{"id":1593,"date":"2019-10-01T13:29:00","date_gmt":"2019-10-01T13:29:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=1593"},"modified":"2026-04-02T11:04:55","modified_gmt":"2026-04-02T11:04:55","slug":"confidence-intervals","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/confidence-intervals\/","title":{"rendered":"Calculation and Interpretation of Confidence Intervals"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n\n  \"name\": \"Sampling and Estimation (2021 Level I CFA\u00ae Exam \u2013 Reading 10)\",\n\n  \"description\": \"This video lesson covers the fundamentals of sampling and estimation, including simple and stratified random sampling, time series vs. cross-sectional data, and the central limit theorem. It explains standard error, confidence intervals, and t-distributions while highlighting common biases in data collection, such as survivorship and look-ahead bias, impacting statistical analysis.\",\n\n  \"uploadDate\": \"2019-12-18T00:00:00+00:00\",\n\n  \"thumbnailUrl\": \"https:\/\/analystprep.com\/path-to-thumbnail\/sampling-estimation-thumbnail.jpg\",\n\n  \"contentUrl\": \"https:\/\/youtu.be\/mgY_3CHHYBw\",\n\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/mgY_3CHHYBw\",\n\n  \"duration\": \"PT32M\",\n\n  \"publisher\": {\n    \"@type\": \"Organization\",\n    \"name\": \"AnalystPrep\",\n    \"logo\": {\n      \"@type\": \"ImageObject\",\n      \"url\": \"https:\/\/analystprep.com\/path-to-logo\/logo.jpg\",\n      \"width\": 600,\n      \"height\": 60\n    }\n  }\n}\n<\/script>\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"@id\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-159.jpg\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-159.jpg\",\n  \"width\": 1647,\n  \"height\": 972,\n  \"caption\": \"Confidence interval construction and interpretation for population parameters\",\n  \"copyrightNotice\": \"\u00a9 AnalystPrep\",\n  \"creditText\": \"AnalystPrep\",\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\": \"ImageObject\",\n  \"@id\": \"https:\/\/cdn.analystprep.com\/study-notes\/wp-content\/uploads\/2020\/11\/08133240\/t-table.png\",\n  \"url\": \"https:\/\/cdn.analystprep.com\/study-notes\/wp-content\/uploads\/2020\/11\/08133240\/t-table.png\",\n  \"width\": 854,\n  \"height\": 820,\n  \"caption\": \"t-distribution critical value table used in confidence interval estimation\",\n  \"copyrightNotice\": \"\u00a9 AnalystPrep\",\n  \"creditText\": \"AnalystPrep\",\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\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"How do you calculate a two-tailed 99% confidence interval for the population mean?\",\n    \"text\": \"Use data from the example above to calculate a two-tailed 99% confidence interval for the population mean.\\n\\nA. (125.3, 127.91)\\nB. (117.9, 135.3)\\nC. (116.6, 136.6)\",\n    \"answerCount\": 3,\n    \"suggestedAnswer\": [\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"A. (125.3, 127.91)\"\n      },\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"B. (117.9, 135.3)\"\n      },\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"C. (116.6, 136.6)\"\n      }\n    ],\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"A. (125.3, 127.91).\",\n      \"commentary\": \"For a two-tailed 99% confidence interval for the population mean (with unknown population variance), use the t-distribution: CI = x\u0304 \u00b1 t\u03b1\/2 \u00d7 (s\/\u221an). Using the same sample inputs from the example and the 99% two-tailed critical value (t\u03b1\/2), the interval evaluates to (125.3, 127.91).\",\n      \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/confidence-intervals\/\"\n    },\n    \"author\": {\n      \"@type\": \"Organization\",\n      \"name\": \"AnalystPrep\"\n    }\n  }\n}\n<\/script>\n\n\n\n<iframe loading=\"lazy\"\n  width=\"611\"\n  height=\"344\"\n  src=\"https:\/\/www.youtube.com\/embed\/mgY_3CHHYBw\"\n  title=\"YouTube video player\"\n  frameborder=\"0\"\n  allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\"\n  referrerpolicy=\"strict-origin-when-cross-origin\"\n  allowfullscreen>\n<\/iframe>\n\n\n<p>Confidence interval (CI) refers to a range of values within which statisticians believe the actual value of a certain population parameter lies. It differs from a point estimate which is a single, specific numerical value.<\/p>\n<p><!--more--><\/p>\n<h2><strong>Breaking Down Confidence Interval<\/strong><\/h2>\n<p>When constructing confidence intervals, we must specify the probability that the interval contains the true value of the parameter of interest. This probability is represented by (1 \u2013 \u03b1), where \u03b1 is the level of significance. In statistical terminology, 1- \u03b1 is called the degree of confidence or certainty.<\/p>\n<p>We define a 100(1 &#8211; \u03b1)% confidence interval for a given parameter, say \u03b8, by specifying two random variables, \u03b8&#8217;<sub>1<\/sub>(X) and \u03b8&#8217;<sub>2<\/sub>(X), such that P{\u03b8&#8217;<sub>1<\/sub>(X?) &lt; \u03b8 &lt; \u03b8&#8217;<sub>2<\/sub>(X)} = 1 &#8211; \u03b1.<\/p>\n<p>It happens that \u03b1 = 0.05 is the most common case in examinations and practice. This leads to a 95% confidence interval.<\/p>\n<p>Consequently, P{\u03b8&#8217;<sub>1<\/sub>(X) &lt; \u03b8 &lt; \u03b8&#8217;<sub>2<\/sub>(X)} = 0.95 specifies {\u03b8&#8217;<sub>1<\/sub>(X), \u03b8&#8217;<sub>2<\/sub>(X)} as a 95% confidence interval for \u03b8. The main task for candidates lies in their ability to construct and interpret a confidence interval. Therefore, the CI for \u03b8 above could be interpreted to mean that if we were to construct similar intervals using samples of equal sizes from the same population, then 95% of the intervals would contain the true parameter value. Only 5% would not contain the true parameter value, hence, the phrase \u201cconfidence\u201d interval.<\/p>\n<div style=\"margin: 20px 0;\"><a style=\"display: block; width: 100%; text-align: center; padding: 10px; border: 2px solid #2f5bea; border-radius: 40px; font-size: 16px; color: #2f5bea; text-decoration: none;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener\"> Practice confidence interval questions with our free trial. <\/a><\/div>\n<h2><strong>Constructing Confidence Intervals<\/strong><\/h2>\n<p>To construct a confidence interval, one must come up with an appropriate value that will be subtracted and added to a point estimate. A confidence interval appears as follows:<\/p>\n<p>$$ \\text C.\\text{I} =\\text{point estimate} \\pm \\text{reliability factor} * \\text{standard error} $$<\/p>\n<p>Where:<\/p>\n<p>Point estimate refers to a calculated value of the sample statistic such as the mean, X.<\/p>\n<p>The reliability factor is a value that depends on the sampling distribution involved and (1 &#8211; \u03b1), the probability that the point estimate is contained in the confidence interval.<\/p>\n<p>$$ \\text{Standard error} =\\text {Standard error of the point estimate} $$<\/p>\n<h2><strong>Different Scenarios<\/strong><\/h2>\n<ol>\n<li>\n<h3><strong> Normal Distribution With a Known Variance<\/strong><\/h3>\n<\/li>\n<\/ol>\n<p>We can calculate the confidence interval for the mean as,<\/p>\n<p>$$ x \\pm z_{\\alpha\/2} * \\frac {\\sigma}{\\sqrt n} $$<\/p>\n<p>Here, the reliability factor is z<sub>\u03b1\/2<\/sub>. The z-score leaves a probability of \u03b1\/2 on the upper tail (right-hand tail) of the standard normal distribution.<\/p>\n<p>The following table represents the standard normal distributions commonly used by analysts.<\/p>\n<p>$$ \\begin{array}{c|c|c} \\text{Degree of confidence} &amp; \\text{Level of significance(one-tailed)} &amp; {z_{\\alpha\/2}} \\\\ \\hline {90\\%} &amp; {10\\%} &amp; {1.645} \\\\ \\hline {95\\%} &amp; {5\\%} &amp; {1.960} \\\\ \\hline {99\\%} &amp; {1\\%} &amp; {2.575} \\\\ \\end{array} $$<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"1647\" height=\"972\" class=\"aligncenter size-full wp-image-17047\" style=\"max-width: 100%;\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-159.jpg\" alt=\"\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-159.jpg 1647w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-159-300x177.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-159-768x453.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-159-1024x604.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-159-400x236.jpg 400w\" sizes=\"auto, (max-width: 1647px) 100vw, 1647px\" \/><\/p>\n<ol start=\"2\">\n<li>\n<h3><strong>Normal Distribution With Unknown Variance<\/strong><\/h3>\n<\/li>\n<\/ol>\n<p>When the variance is unknown, we construct the confidence interval for the mean by replacing the z-score in the first scenario with the t-score. Similarly, we replace the unknown \u03c3 with S, the standard deviation of the sample mean. Thus,<\/p>\n<p>$$ CI = x \\pm t_{\\alpha\/2} * \\frac {S}{\\sqrt n} $$<\/p>\n<p>t<sub>\u03b1\/2 <\/sub>is the t-score that leaves a probability of \u03b1\/2 on the upper tail of the t-distribution. The number of degrees of freedom is determined by the sample size such that the degrees of freedom (df) = n \u2013 1.<\/p>\n<ol start=\"3\">\n<li>\n<h3><strong>The Confidence Interval of the Population Mean When Variance is Unknown, and the Sample Size is Large Enough (Any Distribution)<\/strong><\/h3>\n<\/li>\n<\/ol>\n<p>Thanks to the Central Limit Theorem, we can approximate just about any abnormal distribution as a normal one, provided the sample size is large (n \u2265 30). Therefore, we can use the relevant z-score when constructing a confidence interval for the population mean.<\/p>\n<p>However, some analysts may advocate using the t-distribution in scenarios where the distribution is abnormal, and the population variance is unknown, even if n \u2265 30. Nonetheless, the use of the z statistic would still be justified under such circumstances, provided the Central Limit Theorem is applied correctly.<\/p>\n<h3><strong>Example: Confidence Interval<\/strong><\/h3>\n<p>A teacher draws a sample of five 12-year-old children from a school\u2019s population and records their heights in centimeters as follows:<\/p>\n<p>$$ \\{124, 124, 128, 130, 127\\} $$<\/p>\n<p>Assume that the heights have a normal distribution where both \u03bc and \u03c3 are unknown. Calculate a two-tailed 95% confidence interval for the mean height of the 12-year-old children.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>Since the variance is unknown and the sample size is less than 30, we should use the t-score instead of the z-score, even if the distribution is normal. Therefore, the confidence interval for the mean will take the form illustrated below.<\/p>\n<p>$$ CI = x \\pm t_{\\alpha\/2} * \\frac {S}{\\sqrt n} $$<\/p>\n<p>From the data, X = 126.6 and S<sup>2<\/sup> = 6.8<\/p>\n<p>You can read off the t-score value from the t-distribution table where you will find that,<\/p>\n<p>$$ t_{4, 0.025} = 2.776 $$<\/p>\n<p>Please refer to the t-table below to find the critical t-value.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter size-full wp-image-9733\" style=\"max-width: 80%;\" src=\"https:\/\/cdn.analystprep.com\/study-notes\/wp-content\/uploads\/2020\/11\/08133240\/t-table.png\" alt=\"t-table\" \/><\/p>\n<p>Therefore,<\/p>\n<p>$$ \\begin{align*}<br \/>CI &amp; = 126.6 \\pm 2.776 \\times \\frac {\\sqrt 6.8}{\\sqrt 5} \\\\<br \/>&amp; = 126.6 \\pm 3.2373 \\\\<br \/>\\end{align*} $$<\/p>\n<p>In view of the foregoing, our confidence interval for \u03bc is (123.36, 129.84)<\/p>\n<blockquote>\n<h2><strong>Question<\/strong><\/h2>\n<p>Use data from the example above to calculate a two-tailed 99% confidence interval for the population mean.<\/p>\n<p>A. (125.3, 127.91)<\/p>\n<p>B. (117.9, 135.3)<\/p>\n<p>C. (116.6, 136.6)<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is A.<\/p>\n<p>$$ CI = x \\pm t_{\\alpha\/2} * \\frac {S}{\\sqrt n} $$<\/p>\n<p>$$ t_{4, 0.005 }= 4.604 $$<\/p>\n<p>The other inputs remain the same as in the example above.<\/p>\n<p>Therefore,<\/p>\n<p>$$ \\begin{align*}<br \/>CI &amp; = 126.6 \\pm 4.604* \\frac {\\sqrt 6.8}{\\sqrt 5} \\\\<br \/>&amp; = 126.6 \\pm 5.4391 \\\\<br \/>\\end{align*} $$<\/p>\n<p>The confidence interval for the mean is (121.16, 132.01).<\/p>\n<p>As you might have observed, the interval <strong>widens<\/strong> as the level of confidence <strong>increases<\/strong>.<\/p>\n<\/blockquote>\n<p><em>Reading 10 LOS 10j:<\/em><\/p>\n<p><em>Calculate and interpret a confidence interval for a population mean, given a normal distribution with 1) a known population variance, 2) an unknown population variance, or 3) an unknown variance and a large sample size.<br \/><\/em><\/p>\n<div class=\"notes_inv\"><hr \/>\n<p><a href=\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/learning-sessions-curriculum\/\"><em>Quantitative Methods \u2013 Learning Sessions<\/em><\/a><\/p>\n<\/div>\n<div class=\"notes_inv\"><hr \/>\n<p><a href=\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/learning-sessions-curriculum\/\"><em>Quantitative Methods \u2013 Learning Sessions<\/em><\/a><\/p>\n<div style=\"text-align: center; margin: 40px 0;\"><a style=\"display: inline-block; padding: 10px 26px; background: #3f78d7; color: #fff; border-radius: 40px; font-size: 16px; text-decoration: none;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener\"> Start Free Trial \u2192 <\/a>\n<p style=\"margin-top: 10px; max-width: 600px; margin-left: auto; margin-right: auto; font-size: 14px;\">Solve CFA-style questions on confidence intervals, hypothesis testing, and statistical estimation.<\/p>\n<\/div>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>Confidence interval (CI) refers to a range of values within which statisticians believe the actual value of a certain population parameter lies. It differs from a point estimate which is a single, specific numerical value.<\/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-1593","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>Confidence Interval Example | CFA Level 1 - AnalystPrep<\/title>\n<meta name=\"description\" content=\"A confidence interval estimates the range within which a population parameter is likely to fall, based on sample data and a chosen confidence level.\" \/>\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\/confidence-intervals\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Confidence Interval Example | CFA Level 1 - 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