{"id":2871,"date":"2022-05-09T15:18:57","date_gmt":"2022-05-09T19:18:57","guid":{"rendered":"https:\/\/openbooks.macewan.ca\/rcommander\/?post_type=chapter&p=2871"},"modified":"2025-05-07T18:10:54","modified_gmt":"2025-05-07T22:10:54","slug":"12-6-review-questions","status":"publish","type":"chapter","link":"https:\/\/openbooks.macewan.ca\/introstats\/chapter\/12-6-review-questions\/","title":{"raw":"12.6 Review Questions","rendered":"12.6 Review Questions"},"content":{"raw":"
    \r\n \t
  1. The data on monthly rents, in dollars, for independent random samples of newly completed apartments in the four U.S. regions are presented in the following table.\r\n
    \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    Northeast<\/th>\r\nMidwest<\/th>\r\nSouth<\/th>\r\nWest<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
    1005<\/td>\r\n870<\/td>\r\n891<\/td>\r\n1025<\/td>\r\n<\/tr>\r\n
    898<\/td>\r\n748<\/td>\r\n630<\/td>\r\n1012<\/td>\r\n<\/tr>\r\n
    948<\/td>\r\n699<\/td>\r\n861<\/td>\r\n1090<\/td>\r\n<\/tr>\r\n
    1181<\/td>\r\n814<\/td>\r\n1036<\/td>\r\n926<\/td>\r\n<\/tr>\r\n
    1244<\/td>\r\n721<\/td>\r\n<\/td>\r\n1269<\/td>\r\n<\/tr>\r\n
    <\/td>\r\n606<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nGiven the ANOVA table, test at the 5% significance level whether a difference exists in the mean rent of newly completed apartments in the four U.S. regions.\r\n
    \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    Source<\/th>\r\ndf<\/em><\/th>\r\nSS<\/em><\/th>\r\nMS<\/em> = [latex]\\frac{SS}{df}[\/latex]<\/th>\r\nF-statistic<\/th>\r\nP<\/em>-value<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
    Region<\/th>\r\n3<\/td>\r\nSSTR<\/em> = 400513<\/td>\r\nMSTR<\/em> = 133504<\/td>\r\n[latex]F_o[\/latex] = 7.541<\/td>\r\n0.0023<\/td>\r\n<\/tr>\r\n
    Error<\/th>\r\n16<\/td>\r\nSSE<\/em> = 283265<\/td>\r\nMSE<\/em> = 17704<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
    Total<\/th>\r\n19<\/td>\r\nSST<\/em> = 683778<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div><\/li>\r\n \t
  2. The following table gives the salaries (in thousand dollars) for computer science (CS) majors obtaining a bachelor\u2019s degree, a master\u2019s degree, or a Ph.D.\r\n
    \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    Bachelor<\/th>\r\nMaster<\/th>\r\nPhD<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
    50.8<\/td>\r\n65.8<\/td>\r\n73.3<\/td>\r\n<\/tr>\r\n
    59.4<\/td>\r\n57.5<\/td>\r\n65.7<\/td>\r\n<\/tr>\r\n
    55.9<\/td>\r\n66.9<\/td>\r\n71.7<\/td>\r\n<\/tr>\r\n
    45.1<\/td>\r\n62.8<\/td>\r\n72.5<\/td>\r\n<\/tr>\r\n
    54.1<\/td>\r\n68.5<\/td>\r\n73<\/td>\r\n<\/tr>\r\n
    50.7<\/td>\r\n69.3<\/td>\r\n67.2<\/td>\r\n<\/tr>\r\n
    46.8<\/td>\r\n61.5<\/td>\r\n67.5<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n
      \r\n \t
    1. Fill in missing entries of the following ANOVA table.\r\n
      \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      Source<\/th>\r\ndf<\/em><\/th>\r\nSS<\/em><\/th>\r\nMS<\/em> = [latex]\\frac{SS}{df}[\/latex]<\/th>\r\nF-statistic<\/th>\r\nP<\/em>-value<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
      Group<\/th>\r\nk<\/em>-1 = ?<\/td>\r\nSSTR<\/em> = ?<\/td>\r\n[latex]\\text{MSTR} = \\frac{SSTR}{k-1} = 616.9[\/latex]<\/td>\r\n[latex]F_o = \\frac{MSTR}{MSE} = 34.62[\/latex]<\/td>\r\n<0.0001<\/td>\r\n<\/tr>\r\n
      Error<\/th>\r\nn<\/em>-k<\/em> = ?<\/td>\r\nSSE<\/em> = ?<\/td>\r\n[latex]\\text{MSE} = \\frac{SSE}{n-k} = 17.8[\/latex]<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
      Total<\/th>\r\nn<\/em>-1 = 20<\/td>\r\nSST<\/em> = 1554.5<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div><\/li>\r\n \t
    2. Test at the 1% significance level whether a difference exists in mean salary for computer science (CS) majors obtaining a bachelor\u2019s degree, a master\u2019s degree, or a Ph.D.<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n
      Show\/Hide Answer<\/summary>\r\n
        \r\n \t
      1. From the first table, we have [latex]k=4, n_1=5, n_2=6, n_3=4, n_4=5 \\longrightarrow n=n_1+n_2+n_3+n_4=20[\/latex].\r\nWe assume the assumptions of the one-way ANOVA F test are satisfied. Steps of one-way ANOVA F test:\r\nStep 1: Hypotheses.\r\n[latex]H_0:[\/latex] all means are equal, [latex]\\mu_1=\\mu_2=\\mu_3=\\mu_4[\/latex]versus [latex]H_a:[\/latex]\u00a0not all means are equal, i.e., at least one pair of means are different.\r\nStep 2: Significance level [latex]\\alpha=0.05.[\/latex]\r\nStep 3: Test statistic [latex]F_o[\/latex]=7.541 with degrees of freedom [latex]df_{TR}=k-1=4-1=3, df_E=n-k=20-4=16.[\/latex]\r\nStep 4: P-value=0.0023.\r\nStep 5: Decision. Reject [latex]H_0[\/latex] since P-value=0.0023 <0.01 [latex](\\alpha)[\/latex].\r\nStep 6: Conclusion. At the 5% significance level, we have sufficient evidence that a difference exists in the mean rent of newly completed apartments in the four U.S. regions.<\/li>\r\n \t
      2. \u200c\r\n
          \r\n \t
        1. \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
          Source<\/th>\r\ndf<\/span><\/em><\/th>\r\nSS<\/span><\/em><\/th>\r\nMS<\/em>=[latex]\\frac{SS}{df}[\/latex]<\/span><\/th>\r\nF-statistic<\/th>\r\nP<\/span><\/em>-value<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
          <\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
          Group<\/th>\r\nk-1=?<\/span><\/td>\r\nSSTR=?<\/span><\/td>\r\nMSTR=[latex]\\frac{SSTR}{k-1}=616.9[\/latex]<\/span><\/td>\r\n[latex]F_o=\\frac{MSTR}{MSE}=34.62[\/latex]<\/span><\/td>\r\n<0.0001<\/span><\/td>\r\n<\/tr>\r\n
          <\/th>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
          Error<\/th>\r\nn-k=?<\/span><\/td>\r\nSSE=?<\/span><\/td>\r\nMSE=[latex]\\frac{SSE}{n-k}=17.8[\/latex]<\/span><\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
          <\/th>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
          Total<\/th>\r\nn-1=20<\/span><\/td>\r\nSST=1554.5<\/span><\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

          From the first table, we have [latex]k=3, n_1=n_2=n_3=7 \\Longrightarrow n=n_1+n_2+n_3=21[\/latex].\r\nFrom the second table, we have [latex]df_{T}=n-1=19, SST=1554.5, MSTR=616.0, MSE=17.8[\/latex].\r\n(1) [latex]df_{TR}=k-1=3-1=2.[\/latex]<\/p>\r\n

          (2) [latex]df_E=n-k=21-3=18[\/latex] or [latex]df_E=df_T-df_{TR}=19-2=17.[\/latex]<\/p>\r\n

          (3) [latex]SSTR=MSTR\\times df_{TR}=616.9\\times 2=1233.8.[\/latex]<\/p>\r\n

          (4) [latex]SSE=MSE \\times df_E=17.8 \\times 18=320.4[\/latex] or [latex]SSE=SST-SSTR=1554.5-1233.8=320.7.[\/latex] The difference is due to rounding.<\/p>\r\n<\/li>\r\n \t

        2. We assume the assumptions of the one-way ANOVA F test are satisfied.\r\nSteps of one-way ANOVA F test:\r\n<\/span>Step 1: hypotheses.<\/span>[latex]H_0:[\/latex] all means are equal, [latex]\\mu_1=\\mu_2=\\mu_3[\/latex] versus [latex]H_a:[\/latex]\u00a0not all means are equal, i.e., at least one pair of means are different.\r\nStep 2: Significance level [latex]\\alpha=0.01.[\/latex]\r\nStep 3: Test statistic [latex]F_o=34.62[\/latex] with degrees of freedom [latex]df_{TR}=k-1=3-1=2, df_E=n-k=21-3=18.[\/latex]\r\nStep 4: P-value <0.0001.\r\nStep 5: Decision. Reject [latex]H_0[\/latex] since P-value<0.0001 <0.01 [latex](\\alpha)[\/latex].\r\nStep 6: Conclusion. At the 1% significance level, we have sufficient evidence that a difference exists in mean salary for computer science (CS) majors obtaining a bachelor\u2019s degree, a master\u2019s degree or a Ph.D.<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/details>","rendered":"
            \n
          1. The data on monthly rents, in dollars, for independent random samples of newly completed apartments in the four U.S. regions are presented in the following table.\n
            \n\n\n\n\n\n\n\n\n\n\n
            Northeast<\/th>\nMidwest<\/th>\nSouth<\/th>\nWest<\/th>\n<\/tr>\n<\/thead>\n
            1005<\/td>\n870<\/td>\n891<\/td>\n1025<\/td>\n<\/tr>\n
            898<\/td>\n748<\/td>\n630<\/td>\n1012<\/td>\n<\/tr>\n
            948<\/td>\n699<\/td>\n861<\/td>\n1090<\/td>\n<\/tr>\n
            1181<\/td>\n814<\/td>\n1036<\/td>\n926<\/td>\n<\/tr>\n
            1244<\/td>\n721<\/td>\n<\/td>\n1269<\/td>\n<\/tr>\n
            <\/td>\n606<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

            Given the ANOVA table, test at the 5% significance level whether a difference exists in the mean rent of newly completed apartments in the four U.S. regions.<\/p>\n

            \n\n\n\n\n\n\n\n
            Source<\/th>\ndf<\/em><\/th>\nSS<\/em><\/th>\nMS<\/em> = [latex]\\frac{SS}{df}[\/latex]<\/th>\nF-statistic<\/th>\nP<\/em>-value<\/th>\n<\/tr>\n<\/thead>\n
            Region<\/th>\n3<\/td>\nSSTR<\/em> = 400513<\/td>\nMSTR<\/em> = 133504<\/td>\n[latex]F_o[\/latex] = 7.541<\/td>\n0.0023<\/td>\n<\/tr>\n
            Error<\/th>\n16<\/td>\nSSE<\/em> = 283265<\/td>\nMSE<\/em> = 17704<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
            Total<\/th>\n19<\/td>\nSST<\/em> = 683778<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/li>\n
          2. The following table gives the salaries (in thousand dollars) for computer science (CS) majors obtaining a bachelor\u2019s degree, a master\u2019s degree, or a Ph.D.\n
            \n\n\n\n\n\n\n\n\n\n\n\n
            Bachelor<\/th>\nMaster<\/th>\nPhD<\/th>\n<\/tr>\n<\/thead>\n
            50.8<\/td>\n65.8<\/td>\n73.3<\/td>\n<\/tr>\n
            59.4<\/td>\n57.5<\/td>\n65.7<\/td>\n<\/tr>\n
            55.9<\/td>\n66.9<\/td>\n71.7<\/td>\n<\/tr>\n
            45.1<\/td>\n62.8<\/td>\n72.5<\/td>\n<\/tr>\n
            54.1<\/td>\n68.5<\/td>\n73<\/td>\n<\/tr>\n
            50.7<\/td>\n69.3<\/td>\n67.2<\/td>\n<\/tr>\n
            46.8<\/td>\n61.5<\/td>\n67.5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n
              \n
            1. Fill in missing entries of the following ANOVA table.\n
              \n\n\n\n\n\n\n\n
              Source<\/th>\ndf<\/em><\/th>\nSS<\/em><\/th>\nMS<\/em> = [latex]\\frac{SS}{df}[\/latex]<\/th>\nF-statistic<\/th>\nP<\/em>-value<\/th>\n<\/tr>\n<\/thead>\n
              Group<\/th>\nk<\/em>-1 = ?<\/td>\nSSTR<\/em> = ?<\/td>\n[latex]\\text{MSTR} = \\frac{SSTR}{k-1} = 616.9[\/latex]<\/td>\n[latex]F_o = \\frac{MSTR}{MSE} = 34.62[\/latex]<\/td>\n<0.0001<\/td>\n<\/tr>\n
              Error<\/th>\nn<\/em>–k<\/em> = ?<\/td>\nSSE<\/em> = ?<\/td>\n[latex]\\text{MSE} = \\frac{SSE}{n-k} = 17.8[\/latex]<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
              Total<\/th>\nn<\/em>-1 = 20<\/td>\nSST<\/em> = 1554.5<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/li>\n
            2. Test at the 1% significance level whether a difference exists in mean salary for computer science (CS) majors obtaining a bachelor\u2019s degree, a master\u2019s degree, or a Ph.D.<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n
              \nShow\/Hide Answer<\/summary>\n
                \n
              1. From the first table, we have [latex]k=4, n_1=5, n_2=6, n_3=4, n_4=5 \\longrightarrow n=n_1+n_2+n_3+n_4=20[\/latex].
                \nWe assume the assumptions of the one-way ANOVA F test are satisfied. Steps of one-way ANOVA F test:
                \nStep 1: Hypotheses.
                \n[latex]H_0:[\/latex] all means are equal, [latex]\\mu_1=\\mu_2=\\mu_3=\\mu_4[\/latex]versus [latex]H_a:[\/latex]\u00a0not all means are equal, i.e., at least one pair of means are different.
                \nStep 2: Significance level [latex]\\alpha=0.05.[\/latex]
                \nStep 3: Test statistic [latex]F_o[\/latex]=7.541 with degrees of freedom [latex]df_{TR}=k-1=4-1=3, df_E=n-k=20-4=16.[\/latex]
                \nStep 4: P-value=0.0023.
                \nStep 5: Decision. Reject [latex]H_0[\/latex] since P-value=0.0023 <0.01 [latex](\\alpha)[\/latex].
                \nStep 6: Conclusion. At the 5% significance level, we have sufficient evidence that a difference exists in the mean rent of newly completed apartments in the four U.S. regions.<\/li>\n
              2. \u200c\n
                  \n
                1. \n\n\n\n\n\n\n\n\n\n\n
                  Source<\/th>\ndf<\/span><\/em><\/th>\nSS<\/span><\/em><\/th>\nMS<\/em>=[latex]\\frac{SS}{df}[\/latex]<\/span><\/th>\nF-statistic<\/th>\nP<\/span><\/em>-value<\/th>\n<\/tr>\n<\/thead>\n
                  <\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
                  Group<\/th>\nk-1=?<\/span><\/td>\nSSTR=?<\/span><\/td>\nMSTR=[latex]\\frac{SSTR}{k-1}=616.9[\/latex]<\/span><\/td>\n[latex]F_o=\\frac{MSTR}{MSE}=34.62[\/latex]<\/span><\/td>\n<0.0001<\/span><\/td>\n<\/tr>\n
                  <\/th>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
                  Error<\/th>\nn-k=?<\/span><\/td>\nSSE=?<\/span><\/td>\nMSE=[latex]\\frac{SSE}{n-k}=17.8[\/latex]<\/span><\/td>\n<\/td>\n<\/td>\n<\/tr>\n
                  <\/th>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
                  Total<\/th>\nn-1=20<\/span><\/td>\nSST=1554.5<\/span><\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                  From the first table, we have [latex]k=3, n_1=n_2=n_3=7 \\Longrightarrow n=n_1+n_2+n_3=21[\/latex].
                  \nFrom the second table, we have [latex]df_{T}=n-1=19, SST=1554.5, MSTR=616.0, MSE=17.8[\/latex].
                  \n(1) [latex]df_{TR}=k-1=3-1=2.[\/latex]<\/p>\n

                  (2) [latex]df_E=n-k=21-3=18[\/latex] or [latex]df_E=df_T-df_{TR}=19-2=17.[\/latex]<\/p>\n

                  (3) [latex]SSTR=MSTR\\times df_{TR}=616.9\\times 2=1233.8.[\/latex]<\/p>\n

                  (4) [latex]SSE=MSE \\times df_E=17.8 \\times 18=320.4[\/latex] or [latex]SSE=SST-SSTR=1554.5-1233.8=320.7.[\/latex] The difference is due to rounding.<\/p>\n<\/li>\n

                2. We assume the assumptions of the one-way ANOVA F test are satisfied.
                  \nSteps of one-way ANOVA F test:
                  \n<\/span>Step 1: hypotheses.<\/span>[latex]H_0:[\/latex] all means are equal, [latex]\\mu_1=\\mu_2=\\mu_3[\/latex] versus [latex]H_a:[\/latex]\u00a0not all means are equal, i.e., at least one pair of means are different.
                  \nStep 2: Significance level [latex]\\alpha=0.01.[\/latex]
                  \nStep 3: Test statistic [latex]F_o=34.62[\/latex] with degrees of freedom [latex]df_{TR}=k-1=3-1=2, df_E=n-k=21-3=18.[\/latex]
                  \nStep 4: P-value <0.0001.
                  \nStep 5: Decision. Reject [latex]H_0[\/latex] since P-value<0.0001 <0.01 [latex](\\alpha)[\/latex].
                  \nStep 6: Conclusion. At the 1% significance level, we have sufficient evidence that a difference exists in mean salary for computer science (CS) majors obtaining a bachelor\u2019s degree, a master\u2019s degree or a Ph.D.<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/details>\n","protected":false},"author":19,"menu_order":6,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2871","chapter","type-chapter","status-publish","hentry"],"part":1189,"_links":{"self":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/2871","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/users\/19"}],"version-history":[{"count":69,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/2871\/revisions"}],"predecessor-version":[{"id":5527,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/2871\/revisions\/5527"}],"part":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/parts\/1189"}],"metadata":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/2871\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/media?parent=2871"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapter-type?post=2871"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/contributor?post=2871"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/license?post=2871"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}