{"id":2562,"date":"2019-08-27T17:37:00","date_gmt":"2019-08-27T17:37:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=2562"},"modified":"2026-01-21T17:06:31","modified_gmt":"2026-01-21T17:06:31","slug":"paired-data-tests","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/paired-data-tests\/","title":{"rendered":"Test for Differences Between Means: Paired Data"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"@id\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/images\/p-value-illustration-1\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/08\/cfa-level-1-p-value.png\",\n  \"contentUrl\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/08\/cfa-level-1-p-value.png\",\n  \"caption\": \"Illustration of p-value in hypothesis testing\",\n  \"width\": 534,\n  \"height\": 224,\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\n  \"creditText\": \"AnalystPrep Design Team\",\n  \"creator\": {\n    \"@type\": \"Organization\",\n    \"name\": \"AnalystPrep\",\n    \"url\": \"https:\/\/analystprep.com\/\"\n  }\n}\n<\/script>\n\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"@id\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/images\/p-value-illustration-2\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/08\/cfa-level-1-p-value-5.png\",\n  \"contentUrl\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/08\/cfa-level-1-p-value-5.png\",\n  \"caption\": \"P-value comparison under different test outcomes\",\n  \"width\": 534,\n  \"height\": 224,\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\n  \"creditText\": \"AnalystPrep Design Team\",\n  \"creator\": {\n    \"@type\": \"Organization\",\n    \"name\": \"AnalystPrep\",\n    \"url\": \"https:\/\/analystprep.com\/\"\n  }\n}\n<\/script>\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"Hypothesis test decision at a 0.05% significance level\",\n    \"text\": \"Repeat the hypothesis testing example using a significance level of 0.05% and state the decision rule.\\n\\nA. Reject H\u2080 and conclude that the special diet reduces body weight.\\n\\nB. Do not reject H\u2080 and conclude that the special diet has an insignificant effect on body weight.\\n\\nC. Reject H\u2080 and conclude that the special diet has an insignificant effect on body weight.\",\n    \"answerCount\": 3,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"B. Do not reject H\u2080 and conclude that the special diet has an insignificant effect on body weight.\\n\\nThe observed test statistic is \u22123.591, while the lower critical value at the 0.05% significance level for a t\u2089 distribution is \u22124.781. Because the test statistic does not fall in the rejection region, the null hypothesis is not rejected.\"\n    },\n    \"suggestedAnswer\": [\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"A. Reject H\u2080 and conclude that the special diet reduces body weight.\"\n      },\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"C. Reject H\u2080 and conclude that the special diet has an insignificant effect on body weight.\"\n      }\n    ]\n  }\n}\n<\/script>\n\n\n\n\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/twTTWMuEGx8\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>When testing for the difference between two population means, independent samples may introduce some caveats. The real difference can be masked by variability within each sample caused by factors other than those of interest to the researcher. Random variation within a sample can be so large that it blurs the real difference \u2013 that which has been caused by a phenomenon or factor in which an analyst is interested. For this reason, a paired data test may be preferred.<\/p>\n<p>As a way of controlling the external variability, data analysts often collect a pair of responses from each subject. They then compute and work with the differences within pairs.<\/p>\n<h3><strong>Example 1: Paired Data Test<\/strong><\/h3>\n<p>A group of financial analysts wishes to determine whether the payment of dividends has a significant effect on the share price. Sample data on share prices can be collected from a set of companies prior to the payment of dividends, say, 7 days before dividend announcements. To ensure the samples are independent, the companies should be in the same line of business. This would ensure that all of them are susceptible to similar <strong>specific risks<\/strong> and other business-related conditions. Seven days after all the companies have paid the dividends, the analysts can recollect data from each company. The analysts would end up with paired data from each company. Such data can then be used to establish if any significant differences exist between pre-dividend and post-dividend share prices.<\/p>\n<p>The overriding assumption during paired data tests is that the observed differences constitute a sample from a normal distribution.<\/p>\n<p><strong>Basic Situation<\/strong><\/p>\n<p><strong>H<sub>0<\/sub>: \u03bc<sub>D<\/sub> (= \u03bc<sub>1<\/sub> \u2013 \u03bc<sub>2<\/sub>) = \u03c3 <\/strong><\/p>\n<p>The test statistic is<\/p>\n<p>$$ \\frac { \\bar { D } -\\sigma }{ \\frac { { S }_{ D } }{ \\sqrt { n } } } \\sim { t }_{ n-1 } \\text{ under } { H }_{ 0 } $$<\/p>\n<p>In case <em>n<\/em> is large, the normal distribution assumption is not necessary.<\/p>\n<h3><strong>Example 2: Paired Data Test<\/strong><\/h3>\n<p>Doctors in a certain general hospital wish to determine if a special diet can reduce body weight. To do this, they select ten overweight patients with an average body weight of 77 kg (B&#8217;). After being subjected to the special diet, the average weight reduces to 75 kg (A&#8217;).<\/p>\n<p>Test whether the special diet reduces body weight.<\/p>\n<p>Additional information: \\( \\sum (B_i \u2013 A_i)^2 = 70 \\quad \\quad i = 1, 2, 3, \u2026, 9, 10.\\)<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>As always, begin by stating the hypothesis:<\/p>\n<p>H<sub>0<\/sub>: \u03bc<sub>A<\/sub> = \u03bc<sub>B<\/sub><\/p>\n<p>H<sub>1<\/sub>: \u03bc<sub>A<\/sub> &lt; \u03bc<sub>B<\/sub><\/p>\n<p>Where A stands for \u201cafter\u201d and B is \u201cbefore.\u201d<\/p>\n<p>The test statistic is:<\/p>\n<p>$$ \\frac { \\bar { D } -\\sigma }{ \\frac { { S }_{ D } }{ \\sqrt { n } } } \\sim { t }_{ n-1 }\\quad under\\quad { H }_{ 0 } $$<\/p>\n<p>Where D<sub>i <\/sub>is the difference within each pair (A<sub>i<\/sub> \u2013 B<sub>i<\/sub>).<\/p>\n<p>$$ \\begin{align*} D&#8217; &amp; = A&#8217; &#8211; B&#8217; = 75 \u2013 77 = -2 \\\\ S_D^2 &amp; =\\cfrac {1}{(n \u2013 1)} \\left( \\sum D_i &#8211; D&#8217; \\right)^2 =\\cfrac {1}{(n \u2013 1)} \\left\\{ \\sum D_i^2 &#8211; {nD&#8217;}^2 \\right\\} \\\\ &amp; =\\cfrac {1}{9} \\left\\{ 70 \u2013 10(-2^2) \\right\\} = 3.333 = 1.826^2 \\end{align*} $$<\/p>\n<p>Therefore,<\/p>\n<p>$$ \\text{Test statistic} =\\cfrac {(75 \u2013 77)}{\\left( \\frac {1.826}{\\sqrt{10}} \\right)} = -3.591 $$<\/p>\n<p>The test statistic (-3.591) is lower than the lower 5% point of the t-distribution at 9 degrees of freedom (-1.833).<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-38626\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/08\/cfa-level-1-p-value.png\" alt=\"\" width=\"534\" height=\"224\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/08\/cfa-level-1-p-value.png 534w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/08\/cfa-level-1-p-value-300x126.png 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/08\/cfa-level-1-p-value-400x168.png 400w\" sizes=\"auto, (max-width: 534px) 100vw, 534px\" \/><\/p>\n<p>This gives us sufficient evidence to reject the H<sub>0<\/sub> at 5% significance. Therefore, it would be very reasonable to conclude that the special diet reduces body weight.<\/p>\n<blockquote>\n<h2><strong>Question<\/strong><\/h2>\n<p>Repeat the example above using a significance level of just 0.05% and state the decision rule.<\/p>\n<p>A. Reject the H<sub>0 <\/sub>and conclude that the special diet reduces body weight.<\/p>\n<p>B. Do not reject the H<sub>0<\/sub> and conclude that the special diet has an insignificant effect on body weight.<\/p>\n<p>C. Reject the H<sub>0 <\/sub>and conclude that the special diet has an insignificant effect on body weight.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is B.<\/p>\n<p>The observed test statistic still remains -3.591. The lower 0.05% point of the t<sub>9<\/sub> distribution is -4.781.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-38683\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/08\/cfa-level-1-p-value-5.png\" alt=\"\" width=\"534\" height=\"224\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/08\/cfa-level-1-p-value-5.png 534w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/08\/cfa-level-1-p-value-5-300x126.png 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/08\/cfa-level-1-p-value-5-400x168.png 400w\" sizes=\"auto, (max-width: 534px) 100vw, 534px\" \/><\/p>\n<p>It is clear that the test statistic is way above the t<sub>9<\/sub> distribution, i.e., it lies outside the critical or rejection region. Therefore, we would not reject the H<sub>0<\/sub>.<\/p>\n<\/blockquote>\n<p><em><strong>Note to candidates: <\/strong><\/em>Reducing the significance level basically <strong>reduces the probability of rejecting a true null hypothesis <\/strong>(type I error).<\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[2],"tags":[],"class_list":["post-2562","post","type-post","status-publish","format-standard","hentry","category-quantitative-methods","blog-post","no-post-thumbnail","animate"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.9 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Paired Data Tests | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Learn how paired data tests compare two population means to identify significant differences, aiding statistical analysis and decision-making.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/paired-data-tests\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Paired Data Tests | CFA Level 1\" \/>\n<meta property=\"og:description\" content=\"Learn how paired data tests compare two population means to identify significant differences, aiding statistical analysis and decision-making.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/paired-data-tests\/\" \/>\n<meta property=\"og:site_name\" content=\"AnalystPrep | CFA\u00ae Exam Study Notes\" \/>\n<meta property=\"article:published_time\" content=\"2019-08-27T17:37:00+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2026-01-21T17:06:31+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/08\/cfa-level-1-p-value.png\" \/>\n\t<meta property=\"og:image:width\" content=\"534\" \/>\n\t<meta property=\"og:image:height\" content=\"224\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/png\" \/>\n<meta name=\"author\" content=\"Ngari Joseph\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Ngari Joseph\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/paired-data-tests\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/paired-data-tests\/\"},\"author\":{\"name\":\"Ngari Joseph\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/ddbf15286b86900dfa9f8ac2603c3914\"},\"headline\":\"Test for Differences Between Means: Paired Data\",\"datePublished\":\"2019-08-27T17:37:00+00:00\",\"dateModified\":\"2026-01-21T17:06:31+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/paired-data-tests\/\"},\"wordCount\":650,\"image\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/paired-data-tests\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/08\/cfa-level-1-p-value.png\",\"articleSection\":[\"Quantitative Methods\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/paired-data-tests\/\",\"url\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/paired-data-tests\/\",\"name\":\"Paired Data Tests | CFA Level 1\",\"isPartOf\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/paired-data-tests\/#primaryimage\"},\"image\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/paired-data-tests\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/08\/cfa-level-1-p-value.png\",\"datePublished\":\"2019-08-27T17:37:00+00:00\",\"dateModified\":\"2026-01-21T17:06:31+00:00\",\"author\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/ddbf15286b86900dfa9f8ac2603c3914\"},\"description\":\"Learn how paired data tests compare two population means to identify significant differences, aiding statistical analysis and decision-making.\",\"breadcrumb\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/paired-data-tests\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/paired-data-tests\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/paired-data-tests\/#primaryimage\",\"url\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/08\/cfa-level-1-p-value.png\",\"contentUrl\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/08\/cfa-level-1-p-value.png\",\"width\":534,\"height\":224},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/paired-data-tests\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Test for Differences Between Means: Paired Data\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#website\",\"url\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/\",\"name\":\"AnalystPrep | CFA\u00ae Exam Study Notes\",\"description\":\"\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"},{\"@type\":\"Person\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/ddbf15286b86900dfa9f8ac2603c3914\",\"name\":\"Ngari Joseph\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2016\/08\/male3-512-150x150.png\",\"contentUrl\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2016\/08\/male3-512-150x150.png\",\"caption\":\"Ngari Joseph\"},\"url\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/author\/ngari-joseph\/\"}]}<\/script>\n<meta property=\"og:video\" content=\"https:\/\/www.youtube.com\/embed\/twTTWMuEGx8\" \/>\n<meta property=\"og:video:type\" content=\"text\/html\" \/>\n<meta property=\"og:video:duration\" content=\"3528\" \/>\n<meta property=\"og:video:width\" content=\"480\" \/>\n<meta property=\"og:video:height\" content=\"270\" \/>\n<meta property=\"ya:ovs:adult\" content=\"false\" \/>\n<meta property=\"ya:ovs:upload_date\" content=\"2019-08-27T17:37:00+00:00\" \/>\n<meta property=\"ya:ovs:allow_embed\" content=\"true\" \/>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Paired Data Tests | CFA Level 1","description":"Learn how paired data tests compare two population means to identify significant differences, aiding statistical analysis and decision-making.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/paired-data-tests\/","og_locale":"en_US","og_type":"article","og_title":"Paired Data Tests | CFA Level 1","og_description":"Learn how paired data tests compare two population means to identify significant differences, aiding statistical analysis and decision-making.","og_url":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/paired-data-tests\/","og_site_name":"AnalystPrep | CFA\u00ae Exam Study Notes","article_published_time":"2019-08-27T17:37:00+00:00","article_modified_time":"2026-01-21T17:06:31+00:00","og_image":[{"width":534,"height":224,"url":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/08\/cfa-level-1-p-value.png","type":"image\/png"}],"author":"Ngari Joseph","twitter_card":"summary_large_image","twitter_misc":{"Written by":"Ngari Joseph","Est. reading time":"3 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/paired-data-tests\/#article","isPartOf":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/paired-data-tests\/"},"author":{"name":"Ngari Joseph","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/ddbf15286b86900dfa9f8ac2603c3914"},"headline":"Test for Differences Between Means: Paired Data","datePublished":"2019-08-27T17:37:00+00:00","dateModified":"2026-01-21T17:06:31+00:00","mainEntityOfPage":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/paired-data-tests\/"},"wordCount":650,"image":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/paired-data-tests\/#primaryimage"},"thumbnailUrl":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/08\/cfa-level-1-p-value.png","articleSection":["Quantitative Methods"],"inLanguage":"en-US"},{"@type":"WebPage","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/paired-data-tests\/","url":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/paired-data-tests\/","name":"Paired Data Tests | CFA Level 1","isPartOf":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#website"},"primaryImageOfPage":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/paired-data-tests\/#primaryimage"},"image":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/paired-data-tests\/#primaryimage"},"thumbnailUrl":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/08\/cfa-level-1-p-value.png","datePublished":"2019-08-27T17:37:00+00:00","dateModified":"2026-01-21T17:06:31+00:00","author":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/ddbf15286b86900dfa9f8ac2603c3914"},"description":"Learn how paired data tests compare two population means to identify significant differences, aiding statistical analysis and decision-making.","breadcrumb":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/paired-data-tests\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/paired-data-tests\/"]}]},{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/paired-data-tests\/#primaryimage","url":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/08\/cfa-level-1-p-value.png","contentUrl":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/08\/cfa-level-1-p-value.png","width":534,"height":224},{"@type":"BreadcrumbList","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/paired-data-tests\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/analystprep.com\/cfa-level-1-exam\/"},{"@type":"ListItem","position":2,"name":"Test for Differences Between Means: Paired Data"}]},{"@type":"WebSite","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#website","url":"https:\/\/analystprep.com\/cfa-level-1-exam\/","name":"AnalystPrep | CFA\u00ae Exam Study Notes","description":"","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/analystprep.com\/cfa-level-1-exam\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/ddbf15286b86900dfa9f8ac2603c3914","name":"Ngari Joseph","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/image\/","url":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2016\/08\/male3-512-150x150.png","contentUrl":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2016\/08\/male3-512-150x150.png","caption":"Ngari Joseph"},"url":"https:\/\/analystprep.com\/cfa-level-1-exam\/author\/ngari-joseph\/"}]},"og_video":"https:\/\/www.youtube.com\/embed\/twTTWMuEGx8","og_video_type":"text\/html","og_video_duration":"3528","og_video_width":"480","og_video_height":"270","ya_ovs_adult":"false","ya_ovs_upload_date":"2019-08-27T17:37:00+00:00","ya_ovs_allow_embed":"true"},"_links":{"self":[{"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/posts\/2562","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/comments?post=2562"}],"version-history":[{"count":47,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/posts\/2562\/revisions"}],"predecessor-version":[{"id":58189,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/posts\/2562\/revisions\/58189"}],"wp:attachment":[{"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/media?parent=2562"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/categories?post=2562"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/tags?post=2562"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}