Department of Quantitative Methods & Information Systems Introduction to Business Statistics QM 220 QM 220 Chapter 11 Dr. Mohammad Zainal
Chapter 11: Chi-Square test 11.1 The Chi-Square Distribution � Like the t distribution, the chi-square distribution has only one parameter called the degrees of freedom ( df ). � The shape of a specific chi-square distribution depends on � Th h f ifi hi di t ib ti d d the number of degrees of freedom. � The chi square distribution curve starts at the origin and � The chi-square distribution curve starts at the origin and lies entirely to the right of the vertical axis. 2 QM-220, M. Zainal
Chapter 11: Chi-Square test 11.1 The Chi-Square Distribution � Th � The chi-square distribution has only one parameter, hi di ib i h l called the degrees of freedom. The shape of a chi-square distribution curve is skewed to the right for small df and di t ib ti i k d t th i ht f ll df d becomes symmetric for large df . � The chi-square distribution assumes nonnegative values only, and these are denoted by the symbol χ 2 (read as “chi- square”). � peak (or mode) of a chi-square distribution curve with 1 or 2 degrees of freedom occurs at zero and for a curve with 3 or more degrees of freedom at df − 2. 3 QM-220, M. Zainal
Chapter 11: Chi-Square test 11.1 The Chi-Square Distribution � Example: Find the value of χ 2 for 7 degrees of freedom and an area of 2 f � E l Fi d h l f 7 d f f d d f .10 in the right tail of the chi-square distribution curve. 4 QM-220, M. Zainal
Chapter 11: Chi-Square test 11.1 The Chi-Square Distribution � Example: Find the value of χ 2 for 12 degrees of freedom and an area 2 f � E l Fi d h l f 12 d f f d d of .05 in the left tail of the chi-square distribution curve. 5 QM-220, M. Zainal
Chapter 11: Chi-Square test 11.3 Contingency Tables � Often we may have information on more than one variable for each element. � Such information can be summarized and presented � S h i f ti b i d d t d using a two-way classification table, which is also called a contingency table or cross-tabulation . contingency table or cross tabulation . � Suppose a university has a total of 20,758 students enrolled. 6 QM-220, M. Zainal
Chapter 11: Chi-Square test 11.3 Contingency Tables � By classifying these students based on gender and whether these students are full-time or part-time, we can prepare a contingency table prepare a contingency table. � It is also called a 2 × 2 (read as “two by two”) contingency table. table. � A contingency table can be of any size. � For example it can be 2 × 3 3 × 2 3 × 3 or 4 × 2 � For example, it can be 2 × 3, 3 × 2, 3 × 3, or 4 × 2. � In these notations, the first digit refers to the number of rows in the table, and the second digit refers to the number ows e b e, d e seco d d g e e s o e u be of columns. 7 QM-220, M. Zainal
Chapter 11: Chi-Square test 11.3 Contingency Tables � In general � In general, an R × C table contains R rows and C an R × C table contains R rows and C columns. � Each of the four boxes that contain numbers in the table � Each of the four boxes that contain numbers in the table is called a cell . � The number of cells in a contingency table is obtained by multiplying the number of rows by the number of columns. � In our example we had 2 × 2 = 4 cells. � The subjects that belong to a cell of a contingency table � Th bj t th t b l t ll f ti t bl possess two characteristics. � For example, 2615 students listed in the second cell of the � For example, 2615 students listed in the second cell of the first row in the table 11.5 are male and part-time . 8 QM-220, M. Zainal
Chapter 11: Chi-Square test 11.3 Contingency Tables � The numbers written inside the cells are usually called the joint frequencies . � For example, 2615 students belong to the joint category of � F l 2615 t d t b l t th j i t t f male and part-time . Hence, it is referred to as the joint frequency of this category frequency of this category 9 QM-220, M. Zainal
Chapter 11: Chi-Square test 11.4 A Test of Independence or Homogeneity � In a test of independence for a contingency table, we test the null hypothesis that the two attributes (characteristics) of the elements of a given population are not related (that of the elements of a given population are not related (that is, they are independent) against the alternative hypothesis that the two characteristics are related (that is, they are ( , y dependent). � For example, we may want to test if the affiliation of people with the Democratic and Republican parties is independent of their income levels. We perform such a test b by using the chi-square distribution. sing the chi sq are distrib tion 10 QM-220, M. Zainal
Chapter 11: Chi-Square test 11.4 A Test of Independence or Homogeneity � Another example, we may want to test if there is an association between being a man or a woman and having a preference for watching sports or soap operas on television preference for watching sports or soap operas on television. � Degrees of Freedom for a Test of Independence A test of independence involves a test of the null hypothesis that two independence involves a test of the null hypothesis that two attributes of a population are not related. � The degrees of freedom for a test of independence are g p w e e where R and C are the number of rows and the number of d C e e u be o ows d e u be o columns, respectively, in the given contingency table. 11 QM-220, M. Zainal
Chapter 11: Chi-Square test 11.4 A Test of Independence or Homogeneity � The value of the test statistic χ 2 for a test of independence is calculated as where O and E are the observed and expected frequencies, respectively for a cell respectively, for a cell. � The null hypothesis in a test of independence is always that the two attributes are not related that the two attributes are not related . The alternative The alternative hypothesis is that the two attributes are related . � The frequencies obtained from the performance of an e eque c es ob ed o e pe o ce o experiment for a contingency table are called the observed frequencies. 12 QM-220, M. Zainal
Chapter 11: Chi-Square test 11.4 A Test of Independence or Homogeneity � The expected frequency E for a cell is calculated as � A test of independence is always right-tailed. � To apply a chi-square test of independence, the sample size should be large enough so that the expected frequency for each cell is at least 5 . � If the expected frequency for a cell is not at least 5, we p q y either increase the sample size or combine some categories. 13 QM-220, M. Zainal
Chapter 11: Chi-Square test 11.4 A Test of Independence or Homogeneity � E � Example: A random sample of 300 adults was selected, and they were l A d l f 300 d lt l t d d th asked if they favor giving more freedom to schoolteachers to punish students for violence and lack of discipline. Based on the results of the survey, the two-way classification of the responses of these adults is presented in the following table Does the sample provide sufficient evidence to conclude that the two D th l id ffi i t id t l d th t th t attributes, gender and opinions of adults, are dependent? Use a 1% significance level. 14 QM-220, M. Zainal
Chapter 11: Chi-Square test 11.4 A Test of Independence or Homogeneity Solution � Step 1. State the null and alternative hypotheses . Ho: the two attributes are independent. H 1 : these attributes are dependent. p 1 � Step 2. Select the distribution to use . � We use the chi-square distribution to make a test of � W th hi di t ib ti t k t t f independence for a contingency table. 15 QM-220, M. Zainal
Chapter 11: Chi-Square test 11.4 A Test of Independence or Homogeneity � Step 3 Determine the rejection and nonrejection regions � Step 3. Determine the rejection and nonrejection regions. � The test of independence is always right-tailed � The area of the rejection region is 01 and it falls in � The area of the rejection region is .01, and it falls in the right tail of the chi-square distribution curve. � The contingency table contains two rows and three g y columns. � The degrees of freedom are From Table VI of Appendix C, the critical value of χ 2 for the critical value of χ 2 for df = 2 and α = .01 is 9.210 16 QM-220, M. Zainal
Chapter 11: Chi-Square test 11.4 A Test of Independence or Homogeneity 17 QM-220, M. Zainal
Chapter 11: Chi-Square test 11.4 A Test of Independence or Homogeneity � Step 4 Calculate the value of the test statistic � Step 4. Calculate the value of the test statistic. � Step 5. Make a decision. The value of the test statistic χ 2 = 8.252 is less than the critical value of χ 2 = 9.210, and it falls in the nonrejection region. So, we fail to reject the null hypothesis 18 QM-220, M. Zainal
Chapter 11: Chi-Square test 11.4 A Test of Independence or Homogeneity � E � Example: A researcher wanted to study the relationship between l A h t d t t d th l ti hi b t gender and owning cell phones. She took a sample of 2000 adults and obtained the information given in the following table Own Cell Phones Do Not Own Cell Phones Men Men 640 640 450 450 Women 440 470 At the 5% level of significance, can you conclude that gender and owning a cell phone are related for all adults? 19 QM-220, M. Zainal
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