Section 10.7

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Section 10.7. Chi-Square Test for Association. Objectives. Perform a chi-square test for association. Chi-Square Test for Association. Null and Alternative Hypotheses for a Chi‑Square Test for Association. Chi-Square Test for Association. - PowerPoint PPT Presentation

Transcript of Section 10.7

Section X.X

Section 10.7Chi-Square Test for AssociationHAWKES LEARNING SYSTEMSStudents Matter. Success Counts.Copyright 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.ObjectivesPerform a chi-square test for association.

HAWKES LEARNING SYSTEMSStudents Matter. Success Counts.Copyright 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.Chi-Square Test for AssociationNull and Alternative Hypotheses for a ChiSquare Test for Association

HAWKES LEARNING SYSTEMSStudents Matter. Success Counts.Copyright 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.Chi-Square Test for AssociationExpected Value of a Frequency in a Contingency TableThe expected value of the frequency for the ith possible outcome in a contingency table is given by

where n is the sample size.

HAWKES LEARNING SYSTEMSStudents Matter. Success Counts.Copyright 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.Chi-Square Test for AssociationTest Statistic for a Chi-Square Test for AssociationThe test statistic for a chisquare test for association is given by

where Oi is the observed frequency for the ith possible outcome andEi is the expected frequency for the ith possible outcome.

HAWKES LEARNING SYSTEMSStudents Matter. Success Counts.Copyright 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.Chi-Square Test for AssociationDegrees of Freedom in a Chi-Square Test for AssociationIn a chisquare test for association, the number of degrees of freedom for the chisquare distribution of the test statistic is given bydf = (R 1) (C 1)where R is the number of rows of data in the contingency table (not including the row of totals) and C is the number of columns of data in the contingency table (not including the column of totals).HAWKES LEARNING SYSTEMSStudents Matter. Success Counts.Copyright 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.Chi-Square Test for AssociationRejection Region for ChiSquare Tests for AssociationReject the null hypothesis, H0, if:

HAWKES LEARNING SYSTEMSStudents Matter. Success Counts.Copyright 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.Example 10.32: Performing a Chi-Square Test for AssociationSuppose that the following data were collected in a poll of 13,660 randomly selected voters during the 2008 presidential election campaign.

Is there evidence at the 0.05 level to say that gender and voting choice were related for this election?Observed Sample of 13,660 VotersObamaMcCainOtherTotalMale34552764656284Female35413762737376Total6996652613813,660HAWKES LEARNING SYSTEMSStudents Matter. Success Counts.Copyright 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.Example 10.32: Performing a Chi-Square Test for Association (cont.)Solution Step 1:State the null and alternative hypotheses. We let the null hypothesis be that gender and voting preference are independent of one another.

HAWKES LEARNING SYSTEMSStudents Matter. Success Counts.Copyright 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.Example 10.32: Performing a Chi-Square Test for Association (cont.)Step 2:Determine which distribution to use for the test statistic, and state the level of significance. We wish to determine if there is an association between gender and voting preference. Since we are told that we can safely assume that the necessary conditions have been met for the examples in this section, we will use the chisquare test statistic to test for this association. We are told that the level of significance is a = 0.05. HAWKES LEARNING SYSTEMSStudents Matter. Success Counts.Copyright 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.Example 10.32: Performing a Chi-Square Test for Association (cont.)Step 3:Gather data and calculate the necessary sample statistics. Before we begin to calculate the test statistic, we must calculate the expected value for each cell in the contingency table.

HAWKES LEARNING SYSTEMSStudents Matter. Success Counts.Copyright 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.Example 10.32: Performing a Chi-Square Test for Association (cont.)Lets calculate the c2 -test statistic.

HAWKES LEARNING SYSTEMSStudents Matter. Success Counts.Copyright 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.Example 10.32: Performing a Chi-Square Test for Association (cont.)Step 4:Draw a conclusion and interpret the decision. The number of degrees of freedom for this test is df = (2 1) (3 1) = 2 and a = 0.05. Using the table, we find that the critical value is c20.050 = 5.991. Comparing the test statistic to the critical value, we have 67.276 > 5.991, so c2 c20.050 , and thus we must reject the null hypothesis. In other words, at the 0.05 level of significance, we can conclude that gender and voting choice were related for the 2008 presidential election.HAWKES LEARNING SYSTEMSStudents Matter. Success Counts.Copyright 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.Example 10.33: Performing a Chi-Square Test for Association Using a TI83/84 Plus CalculatorA local hairdresser is curious about whether there is a relationship between hair color and the combination of gender and marital status among his clients. He collects data from a random sample of his clients and records the data in the following contingency table.HAWKES LEARNING SYSTEMSStudents Matter. Success Counts.Copyright 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.Example 10.33: Performing a Chi-Square Test for Association Using a TI83/84 Plus Calculator (cont.)Based on these data, is there enough evidence at the 0.10 level of significance to say that there is a relationship between a persons hair color and the combination of gender and marital status for this hairdressers clients?Observed Sample of 232 ClientsBlondeBrownRedBlackTotalSingle Women 181981459Married Women201891764Single Men132241655Married Men122431554Total63832462232HAWKES LEARNING SYSTEMSStudents Matter. Success Counts.Copyright 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.Example 10.33: Performing a Chi-Square Test for Association Using a TI83/84 Plus Calculator (cont.)SolutionFor this example, we will use a TI83/84 Plus calculator to calculate the test statistic and draw our conclusion. As we saw in previous sections when using technology, we must begin by doing the first few steps by hand.HAWKES LEARNING SYSTEMSStudents Matter. Success Counts.Copyright 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.Example 10.33: Performing a Chi-Square Test for Association Using a TI83/84 Plus Calculator (cont.)Step 1:State the null and alternative hypotheses. As always in a test for association, the null hypothesis is that the two variables, hair color and the combination of gender and marital status in this example, are independent.

HAWKES LEARNING SYSTEMSStudents Matter. Success Counts.Copyright 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.Example 10.33: Performing a Chi-Square Test for Association Using a TI83/84 Plus Calculator (cont.)Step 2:Determine which distribution to use for the test statistic, and state the level of significance. We wish to determine if there is an association between hair color and the combination of gender and marital status. Since we have been told that we can safely assume that the necessary criteria are met for the examples in this section, we will use the chisquare test statistic to test for this association. We are given a level of significance of a = 0.10.HAWKES LEARNING SYSTEMSStudents Matter. Success Counts.Copyright 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.Example 10.33: Performing a Chi-Square Test for Association Using a TI83/84 Plus Calculator (cont.)Step 3:Gather data and calculate the necessary sample statistics. The data have been gathered and presented in the form of a contingency table. When using a TI83/84 Plus calculator, you do not have to calculate the expected values as you would if you were performing a chisquare test for association by hand.HAWKES LEARNING SYSTEMSStudents Matter. Success Counts.Copyright 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.Example 10.33: Performing a Chi-Square Test for Association Using a TI83/84 Plus Calculator (cont.)To use a TI-83/84 Plus calculator, start by entering the table of observed values into the calculator in the form of a matrix. Press and then to access the MATRIX menu. Scroll over to EDIT and choose option 1:[A], which is the name of the first matrix. Now you need to enter the size of the matrix in the form of (Number of Rows)x (Number of Columns).

HAWKES LEARNING SYSTEMSStudents Matter. Success Counts.Copyright 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.Example 10.33: Performing a Chi-Square Test for Association Using a TI83/84 Plus Calculator (cont.)It is important to know that when using the calculator, you should not enter the total row or column! For our table of observed values, there are 4 rows and 4 columns (omitting the totals), so the size of the matrix is 4 4. Now enter the data from the table, as shown in the screenshot.

HAWKES LEARNING SYSTEMSStudents Matter. Success Counts.Copyright 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.Example 10.33: Performing a Chi-Square Test for Association Using a TI83/84 Plus Calculator (cont.)Next press and scroll to TESTS. Choose option C:c2-Test. We must then enter the name of the matrix containing the observed values and the name of th