{"id":33586,"date":"2021-10-07T03:37:14","date_gmt":"2021-10-07T03:37:14","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=33586"},"modified":"2026-02-26T09:10:26","modified_gmt":"2026-02-26T09:10:26","slug":"test-statistic-type-i-and-type-ii-errors-power-of-a-test-and-significance-levels","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/test-statistic-type-i-and-type-ii-errors-power-of-a-test-and-significance-levels\/","title":{"rendered":"Test Statistic, Type I and Type II Errors, Power of a Test, and Significance Levels"},"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  \"contentUrl\": \"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\n\n\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/twTTWMuEGx8?si=T-vP3f-Hu49ieGhb\" 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\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 a null hypothesis. As such, it is a major determinant of the decision to either accept or reject H<sub>0<\/sub>, the null hypothesis.<\/p>\n\n\n\n<!--more-->\n\n\n\n<p>We use 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\n\n\n<p>$$ \\text{Test statistic} = \\cfrac { (\\text{Sample statistic} \u2013 \\text{Hypothesized value})}{\\text{Standard error of the sample statistic}} $$<\/p>\n\n\n\n<p>The test statistic is a random variable that changes from one sample to another. The following table&nbsp; gives a brief outline of the various test statistics used regularly, based on the distribution the data presumably follows:<\/p>\n\n\n\n<p>$$<br>\\begin{array}{c|c}<br>\\textbf{Hypothesis Test} &amp; \\textbf{Test statistic} \\\\ \\hline<br>\\text{Z-test} &amp; \\text{Z-statistic (normal distribution)} \\\\ \\hline<br>\\text{Chi-square test} &amp; \\text{Chi-square statistic} \\\\ \\hline<br>\\text{t-test} &amp; \\text{t-statistic} \\\\ \\hline<br>\\text{ANOVA} &amp; \\text{F-statistic} \\\\<br>\\end{array}<br>$$<\/p>\n\n\n\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\n\n\n<div style=\"margin:18px 0;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\"\n     target=\"_blank\"\n     rel=\"noopener noreferrer\"\n     style=\"\n        display:block;\n        text-align:center;\n        padding:14px 18px;\n        border:2px solid #2F5BFF;\n        border-radius:18px;\n        color:#2F5BFF;\n        font-weight:600;\n        font-size:16px;\n        text-decoration:none;\n        background-color:#f9faff;\n     \">\n     Practice CFA questions on hypothesis testing concepts.\n  <\/a>\n<\/div>\n\n\n<p>As is the case 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<p>The following is the list of test statistics and their distributions:<\/p>\n<p>$$\\begin{array} {c|c|c|c}\u00a0 {}&amp;{}&amp;\\textbf{Test Statistic}&amp;\\textbf{Number of Degrees of}\\\\ \\textbf{Test Subject}&amp;\\textbf{Test Statistic Formula}&amp;\\textbf{Distribution}&amp;\\textbf{Freedom}\\\\ \\hline \\text{Single mean }&amp;t=\\frac{\\bar{X}-\\mu_{0}}{\\frac{s }{\\sqrt{n}}}&amp;\\text{t-distribution}&amp;n-1\\\\ \\hline \\text{Difference in means}&amp;t=\\frac{\\left(\\bar{X}_{1}-\\bar{X}_{2}\\right)-\\left(\\mu_{1}-\\mu_{2}\\right)}{\\sqrt{\\frac{s_{p}^{2}}{n_{1}}+\\frac{s_{p}^{2}}{n_{2}}}}&amp;\\text{t-Distribution}&amp;n_1+n_2-2\\\\ \\hline \\text{Mean of differences}&amp;t=\\frac{\\bar{d}-\\mu_{d 0}}{s_{\\bar{d}}}&amp;\\text{t-Distribution}&amp;n-1\\\\ \\hline \\text{Single variance}&amp;\\chi^{2}=\\frac{s^{2}(n-1)}{\\sigma_{0}^{2}}&amp;\\text{Chi-square distribution}&amp;n-1\\\\ \\hline \\text{Differences in variances}&amp;F=\\frac{s_{1}^{2}}{s_{2}^{2}} &amp;\\text{F-distribution}&amp;n_1-1, n_2-1\\\\ \\hline \\text{Correlation}&amp;t=\\frac{r \\sqrt{n-2}}{\\sqrt{1-r^{2}}}&amp;\\text{t-Distribution}&amp;n-2\\\\ \\hline \\text{Independence (categorical data)}&amp;\\chi^{2}=\\sum_{i=1}^{m} \\frac{\\left(O_{i j}-E_{i j}\\right)^{2}}{E_{i j}}&amp; \\text{Chi-square distribution}&amp;(r-1)(c-1)\\\\\u00a0 \\end{array}$$<\/p>\n<p>Where:<\/p>\n<p>\\(\\mu_o, \\mu_{d_0}\\) and \\(\\sigma^2_0\\) denote hypothesized values of the mean, mean difference, and variance, in that order.<\/p>\n<p>\\(\\bar{x}, \\bar{d}, s^2, s\\) and \\(r\\) denote sample mean of the differences, sample variance, sample standard deviation, and correlation, in that order.<\/p>\n<p>\\(O_{ij}\\) and \\(E_ij\\) are observed and expected frequencies, respectively, with <em>r<\/em> indicating the number of rows and <em>c<\/em> indicating the number of columns in the contingency table.<\/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. This brings us to the two errors highlighted below.<\/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 the rejection of 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 evidence the test provides is insufficient and, as such, cannot justify the rejection of 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 is actually true. Consequently, \u03b2, the direct opposite of \u03b1, 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 \u03b1 to determine critical values that subdivide a 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<h2><strong>The Power of a Test<\/strong><\/h2>\n<p>The power of a test is the direct opposite of the level of significance. While the level of significance gives us the probability of rejecting the null hypothesis when it is, in fact, true, the power of a test gives us the probability of correctly discrediting and rejecting the null hypothesis when it is false. In other words, it gives the likelihood of rejecting \\(H_0\\) when, indeed, it is false. Expressed mathematically,<\/p>\n<p>$$\\text{Power of a test = 1- \u03b2 = 1-P(type II error)}$$<\/p>\n<p>When presented with a situation where there are multiple test results for the same purpose, it is the test with the highest power is considered the best.<\/p>\n<div class=\"notes_inv\">\n<hr \/><\/div>\n\n\n<div style=\"text-align:center; margin:40px 0 20px;\">\n\n <div style=\"text-align:center; margin:40px 0 10px 0;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener\"\n     style=\"display:inline-block; padding:14px 36px; background:#1a73e8; color:#ffffff; border-radius:999px; font-weight:700; font-size:16px; text-decoration:none;\">\n    Start Free Trial \u2192\n  <\/a>\n<\/div>\n\n  <p style=\"\n        margin-top:18px;\n        font-size:16px;\n        line-height:1.6;\n        max-width:700px;\n        margin-left:auto;\n        margin-right:auto;\n        color:#333333;\n     \">\n     Strengthen your CFA Level I Quantitative Methods preparation with exam-style questions, QBank drills, and full mock exams designed to help you apply test statistics, interpret error types, and evaluate statistical significance under exam conditions.\n  <\/p>\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"<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 a null hypothesis. As such, it is a major determinant of the decision to&#8230;<\/p>\n","protected":false},"author":15,"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-33586","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>Test Statistic &amp; Type I vs. Type II Errors | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Learn about test statistics, Type I and Type II errors, significance levels, and the power of a test in hypothesis testing.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" 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