{"id":2186,"date":"2021-10-05T18:34:24","date_gmt":"2021-10-05T18:34:24","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=2186"},"modified":"2022-04-25T14:28:37","modified_gmt":"2022-04-25T14:28:37","slug":"writing-comparisons-between-two-population-parameters","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/writing-comparisons-between-two-population-parameters\/","title":{"raw":"Writing Comparisons Between Two Population Parameters","rendered":"Writing Comparisons Between Two Population Parameters"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Express verbal comparisons of two population parameters as mathematical expressions<\/li>\r\n<\/ul>\r\n<\/div>\r\nSometimes we are interested in comparing two population parameters. For example, a researcher may want to test whether a drug is effective in lowering blood pressure. Participants in a drug trial are randomly assigned to treatment and control groups. The treatment group will take the new drug for six weeks, while the control group will receive standard care. The blood pressures of participants will be recorded at the end of six weeks and the means for the two groups will be compared.\r\n\r\nThe researcher\u2019s initial hypothesis might be stated: the mean blood pressure for the treatment group is lower than the mean blood pressure for the control group. In symbols, we might write\r\n<p style=\"text-align: center;\">[latex]\\large\\mu _\\mathrm{treatment} &lt; \\mu _\\mathrm{control}[\/latex].<\/p>\r\nWe previously saw the connection between verbal statements about the relationship between a variable <em>x<\/em> and a constant <em>k<\/em>\u00a0and the corresponding mathematical statement.\r\n<div align=\"left\">\r\n<table style=\"border-collapse: collapse; width: 100%; height: 294px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 28px;\">\r\n<td style=\"width: 50%; height: 28px; text-align: center;\">Verbal statement\r\n<em>x<\/em> is ...<\/td>\r\n<td style=\"width: 50%; height: 28px; text-align: center;\">Mathematical Statement<\/td>\r\n<\/tr>\r\n<tr style=\"height: 42px;\">\r\n<td style=\"width: 50%; height: 42px; text-align: left;\">greater than or equal to <em>k<\/em>\r\nat least <em>k<\/em>\r\nnot less than <em>k<\/em><\/td>\r\n<td style=\"width: 50%; height: 42px; text-align: center;\">[latex]x \\geq k[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 42px;\">\r\n<td style=\"width: 50%; height: 42px; text-align: left;\">less than or equal to <em>k<\/em>\r\nat most <em>k<\/em>\r\nnot more than <em>k<\/em><\/td>\r\n<td style=\"width: 50%; height: 42px; text-align: center;\">[latex]x \\leq k[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 42px;\">\r\n<td style=\"width: 50%; height: 42px; text-align: left;\">less than <em>k<\/em>\r\nbelow <em>k<\/em>\r\nfewer than <em>k<\/em><\/td>\r\n<td style=\"width: 50%; height: 42px; text-align: center;\">[latex]x &lt; k[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 42px;\">\r\n<td style=\"width: 50%; height: 42px; text-align: left;\">greater than <em>k<\/em>\r\nmore than <em>k<\/em>\r\nabove <em>k<\/em><\/td>\r\n<td style=\"width: 50%; height: 42px; text-align: center;\">[latex]x &gt; k[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 56px;\">\r\n<td style=\"width: 50%; height: 56px; text-align: left;\">equal to <em>k<\/em>\r\nis <em>k<\/em>\r\nexactly <em>k<\/em>\r\nthe same as <em>k<\/em><\/td>\r\n<td style=\"width: 50%; height: 56px; text-align: center;\">[latex]x=k[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 42px;\">\r\n<td style=\"width: 50%; height: 42px; text-align: left;\">not equal to <em>k<\/em>\r\nnot <em>k<\/em>\r\ndifferent from <em>k<\/em><\/td>\r\n<td style=\"width: 50%; height: 42px; text-align: center;\">[latex]x \\neq k[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe can apply the same principles to comparisons involving two population parameters.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<table style=\"border-collapse: collapse; width: 100%; height: 60px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px; text-align: center;\">Verbal Statement<\/td>\r\n<td style=\"width: 50%; height: 12px; text-align: center;\">Mathematical Statement<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">Proportion A is at least proportion B.<\/td>\r\n<td style=\"width: 50%; height: 12px; text-align: center;\">[latex]p_A \\geq p_B[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">Mean 1 is no greater than mean 2.<\/td>\r\n<td style=\"width: 50%; height: 12px; text-align: center;\">[latex]\\mu _1 \\leq \\mu _2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">The average time for dogs and cats is the same.<\/td>\r\n<td style=\"width: 50%; height: 12px; text-align: center;\">[latex]\\mu _\\mathrm{dogs} = \\mu _\\mathrm{cats}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">The mean for college X is more than the mean for college Y.<\/td>\r\n<td style=\"width: 50%; height: 12px; text-align: center;\">[latex]\\mu _X &gt; \\mu _Y[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nWrite the following statement as a comparison between two population parameters.\r\n\r\nA researcher states the mean sodium content of soup from manufacturer A is no more than the mean sodium content of soup from manufacturer B.\r\n\r\n[reveal-answer q=\"321342\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"321342\"]\r\n\r\nSince the statement refers to two population means, we can represent the mean content of soup from manufacturer A as [latex]\\mu _A[\/latex]\u00a0and the mean content of soup from manufacturer B as [latex]\\mu _B[\/latex].\u00a0The comparison between these means is \u201cno more than,\u201d which is represented by [latex]\\leq[\/latex].\u00a0So we write the comparison mathematically as\r\n<p style=\"text-align: center;\">[latex]\\mu _A \\leq \\mu _B[\/latex].<\/p>\r\nThis can also be written\r\n<p style=\"text-align: center;\">[latex]\\mu _A - \\mu _B \\leq 0[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nWrite the following statement as a comparison between two population parameters.\r\n\r\nA marketing researcher states the percentage of Michigan residents that regularly watch professional football is 5 percentage points lower than the percentage of Wisconsin residents who regularly watch professional football.\r\n\r\n[reveal-answer q=\"320647\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"320647\"]\r\n\r\nSince the statement refers to two population proportions, we can represent the proportion of Michigan residents that regularly watch professional football as [latex]p_\\mathrm{Michigan}[\/latex]\u00a0and the proportion of Wisconsin residents that regularly watch professional football as [latex]p_\\mathrm{Wisconsin}[\/latex].\u00a0The comparison between these proportions \u201cis\u201d corresponds to [latex]=[\/latex].\u00a0Since Michigan\u2019s proportion is [latex]5%=0.05[\/latex] lower than Wisconsin\u2019s proportion, we write\r\n<p style=\"text-align: center;\">[latex]p_\\mathrm{Michigan} = p_\\mathrm{Wisconsin} - 0.05[\/latex].<\/p>\r\nThis can also be written as\r\n<p style=\"text-align: center;\">[latex]p_\\mathrm{Michigan} - p_\\mathrm{Wisconsin} = -0.05[\/latex]<\/p>\r\nor\r\n<p style=\"text-align: center;\">[latex]p_\\mathrm{Wisconsin} - p_\\mathrm{Michigan} = 0.05[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nWrite the following statement as a comparison between two population parameters.\r\n\r\nA physical therapist suggests that the average recovery time for patients is lower for those patients receiving massage therapy in addition to standard treatment as compared to patients receiving standard treatment alone.\r\n\r\n[reveal-answer q=\"817297\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"817297\"]\r\n\r\nSince the statement refers to two population means, we can represent the mean recovery time for patients who receive massage therapy as [latex]\\mu _\\mathrm{massage}[\/latex]\u00a0and the mean recovery time for patients who receive standard treatment only as [latex]\\mu _\\mathrm{standard}[\/latex].\u00a0The comparison between these means is \u201cless than,\u201d which is represented by [latex]&lt;[\/latex]. So we write the comparison mathematically as\r\n<p style=\"text-align: center;\">[latex]\\mu _\\mathrm{massage} &lt; \\mu _\\mathrm{standard}[\/latex].<\/p>\r\nThis can also be written as\r\n<p style=\"text-align: center;\">[latex]\\mu _\\mathrm{massage} - \\mu _\\mathrm{standard} &lt; 0[\/latex]<\/p>\r\nor\r\n<p style=\"text-align: center;\">[latex]\\mu_\\mathrm{standard} - \\mu _\\mathrm{massage} &gt; 0[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Express verbal comparisons of two population parameters as mathematical expressions<\/li>\n<\/ul>\n<\/div>\n<p>Sometimes we are interested in comparing two population parameters. For example, a researcher may want to test whether a drug is effective in lowering blood pressure. Participants in a drug trial are randomly assigned to treatment and control groups. The treatment group will take the new drug for six weeks, while the control group will receive standard care. The blood pressures of participants will be recorded at the end of six weeks and the means for the two groups will be compared.<\/p>\n<p>The researcher\u2019s initial hypothesis might be stated: the mean blood pressure for the treatment group is lower than the mean blood pressure for the control group. In symbols, we might write<\/p>\n<p style=\"text-align: center;\">[latex]\\large\\mu _\\mathrm{treatment} < \\mu _\\mathrm{control}[\/latex].<\/p>\n<p>We previously saw the connection between verbal statements about the relationship between a variable <em>x<\/em> and a constant <em>k<\/em>\u00a0and the corresponding mathematical statement.<\/p>\n<div style=\"text-align: left;\">\n<table style=\"border-collapse: collapse; width: 100%; height: 294px;\">\n<tbody>\n<tr style=\"height: 28px;\">\n<td style=\"width: 50%; height: 28px; text-align: center;\">Verbal statement<br \/>\n<em>x<\/em> is &#8230;<\/td>\n<td style=\"width: 50%; height: 28px; text-align: center;\">Mathematical Statement<\/td>\n<\/tr>\n<tr style=\"height: 42px;\">\n<td style=\"width: 50%; height: 42px; text-align: left;\">greater than or equal to <em>k<\/em><br \/>\nat least <em>k<\/em><br \/>\nnot less than <em>k<\/em><\/td>\n<td style=\"width: 50%; height: 42px; text-align: center;\">[latex]x \\geq k[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 42px;\">\n<td style=\"width: 50%; height: 42px; text-align: left;\">less than or equal to <em>k<\/em><br \/>\nat most <em>k<\/em><br \/>\nnot more than <em>k<\/em><\/td>\n<td style=\"width: 50%; height: 42px; text-align: center;\">[latex]x \\leq k[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 42px;\">\n<td style=\"width: 50%; height: 42px; text-align: left;\">less than <em>k<\/em><br \/>\nbelow <em>k<\/em><br \/>\nfewer than <em>k<\/em><\/td>\n<td style=\"width: 50%; height: 42px; text-align: center;\">[latex]x < k[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 42px;\">\n<td style=\"width: 50%; height: 42px; text-align: left;\">greater than <em>k<\/em><br \/>\nmore than <em>k<\/em><br \/>\nabove <em>k<\/em><\/td>\n<td style=\"width: 50%; height: 42px; text-align: center;\">[latex]x > k[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 56px;\">\n<td style=\"width: 50%; height: 56px; text-align: left;\">equal to <em>k<\/em><br \/>\nis <em>k<\/em><br \/>\nexactly <em>k<\/em><br \/>\nthe same as <em>k<\/em><\/td>\n<td style=\"width: 50%; height: 56px; text-align: center;\">[latex]x=k[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 42px;\">\n<td style=\"width: 50%; height: 42px; text-align: left;\">not equal to <em>k<\/em><br \/>\nnot <em>k<\/em><br \/>\ndifferent from <em>k<\/em><\/td>\n<td style=\"width: 50%; height: 42px; text-align: center;\">[latex]x \\neq k[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We can apply the same principles to comparisons involving two population parameters.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<table style=\"border-collapse: collapse; width: 100%; height: 60px;\">\n<tbody>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px; text-align: center;\">Verbal Statement<\/td>\n<td style=\"width: 50%; height: 12px; text-align: center;\">Mathematical Statement<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">Proportion A is at least proportion B.<\/td>\n<td style=\"width: 50%; height: 12px; text-align: center;\">[latex]p_A \\geq p_B[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">Mean 1 is no greater than mean 2.<\/td>\n<td style=\"width: 50%; height: 12px; text-align: center;\">[latex]\\mu _1 \\leq \\mu _2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">The average time for dogs and cats is the same.<\/td>\n<td style=\"width: 50%; height: 12px; text-align: center;\">[latex]\\mu _\\mathrm{dogs} = \\mu _\\mathrm{cats}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">The mean for college X is more than the mean for college Y.<\/td>\n<td style=\"width: 50%; height: 12px; text-align: center;\">[latex]\\mu _X > \\mu _Y[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Write the following statement as a comparison between two population parameters.<\/p>\n<p>A researcher states the mean sodium content of soup from manufacturer A is no more than the mean sodium content of soup from manufacturer B.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q321342\">Show Answer<\/span><\/p>\n<div id=\"q321342\" class=\"hidden-answer\" style=\"display: none\">\n<p>Since the statement refers to two population means, we can represent the mean content of soup from manufacturer A as [latex]\\mu _A[\/latex]\u00a0and the mean content of soup from manufacturer B as [latex]\\mu _B[\/latex].\u00a0The comparison between these means is \u201cno more than,\u201d which is represented by [latex]\\leq[\/latex].\u00a0So we write the comparison mathematically as<\/p>\n<p style=\"text-align: center;\">[latex]\\mu _A \\leq \\mu _B[\/latex].<\/p>\n<p>This can also be written<\/p>\n<p style=\"text-align: center;\">[latex]\\mu _A - \\mu _B \\leq 0[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Write the following statement as a comparison between two population parameters.<\/p>\n<p>A marketing researcher states the percentage of Michigan residents that regularly watch professional football is 5 percentage points lower than the percentage of Wisconsin residents who regularly watch professional football.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q320647\">Show Answer<\/span><\/p>\n<div id=\"q320647\" class=\"hidden-answer\" style=\"display: none\">\n<p>Since the statement refers to two population proportions, we can represent the proportion of Michigan residents that regularly watch professional football as [latex]p_\\mathrm{Michigan}[\/latex]\u00a0and the proportion of Wisconsin residents that regularly watch professional football as [latex]p_\\mathrm{Wisconsin}[\/latex].\u00a0The comparison between these proportions \u201cis\u201d corresponds to [latex]=[\/latex].\u00a0Since Michigan\u2019s proportion is [latex]5%=0.05[\/latex] lower than Wisconsin\u2019s proportion, we write<\/p>\n<p style=\"text-align: center;\">[latex]p_\\mathrm{Michigan} = p_\\mathrm{Wisconsin} - 0.05[\/latex].<\/p>\n<p>This can also be written as<\/p>\n<p style=\"text-align: center;\">[latex]p_\\mathrm{Michigan} - p_\\mathrm{Wisconsin} = -0.05[\/latex]<\/p>\n<p>or<\/p>\n<p style=\"text-align: center;\">[latex]p_\\mathrm{Wisconsin} - p_\\mathrm{Michigan} = 0.05[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Write the following statement as a comparison between two population parameters.<\/p>\n<p>A physical therapist suggests that the average recovery time for patients is lower for those patients receiving massage therapy in addition to standard treatment as compared to patients receiving standard treatment alone.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q817297\">Show Answer<\/span><\/p>\n<div id=\"q817297\" class=\"hidden-answer\" style=\"display: none\">\n<p>Since the statement refers to two population means, we can represent the mean recovery time for patients who receive massage therapy as [latex]\\mu _\\mathrm{massage}[\/latex]\u00a0and the mean recovery time for patients who receive standard treatment only as [latex]\\mu _\\mathrm{standard}[\/latex].\u00a0The comparison between these means is \u201cless than,\u201d which is represented by [latex]<[\/latex]. So we write the comparison mathematically as\n\n\n<p style=\"text-align: center;\">[latex]\\mu _\\mathrm{massage} < \\mu _\\mathrm{standard}[\/latex].<\/p>\n<p>This can also be written as<\/p>\n<p style=\"text-align: center;\">[latex]\\mu _\\mathrm{massage} - \\mu _\\mathrm{standard} < 0[\/latex]<\/p>\n<p>or<\/p>\n<p style=\"text-align: center;\">[latex]\\mu_\\mathrm{standard} - \\mu _\\mathrm{massage} > 0[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-2186\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li><strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":169134,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2186","chapter","type-chapter","status-publish","hentry"],"part":285,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2186","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/users\/169134"}],"version-history":[{"count":13,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2186\/revisions"}],"predecessor-version":[{"id":4037,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2186\/revisions\/4037"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/parts\/285"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2186\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=2186"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=2186"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=2186"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=2186"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}