{"id":1232,"date":"2019-10-10T20:04:00","date_gmt":"2019-10-10T20:04:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=1232"},"modified":"2025-12-30T12:58:27","modified_gmt":"2025-12-30T12:58:27","slug":"confidence-interval-example","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/confidence-interval-example\/","title":{"rendered":"Confidence Intervals"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Common Probability Distributions (2021 Level I CFA\u00ae Exam \u2013 Reading 9)\",\n  \"description\": \"This video lesson covers common probability distributions for the CFA Level I exam, including discrete vs. continuous variables, Bernoulli and binomial trials, normal and log-normal distributions, Z-scores, confidence intervals, shortfall risk, the safety-first criterion, and Monte Carlo simulations\u2014essential tools for financial modeling and risk assessment.\",\n  \"uploadDate\": \"2019-12-18T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/i.ytimg.com\/vi\/KbEfz3KiJDo\/hqdefault.jpg\",\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=KbEfz3KiJDo\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/KbEfz3KiJDo\",\n  \"duration\": \"PT45M31S\"\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\": \"Repeat the question above assuming the client is interested in finding a 99% confidence interval for the expected return.\",\n    \"text\": \"Question\\nRepeat the question above assuming the client is interested in finding a 99% confidence interval for the expected return.\\n\\nA. {0.08, 0.49}\\nB. {0.09, 0.39}\\nC. {0.09, 0.9}\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is B. For a 99% confidence interval, 99% of all observations fall within the interval \u03bc \u00b1 3\u03c3. Therefore, the confidence interval is {0.09, 0.39}.\"\n    }\n  }\n}\n<\/script>\n\n\n\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/KbEfz3KiJDo?si=d6MITKIgrEFqHrBb\" 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>A confidence interval (CI) gives an \u201cinterval estimate\u201d of an unknown population parameter such as the mean. It gives us the probability that the parameter lies within the stated interval (range). The precision or accuracy of the estimate depends on the width of the interval.<\/p>\n<p><!--more--><\/p>\n<p>For us to define a <em>100 (1 \u2013 \u03b1)%<\/em> confidence interval for <em>\u03b8<\/em>, we must specify two random variables <em>\u03b8<sub>1<\/sub>(X)<\/em> and <em>\u03b8<sub>2<\/sub>(X)<\/em> such that <em>P(\u03b8<sub>1<\/sub>(X) &lt; \u03b8 &lt; \u03b8<sub>2<\/sub>(X)) = 1 &#8211; \u03b1.<\/em><\/p>\n<p>The most common value for <em>\u03b1<\/em> is 0.05, which leads us to a 95% confidence interval. As such, <em>P(\u03b8<sub>1<\/sub>(X) &lt; \u03b8 &lt; \u03b8<sub>2<\/sub>(X)) = 0.95<\/em> specifies <em>\u03b8<sub>1<\/sub>(X)<\/em> and <em>\u03b8<sub>2 <\/sub>(X)<\/em> such that there is a 95% chance of finding the <strong>true value<\/strong> of \u03b8 in the interval. Alternatively, we could say that 5% of the realizations of such intervals <strong>would not<\/strong> contain the true value of <em>\u03b8<\/em>.<\/p>\n<h2><strong>The Importance of a Confidence Interval<\/strong><\/h2>\n<p>It is not always possible to know the exact values of the population mean or the population standard deviation. This is because most populations are too large, making data collection from every subject in the population not only economically unviable but also very time-consuming. This makes confidence intervals very important.<\/p>\n<p>Confidence intervals can also be used to predict the value of a given parameter. We usually assume that the underlying random variable has a normal distribution. In this regard, the central limit theorem (the assertion that most distributions tend to adopt a normal distribution when n is large) is a very important tool.<\/p>\n<p>The following statements are true for any random variable that assumes a normal distribution:<\/p>\n<ol>\n<li>50% CI: aproximately 50% of all observations fall in the range \u03bc \u00b1 (2\/3)\u03c3;<\/li>\n<li>68% CI: approximately 68% of all the observations fall in the range \u03bc \u00b1 \u03c3;<\/li>\n<li>95% CI: approximately 95% of all observations fall in the interval \u03bc \u00b1 2\u03c3; and<\/li>\n<li>99% CI: approximately 99% of all the observations fall in the interval \u03bc \u00b1 3\u03c3.<\/li>\n<\/ol>\n<p>Note: the words \u201cinterval\u201d and \u201crange\u201d have been used interchangeably in this context.<\/p>\n<h3><strong>Example: Confidence Interval<\/strong><\/h3>\n<p>A financial analyst encounters a client whose portfolio return has a mean yearly return of 24% and a standard deviation of 5%. Then, assuming a normal distribution, compute a 50% confidence interval for the expected return.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>$$ \\begin{align*}<br \/>\\text{The CI} &amp; = \\left\\{ 0.24 \u2013 \\cfrac {2}{3} * 0.05, 0.24 + \\cfrac {2}{3} * 0.05 \\right\\} \\\\<br \/>&amp; = \\left\\{ 0.207, 0.273 \\right\\} \\\\<br \/>\\end{align*} $$<\/p>\n<blockquote>\n<h2><strong>Question<\/strong><\/h2>\n<p>Repeat the question above assuming the client is interested in finding a 99% confidence interval for the expected return.<\/p>\n<p>A. {0.08, 0.49}<\/p>\n<p>B. {0.09, 0.39}<\/p>\n<p>C. {0.09, 0.9}<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is B.<\/p>\n<p>For a 99% CI, 99% of all the observations fall in the interval \u03bc \u00b1 3\u03c3.<\/p>\n<p>Therefore,<\/p>\n<p>$$ \\begin{align*} \\text{the CI} &amp; = \\left\\{ 0.24 \u2013 3 * 0.05, 0.24 + 3 * 0.05 \\right\\} \\\\<br \/>&amp; = \\left\\{ 0.09, 0.39 \\right\\} \\\\<br \/>\\end{align*} $$<\/p>\n<\/blockquote>\n<div class=\"notes_inv\"><hr \/><\/div>","protected":false},"excerpt":{"rendered":"<p>A confidence interval (CI) gives an \u201cinterval estimate\u201d of an unknown population parameter such as the mean. It gives us the probability that the parameter lies within the stated interval (range). The precision or accuracy of the estimate depends on&#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-1232","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 Examples | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Learn how to calculate and interpret confidence intervals with real exam-style questions and examples tailored for CFA Level 1 candidates.\" \/>\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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