{"id":5385,"date":"2022-08-20T21:23:23","date_gmt":"2022-08-20T21:23:23","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/?post_type=chapter&p=5385"},"modified":"2022-08-20T21:24:48","modified_gmt":"2022-08-20T21:24:48","slug":"11f-in-class-activity","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/11f-in-class-activity\/","title":{"raw":"11F In-Class Activity","rendered":"11F In-Class Activity"},"content":{"raw":"In 2004, two University of Chicago economists (Marianne Bertrand and Sendhil Mullainathan) decided to conduct an experiment[footnote]Bertrand, M. & Mullainathan, S. (2003, July). Are Emily and Greg more employable than Lakisha and Jamal? A field experiment on labor market discrimination<\/em>. National Bureau of Economic Research. https:\/\/www.nber.org\/papers\/w9873 [\/footnote] to test for labor market discrimination.\r\n\r\nThe investigators created 4,890 mock identical resum\u00e9s, which were sent to job placement ads in Chicago and Boston. To gauge market racial discrimination, each resum\u00e9 was randomly assigned either a commonly-white or commonly-black name. The experimenters then measured the proportion of resum\u00e9s from each group (white and black) that received callbacks.[footnote]Lesson adapted from Skew The Script<\/em>. https:\/\/skewthescript.org\/7-8 [\/footnote]\r\n\r\n\"\"\r\n\r\nCredit: iStock\/Mitoria\r\n
\r\n

Question 1<\/h3>\r\nConducting this experiment took a lot of work and resources. Why didn\u2019t the investigators just compare observed black and white wages in current data? Why do you believe they went through all the trouble of conducting this experiment?\r\n\r\n<\/div>\r\nQuestions 2\u20136: These questions reference the study results, which are summarized in the following table:\r\n
\r\n\r\n\r\n\r\n\r\n\r\n\r\n
<\/td>\r\nCommonly-White Names<\/td>\r\nCommonly-Black Names<\/td>\r\nTotal<\/td>\r\n<\/tr>\r\n
Called back<\/td>\r\n246<\/td>\r\n164<\/td>\r\n410<\/td>\r\n<\/tr>\r\n
Not called back<\/td>\r\n2,199<\/td>\r\n2,281<\/td>\r\n4,480<\/td>\r\n<\/tr>\r\n
Total<\/td>\r\n2,445<\/td>\r\n2,445<\/td>\r\n4,890<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n[latex] p_{1}[\/latex] = true proportion of commonly-white-named resum\u00e9s that received callbacks\r\n\r\n[latex] p_{2}[\/latex]\u00a0= true proportion of commonly-black-named resum\u00e9s that received callbacks\r\n
\r\n

Question 2<\/h3>\r\nBefore conducting a test, let\u2019s explore the results.\r\n
    \r\n \t
  1. Part A: Calculate\u00a0 and\u00a0 (the sample proportions of white\/black-named resum\u00e9s that received callbacks, respectively).<\/li>\r\n \t
  2. Part B: Are these two sample proportions different enough to show that the\u00a0 difference didn\u2019t occur by chance alone? Justify using basic reasoning\u00a0 (without conducting a hypothesis test).<\/li>\r\n \t
  3. Part C: If the sample sizes were only 20 resum\u00e9s per group, would you be more\u00a0 likely to believe the difference in proportions could have occurred by chance\u00a0 alone? Explain.<\/li>\r\n<\/ol>\r\n<\/div>\r\n
    \r\n

    Question 3<\/h3>\r\nIf we wanted to test if the difference in callback rates is statistically significant,\u00a0 should we use a one-sample or two-sample inference procedure? Explain.\r\n\r\n<\/div>\r\n
    \r\n

    Question 4<\/h3>\r\nWe will now conduct a two-sample z-test of proportions. Here are the hypotheses for\u00a0 this test:\r\n\r\n*MISSING LaTeX*\r\n
      \r\n \t
    1. Part A: In which hypothesis are the callback rates between the groups equal? In\u00a0 which hypothesis could the commonly-white names get higher callback\u00a0 rates?<\/li>\r\n \t
    2. Part B: What is the null hypothesis value of the difference in proportions?<\/li>\r\n<\/ol>\r\n<\/div>\r\n
      \r\n

      Question 5<\/h3>\r\nTo ensure our inferences are accurate, we must check certain conditions for\u00a0 conducting a two-sample z-test of proportions:\r\n
        \r\n \t
      1. Part A: Confirm that the data were collected via a random sample or an experiment\u00a0 with random assignment to treatment. This would allow us to propose that\u00a0 we have comparable experimental groups.<\/li>\r\n \t
      2. Part B: Confirm that we have a large enough sample size to meet the \u201csample size\u201d\u00a0 condition. This will help ensure that the underlying sampling distribution we\u00a0 use to calculate the P-value can be modeled with a normal curve. The\u00a0 conditions are:\u00a0\"\"<\/li>\r\n<\/ol>\r\nTo calculate the combined (\u201cpooled\u201d) sample proportion, add the total\u00a0 number of candidates that received callbacks in both groups and divide this\u00a0 sum by the total number of people in the study:\r\n\r\n[latex] \\hat{p_{c}} \\frac{x_{1}+x_{2}}{n_{1}+n_{2}},\r\n\r\nwhere\u00a0[latex] x_{1} [\/latex] and\u00a0[latex] x_{2} [\/latex] are the number of \u201csuccesses\u201d from Groups 1 and 2, respectively.\r\n\r\n<\/div>\r\nNote: The final condition is that the sample sizes are each less than a tenth of the size\u00a0 of the populations from which they\u2019re drawn [[latex] n_{1} < 0.10(N_{1}) [\/latex] and [latex] n_{2} < 0.10(N_{2}) [\/latex]]. This\u00a0 helps ensure our estimates for the standard errors are accurate. However, this condition\u00a0 does not need to be checked in the case of a randomized experiment.\r\n
        \r\n\r\nConditions for Two-Sample Z-Test of Proportions\r\n
          \r\n \t
        1. Large Counts: Check that *MISSING LATEX*<\/li>\r\n \t
        2. Random Samples\/Assignment: Check that the two samples\u00a0 are independent and random samples or that they come from\u00a0 randomly assigned groups in an experiment.<\/li>\r\n \t
        3. 10%: Check that *MISSING LATEX*.<\/li>\r\n<\/ol>\r\n<\/div>\r\n
          \r\n

          Question 6<\/h3>\r\nGo to the DCMP Compare Two Population Proportions tool at\r\n\r\nhttps:\/\/dcmathpathways.shinyapps.io\/2sample_prop\/. Select the \u201cNumber of\u00a0 successes\u201d option and input the relevant data from the study. Under the type of\u00a0 inference section, select \u201cSignificance Test\u201d and the appropriate alternative\u00a0 hypothesis.\r\n
            \r\n \t
          1. Part A: Interpret the \u201cobserved difference\u201d value. How was this calculated and what\u00a0 does it mean?<\/li>\r\n \t
          2. Part B: State and interpret the z-test statistic value.<\/li>\r\n \t
          3. Part C: State and interpret the P-value.<\/li>\r\n \t
          4. Part D: Using the previous information, draw a conclusion for this test. State what\u00a0 your conclusion means in the context of the study. Use significance level\u00a0[latex] \\alpha =0.05 [\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n
            \r\n

            Question 7<\/h3>\r\nWe found that the difference between callback rates was statistically significant, but\u00a0 is it practically significant? Justify your answer using the observed difference in\u00a0 sample proportions.\r\n\r\n<\/div>\r\n \r\n\r\n ","rendered":"

            In 2004, two University of Chicago economists (Marianne Bertrand and Sendhil Mullainathan) decided to conduct an experiment[1]<\/sup><\/a> to test for labor market discrimination.<\/p>\n

            The investigators created 4,890 mock identical resum\u00e9s, which were sent to job placement ads in Chicago and Boston. To gauge market racial discrimination, each resum\u00e9 was randomly assigned either a commonly-white or commonly-black name. The experimenters then measured the proportion of resum\u00e9s from each group (white and black) that received callbacks.[2]<\/sup><\/a><\/p>\n

            \"\"<\/p>\n

            Credit: iStock\/Mitoria<\/p>\n

            \n

            Question 1<\/h3>\n

            Conducting this experiment took a lot of work and resources. Why didn\u2019t the investigators just compare observed black and white wages in current data? Why do you believe they went through all the trouble of conducting this experiment?<\/p>\n<\/div>\n

            Questions 2\u20136: These questions reference the study results, which are summarized in the following table:<\/p>\n

            \n\n\n\n\n\n\n
            <\/td>\nCommonly-White Names<\/td>\nCommonly-Black Names<\/td>\nTotal<\/td>\n<\/tr>\n
            Called back<\/td>\n246<\/td>\n164<\/td>\n410<\/td>\n<\/tr>\n
            Not called back<\/td>\n2,199<\/td>\n2,281<\/td>\n4,480<\/td>\n<\/tr>\n
            Total<\/td>\n2,445<\/td>\n2,445<\/td>\n4,890<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

            [latex]p_{1}[\/latex] = true proportion of commonly-white-named resum\u00e9s that received callbacks<\/p>\n

            [latex]p_{2}[\/latex]\u00a0= true proportion of commonly-black-named resum\u00e9s that received callbacks<\/p>\n

            \n

            Question 2<\/h3>\n

            Before conducting a test, let\u2019s explore the results.<\/p>\n

              \n
            1. Part A: Calculate\u00a0 and\u00a0 (the sample proportions of white\/black-named resum\u00e9s that received callbacks, respectively).<\/li>\n
            2. Part B: Are these two sample proportions different enough to show that the\u00a0 difference didn\u2019t occur by chance alone? Justify using basic reasoning\u00a0 (without conducting a hypothesis test).<\/li>\n
            3. Part C: If the sample sizes were only 20 resum\u00e9s per group, would you be more\u00a0 likely to believe the difference in proportions could have occurred by chance\u00a0 alone? Explain.<\/li>\n<\/ol>\n<\/div>\n
              \n

              Question 3<\/h3>\n

              If we wanted to test if the difference in callback rates is statistically significant,\u00a0 should we use a one-sample or two-sample inference procedure? Explain.<\/p>\n<\/div>\n

              \n

              Question 4<\/h3>\n

              We will now conduct a two-sample z-test of proportions. Here are the hypotheses for\u00a0 this test:<\/p>\n

              *MISSING LaTeX*<\/p>\n

                \n
              1. Part A: In which hypothesis are the callback rates between the groups equal? In\u00a0 which hypothesis could the commonly-white names get higher callback\u00a0 rates?<\/li>\n
              2. Part B: What is the null hypothesis value of the difference in proportions?<\/li>\n<\/ol>\n<\/div>\n
                \n

                Question 5<\/h3>\n

                To ensure our inferences are accurate, we must check certain conditions for\u00a0 conducting a two-sample z-test of proportions:<\/p>\n

                  \n
                1. Part A: Confirm that the data were collected via a random sample or an experiment\u00a0 with random assignment to treatment. This would allow us to propose that\u00a0 we have comparable experimental groups.<\/li>\n
                2. Part B: Confirm that we have a large enough sample size to meet the \u201csample size\u201d\u00a0 condition. This will help ensure that the underlying sampling distribution we\u00a0 use to calculate the P-value can be modeled with a normal curve. The\u00a0 conditions are:\u00a0\"\"<\/li>\n<\/ol>\n

                  To calculate the combined (\u201cpooled\u201d) sample proportion, add the total\u00a0 number of candidates that received callbacks in both groups and divide this\u00a0 sum by the total number of people in the study:<\/p>\n

                  [latex]\\hat{p_{c}} \\frac{x_{1}+x_{2}}{n_{1}+n_{2}}, where\u00a0[latex] x_{1}[\/latex] and\u00a0[latex]x_{2}[\/latex] are the number of \u201csuccesses\u201d from Groups 1 and 2, respectively.<\/p>\n<\/div>\n

                  Note: The final condition is that the sample sizes are each less than a tenth of the size\u00a0 of the populations from which they\u2019re drawn [[latex]n_{1} < 0.10(N_{1})[\/latex] and [latex]n_{2} < 0.10(N_{2})[\/latex]]. This\u00a0 helps ensure our estimates for the standard errors are accurate. However, this condition\u00a0 does not need to be checked in the case of a randomized experiment.\n\n\n

                  \n

                  Conditions for Two-Sample Z-Test of Proportions<\/p>\n

                    \n
                  1. Large Counts: Check that *MISSING LATEX*<\/li>\n
                  2. Random Samples\/Assignment: Check that the two samples\u00a0 are independent and random samples or that they come from\u00a0 randomly assigned groups in an experiment.<\/li>\n
                  3. 10%: Check that *MISSING LATEX*.<\/li>\n<\/ol>\n<\/div>\n
                    \n

                    Question 6<\/h3>\n

                    Go to the DCMP Compare Two Population Proportions tool at<\/p>\n

                    https:\/\/dcmathpathways.shinyapps.io\/2sample_prop\/. Select the \u201cNumber of\u00a0 successes\u201d option and input the relevant data from the study. Under the type of\u00a0 inference section, select \u201cSignificance Test\u201d and the appropriate alternative\u00a0 hypothesis.<\/p>\n

                      \n
                    1. Part A: Interpret the \u201cobserved difference\u201d value. How was this calculated and what\u00a0 does it mean?<\/li>\n
                    2. Part B: State and interpret the z-test statistic value.<\/li>\n
                    3. Part C: State and interpret the P-value.<\/li>\n
                    4. Part D: Using the previous information, draw a conclusion for this test. State what\u00a0 your conclusion means in the context of the study. Use significance level\u00a0[latex]\\alpha =0.05[\/latex].<\/li>\n<\/ol>\n<\/div>\n
                      \n

                      Question 7<\/h3>\n

                      We found that the difference between callback rates was statistically significant, but\u00a0 is it practically significant? Justify your answer using the observed difference in\u00a0 sample proportions.<\/p>\n<\/div>\n

                       <\/p>\n

                       <\/p>\n

                      1. Bertrand, M. & Mullainathan, S. (2003, July). Are Emily and Greg more employable than Lakisha and Jamal? A field experiment on labor market discrimination<\/em>. National Bureau of Economic Research. https:\/\/www.nber.org\/papers\/w9873 ↵<\/a><\/li>
                      2. Lesson adapted from Skew The Script<\/em>. https:\/\/skewthescript.org\/7-8 ↵<\/a><\/li><\/ol><\/div>","protected":false},"author":574340,"menu_order":17,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-5385","chapter","type-chapter","status-publish","hentry"],"part":5305,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5385","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/users\/574340"}],"version-history":[{"count":3,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5385\/revisions"}],"predecessor-version":[{"id":5391,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5385\/revisions\/5391"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/parts\/5305"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5385\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=5385"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=5385"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=5385"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=5385"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}