statistic is: You may perform a hypothesis test using Table 6 Appendix B to find critical values 4 Larson/Farber The test is always a right-tailed test. O = Observed frequency in each category E = Expected frequency of each category

Example1: Goodness of Fit Test Use the music preference data to perform a chi-square goodness-of-fit test to test whether the distributions are different. Use α = Survey results (n = 500) Classical8 Country210 Gospel72 Oldies10 Pop75 Rock125 Distribution of music preferences Classical4% Country36% Gospel11% Oldies2% Pop18% Rock29% 5 Larson/Farber 4th ed H 0 : music preference is 4% classical, 36% country, 11% gospel, 2% oldies, 18% pop, and 29% rock H a :music preference differs from the claimed or expected distribution  =.01 d.f. = n –1 = 6 –1 = 5 01 χ2χ Type of music Observed frequency Expected frequency Classical820 Country Gospel7255 Oldies10 Pop7590 Rock Decision: Reject H 0 There is enough evidence to conclude that the distribution of music preferences differs from the claimed distribution.

Example2: Goodness of Fit Test The manufacturer of M&M’s candies claims that the number of different- colored candies in bags of dark chocolate M&M’s is uniformly distributed. To test this claim, you randomly select a bag that contains 500 dark chocolate M&M’s. The results are shown in the table. Using α = 0.10, perform a chi- square goodness-of-fit test to test the claimed or expected distribution. What can you conclude? (Adapted from Mars Incorporated) 6 Larson/Farber 4th ed ColorFrequency Brown80 Yellow95 Red88 Blue83 Orange76 Green78 d.f. = 6 –1 = 5 n = 500 The claim is that the distribution is uniform, so the expected frequencies of the colors are equal. To find each expected frequency, divide the sample size by the number of colors. E = 500/6 ≈ 83.3 H 0 : distribution of different-colored candies in bags of dark chocolate M & Ms is uniform. H a :distribution of different-colored candies in bags of dark chocolate M & Ms is not uniform χ2χ Expected Frequency 83.3 Decision: Fail to Reject H 0 – there is not enough evidence to dispute the claim that the distribution is uniform.