Lecture 28 Chi-Square Analysis

Transcription

1 Lecture 28 STAT 225 Introduction to Probability Models April 23, 2014 Whitney Huang Purdue University 28.1

2 χ 2 test for For a given contingency table, we want to test if two have a relationship or not To answer this question, we need to perform a statistical hypothesis test A statistical hypothesis test is a method of making decisions using data from a scientific study A χ 2 test is suitable for answering whether two qualitative have a relationship or not 28.2

3 χ 2 test for For a given contingency table, we want to test if two have a relationship or not To answer this question, we need to perform a statistical hypothesis test A statistical hypothesis test is a method of making decisions using data from a scientific study A χ 2 test is suitable for answering whether two qualitative have a relationship or not 28.2

4 χ 2 test for For a given contingency table, we want to test if two have a relationship or not To answer this question, we need to perform a statistical hypothesis test A statistical hypothesis test is a method of making decisions using data from a scientific study A χ 2 test is suitable for answering whether two qualitative have a relationship or not 28.2

5 χ 2 test for For a given contingency table, we want to test if two have a relationship or not To answer this question, we need to perform a statistical hypothesis test A statistical hypothesis test is a method of making decisions using data from a scientific study A χ 2 test is suitable for answering whether two qualitative have a relationship or not 28.2

6 χ 2 test The procedure of χ 2 test for : 1 Define the Null (H 0 ) and Alternative (H A ) hypotheses H 0 : there is no relationship between the 2 H A : there is a relationship between the 2 2 (If necessary) Calculate the marginal totals, and the grand total 3 Calculate the expected cell counts expected cell count = Row Total Column Total Grand Total 4 Calculate the partial χ 2 values (a χ 2 value for each cell of the table) partial χ 2 value = (observed - expected)2 expected 28.3

7 χ 2 test The procedure of χ 2 test for : 1 Define the Null (H 0 ) and Alternative (H A ) hypotheses H 0 : there is no relationship between the 2 H A : there is a relationship between the 2 2 (If necessary) Calculate the marginal totals, and the grand total 3 Calculate the expected cell counts expected cell count = Row Total Column Total Grand Total 4 Calculate the partial χ 2 values (a χ 2 value for each cell of the table) partial χ 2 value = (observed - expected)2 expected 28.3

8 χ 2 test The procedure of χ 2 test for : 1 Define the Null (H 0 ) and Alternative (H A ) hypotheses H 0 : there is no relationship between the 2 H A : there is a relationship between the 2 2 (If necessary) Calculate the marginal totals, and the grand total 3 Calculate the expected cell counts expected cell count = Row Total Column Total Grand Total 4 Calculate the partial χ 2 values (a χ 2 value for each cell of the table) partial χ 2 value = (observed - expected)2 expected 28.3

9 χ 2 test The procedure of χ 2 test for : 1 Define the Null (H 0 ) and Alternative (H A ) hypotheses H 0 : there is no relationship between the 2 H A : there is a relationship between the 2 2 (If necessary) Calculate the marginal totals, and the grand total 3 Calculate the expected cell counts expected cell count = Row Total Column Total Grand Total 4 Calculate the partial χ 2 values (a χ 2 value for each cell of the table) partial χ 2 value = (observed - expected)2 expected 28.3

10 χ 2 test cont d 5 Calculate the χ 2 statistic χ 2 = partial χ 2 value 6 Calculate the degrees of freedom (df ) df = (#of rows 1) (#of columns 1) 7 Find the χ 2 critical value with respect to α from the χ 2 table 8 Draw your conclusion: Reject H 0 if your χ 2 statistic is bigger than the χ 2 critical value There is an statistical evidence that there is a relationship between the 2 at α level 28.4

11 χ 2 test cont d 5 Calculate the χ 2 statistic χ 2 = partial χ 2 value 6 Calculate the degrees of freedom (df ) df = (#of rows 1) (#of columns 1) 7 Find the χ 2 critical value with respect to α from the χ 2 table 8 Draw your conclusion: Reject H 0 if your χ 2 statistic is bigger than the χ 2 critical value There is an statistical evidence that there is a relationship between the 2 at α level 28.4

12 χ 2 test cont d 5 Calculate the χ 2 statistic χ 2 = partial χ 2 value 6 Calculate the degrees of freedom (df ) df = (#of rows 1) (#of columns 1) 7 Find the χ 2 critical value with respect to α from the χ 2 table 8 Draw your conclusion: Reject H 0 if your χ 2 statistic is bigger than the χ 2 critical value There is an statistical evidence that there is a relationship between the 2 at α level 28.4

13 χ 2 test cont d 5 Calculate the χ 2 statistic χ 2 = partial χ 2 value 6 Calculate the degrees of freedom (df ) df = (#of rows 1) (#of columns 1) 7 Find the χ 2 critical value with respect to α from the χ 2 table 8 Draw your conclusion: Reject H 0 if your χ 2 statistic is bigger than the χ 2 critical value There is an statistical evidence that there is a relationship between the 2 at α level 28.4

14 Example 67 A 2011 study was conducted in Kalamazoo, Michigan. The objective was to determine if parents marital status affects children s marital status later in their life. In total, 2,000 children were interviewed. The columns refer to the parents marital status. Use the contingency table below to conduct a χ 2 test from beginning to end. Use α =.10 (Observed) Married Divorced Total Married Divorced Total 28.5

15 Example 67 cont d 1 Define the Null and Alternative hypotheses: H 0 : there is no relationship between parents marital status and childrens marital status H A : there is a relationship between parents marital status and childrens marital status 2 Calculate the marginal totals, and the grand total (Observed) Married Divorced Total Married Divorced Total

16 Example 67 cont d 1 Define the Null and Alternative hypotheses: H 0 : there is no relationship between parents marital status and childrens marital status H A : there is a relationship between parents marital status and childrens marital status 2 Calculate the marginal totals, and the grand total (Observed) Married Divorced Total Married Divorced Total

17 Example 67 cont d 3 Calculate the expected cell counts (Expected) Married Divorced Married 2000 = = Divorced 2000 = = Calculate the partial χ 2 values partial χ 2 Married Divorced Married Divorced ( ) = 1.39 ( ) = 1.60 ( ) = 1.50 ( ) =

18 Example 67 cont d 3 Calculate the expected cell counts (Expected) Married Divorced Married 2000 = = Divorced 2000 = = Calculate the partial χ 2 values partial χ 2 Married Divorced Married Divorced ( ) = 1.39 ( ) = 1.60 ( ) = 1.50 ( ) =

19 Example 67 cont d 5 Calculate the χ 2 statistic χ 2 = = Calculate the degrees of freedom (df ) The df is (2 1) (2 1) = 1 7 Find the χ 2 critical value with respect to α from the χ 2 table The χ 2 α=0.1,df =1 = Draw your conclusion: We reject H 0 and conclude that there is a relationship between parents marital status and childrens marital status. 28.8

20 Example 67 cont d 5 Calculate the χ 2 statistic χ 2 = = Calculate the degrees of freedom (df ) The df is (2 1) (2 1) = 1 7 Find the χ 2 critical value with respect to α from the χ 2 table The χ 2 α=0.1,df =1 = Draw your conclusion: We reject H 0 and conclude that there is a relationship between parents marital status and childrens marital status. 28.8

21 Example 67 cont d 5 Calculate the χ 2 statistic χ 2 = = Calculate the degrees of freedom (df ) The df is (2 1) (2 1) = 1 7 Find the χ 2 critical value with respect to α from the χ 2 table The χ 2 α=0.1,df =1 = Draw your conclusion: We reject H 0 and conclude that there is a relationship between parents marital status and childrens marital status. 28.8

22 Example 67 cont d 5 Calculate the χ 2 statistic χ 2 = = Calculate the degrees of freedom (df ) The df is (2 1) (2 1) = 1 7 Find the χ 2 critical value with respect to α from the χ 2 table The χ 2 α=0.1,df =1 = Draw your conclusion: We reject H 0 and conclude that there is a relationship between parents marital status and childrens marital status. 28.8

23 Example 68 The following contingency table contains enrollment data for a random sample of students from several colleges at Purdue University during the academic year. The table lists the number of male and female students enrolled in each college. Use the two-way table to conduct a χ 2 test from beginning to end. Use α =.01 (Observed) Female Male Total Liberal Arts Science Engineering Total

24 Example 68 cont d (Expected) Female Male Liberal Arts 1528 = Science = Engineering = = = = partial χ 2 Female Male Lib Arts Sci Eng ( ) = ( ) = 0.26 ( ) = ( ) = ( ) = 0.16 ( ) = χ 2 = = The df = (3 1) (2 1) = 2 The χ 2 α=.01,df =2 = 9.21 We reject H 0 and conclude that there is a relationship between gender and major