{"id":2516,"date":"2019-08-27T17:37:00","date_gmt":"2019-08-27T17:37:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=2516"},"modified":"2026-03-10T17:26:55","modified_gmt":"2026-03-10T17:26:55","slug":"understanding-test-statistics","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/understanding-test-statistics\/","title":{"rendered":"Understanding Test Statistics"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/08\/page-168.jpg\",\n  \"caption\": \"Two Tailed Test\",\n  \"width\": 1473,\n  \"height\": 900,\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  }\n}\n<\/script>\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Hypothesis Testing (2021 Level I CFA\u00ae Exam \u2013 Reading 11)\",\n  \"description\": \"This video lesson covers CFA Level 1 Quantitative Methods on hypothesis testing. It explains key concepts such as null and alternative hypotheses, test statistics, p-values, Type I and II errors, one- and two-tailed tests, chi-square and F-distributions, and the differences between statistical and economic significance.\",\n  \"uploadDate\": \"2019-12-18T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/i.ytimg.com\/vi\/kvocth0y_tQ\/hqdefault.jpg\",\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=kvocth0y_tQ\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/kvocth0y_tQ\",\n  \"duration\": \"PT38M55S\"\n}\n<\/script>\n\n\n\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/kvocth0y_tQ?si=dCtoSskOSirl4H8-\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n\n\n<p>\u00a0<\/p>\n<p>A test statistic is a standardized value computed from sample information when testing hypotheses. It compares the given data with what an analyst would expect under the null hypothesis. As such, it is a major determinant when deciding whether to reject H<sub>0<\/sub>, the null hypothesis.<\/p>\n<p><!--more--><\/p>\n<p>We use the test statistic to gauge the degree of agreement between sample data and the null hypothesis. Analysts use the following formula when calculating the test statistic.<\/p>\n<p>$$ \\text{Test statistic} = \\cfrac { (\\text{sample statistic} \u2013 \\text{hypothesized value})}{\\text{Standard error of the sample statistic}} $$<\/p>\n<p>The test statistic is a random variable that changes from one sample to another. The following table\u00a0 gives a brief outline of the various test statistics used regularly, based on the distribution the data presumably follows:<\/p>\n<p>$$<br \/>\\begin{array}{l|r}<br \/>\\text{Hypothesis Test} &amp; \\text{Test statistic} \\\\<br \/>\\text{Z-test} &amp; \\text{Z-statistic (normal distribution)} \\\\<br \/>\\text{Chi-square test} &amp; \\text{Chi-square statistic} \\\\<br \/>\\text{t-test} &amp; \\text{t-statistic} \\\\<br \/>\\text{ANOVA} &amp; \\text{F-statistic} \\\\<br \/>\\end{array}<br \/>$$<\/p>\n<p>We can subdivide the set of values that the test statistic can take into two regions: the non-rejection region, which is consistent with the H<sub>0,<\/sub> and the rejection region (critical region), which is inconsistent with the H<sub>0<\/sub>. If the test statistic has a value found within the critical region, we reject the H<sub>0<\/sub>.<\/p>\n<p>Like with any other statistic, the distribution of the test statistic must be completely specified under the H<sub>0 <\/sub>when the H<sub>0 <\/sub>is true.<\/p>\n<h2><strong>Type I and Type II Errors<\/strong><\/h2>\n<p>While using sample statistics to draw conclusions about the parameters of an entire population, there is always the possibility that the sample collected does not accurately represent the population. Consequently, statistical tests carried out using such sample data may yield incorrect results that may lead to erroneous rejection (or lack thereof) of the null hypothesis. We have two types of errors:<\/p>\n<h3><strong>Type I Error<\/strong><\/h3>\n<p>Type I error occurs when we reject a true null hypothesis. For example, a type I error would manifest in rejecting H<sub>0<\/sub> = 0 when it is actually zero.<\/p>\n<h3><strong>Type II Error<\/strong><\/h3>\n<p>Type II error occurs when we fail to reject a false null hypothesis. In such a scenario, the test provides insufficient evidence to reject the null hypothesis when it is actually false.<\/p>\n<p>The level of significance, denoted by \u03b1, represents the probability of making a type I error, i.e., rejecting the null hypothesis when it\u2019s true. \u03b1 is the direct opposite of \u03b2, which is the probability of making a type II error within the bounds of statistical testing. The ideal but practically impossible statistical test would be one that <strong>simultaneously <\/strong>minimizes \u03b1 and \u03b2. We use\u00a0\u03b1 to determine critical values that subdivide the distribution into the rejection and the non-rejection regions. The figure below gives an example of the critical regions under a two-tailed normal distribution and 5% significance level:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-17052\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/08\/page-168.jpg\" alt=\"two-tailed-test\" width=\"1473\" height=\"900\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/08\/page-168.jpg 1473w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/08\/page-168-300x183.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/08\/page-168-768x469.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/08\/page-168-1024x626.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/08\/page-168-400x244.jpg 400w\" sizes=\"auto, (max-width: 1473px) 100vw, 1473px\" \/><\/p>\n<div class=\"notes_inv\"><hr \/><\/div>","protected":false},"excerpt":{"rendered":"<p>\u00a0 A test statistic is a standardized value computed from sample information when testing hypotheses. It compares the given data with what an analyst would expect under the null hypothesis. As such, it is a major determinant when deciding whether&#8230;<\/p>\n","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-2516","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>Understanding Test Statistics | CFA Level 1 - AnalystPrep<\/title>\n<meta name=\"description\" content=\"Learn how test statistics are used in hypothesis testing, including key formulas, interpretation, and applications in statistical analysis.\" \/>\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\/understanding-test-statistics\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Understanding Test Statistics | CFA Level 1 - 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