{"id":29807,"date":"2021-09-09T04:43:22","date_gmt":"2021-09-09T04:43:22","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=29807"},"modified":"2026-03-09T19:34:51","modified_gmt":"2026-03-09T19:34:51","slug":"normal-distribution-and-confidence-intervals","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/normal-distribution-and-confidence-intervals\/","title":{"rendered":"Normal Distribution and Confidence Intervals"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"A financial analyst encounters a client whose portfolio return has a mean yearly return of 24% and a standard deviation of 5%. Assuming a normal distribution, a 99% confidence interval for the expected return is closest to which of the following?\",\n    \"text\": \"A financial analyst encounters a client whose portfolio return has a mean yearly return of 24% and a standard deviation of 5%. Assuming a normal distribution, a 99% confidence interval for the expected return is closest to which of the following?\\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, approximately 99% of observations fall within \u03bc \u00b1 3\u03c3. Therefore, the interval is {0.24 \u2212 3 \u00d7 0.05, 0.24 + 3 \u00d7 0.05} = {0.09, 0.39}. Option A assumes a 95% confidence interval using \u03bc \u00b1 2\u03c3. Option C assumes a 68% confidence interval using \u03bc \u00b1 1\u03c3.\"\n    }\n  }\n}\n<\/script>\n\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\/10\/page-123.jpg\",\n  \"caption\": \"Shape of the Normal Distribution\",\n  \"width\": 1463,\n  \"height\": 1065,\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<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/TXO9ODcLWiU\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\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>For us to define a \\(100(1 \u2013 \u03b1)%\\) confidence interval for \\(\u03b8\\), we must specify two random variables \\(\u03b8_1(X)\\) and \\(\u03b8_2(X)\\) such that \\(P(\u03b8_1(X) &lt; \u03b8 &lt; \u03b8_2(X)) = 1 &#8211; \u03b1\\).<\/p>\n<p>The most common value for \\(\u03b1\\) is 0.05, which leads us to a 95% confidence interval. As such, \\(P(\u03b8_1(X) &lt; \u03b8 &lt; \u03b8_2&gt;(X)) = 0.95\\) specifies \\(\u03b8_1(X)\\) and \\(\u03b8_2(X)\\) 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 \\(\u03b8\\).<\/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 to allow data collection from every subject. In fact, the collection of data from every subject in a large population is 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><strong>50% CI<\/strong>: Approximately 50% of all observations fall in the range \\(\u03bc \u00b1 \\left(\\frac{2}{3}\\right)\u03c3\\).<\/li>\n<li><strong>68% CI<\/strong>: Approximately 68.3% of all the observations fall in the range \\(\u03bc \u00b1 \u03c3\\).<\/li>\n<li><strong>95% CI<\/strong>: Approximately 95.5% of all observations fall in the interval \\(\u03bc \u00b1 2\u03c3\\).<\/li>\n<li><strong>99% CI<\/strong>: Approximately 99.7% of all the observations fall in the interval \\(\u03bc \u00b1 3\u03c3\\).<\/li>\n<\/ol>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-17012\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-123.jpg\" alt=\"shape-of-the-normal-distribution\" width=\"1463\" height=\"1065\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-123.jpg 1463w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-123-300x218.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-123-768x559.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-123-1024x745.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-123-400x291.jpg 400w\" sizes=\"auto, (max-width: 1463px) 100vw, 1463px\" \/><\/p>\n<p><em><strong>Note to candidates<\/strong><\/em>: The words \u201cinterval\u201d and \u201crange\u201d have been used interchangeably in this context.<\/p>\n<h4><strong>Example: Confidence Interval<\/strong><\/h4>\n<p>A financial analyst encounters a client whose portfolio return has a mean yearly return of 24% and a standard deviation of 5%. Assuming a normal distribution, the 50% confidence interval for the expected return is <em>closest<\/em> to:<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>$$ \\begin{align*}<br \/>\n\\text{Confidence interval at 50%} &amp; = \\left\\{ 0.24 \u2013 \\cfrac {2}{3} \\times 0.05, 0.24 + \\cfrac {2}{3} \\times 0.05 \\right\\} \\\\<br \/>\n&amp; = \\left\\{ 0.207, 0.273 \\right\\} \\\\<br \/>\n\\end{align*} $$<\/p>\n<blockquote>\n<h2><strong>Question<\/strong><\/h2>\n<p>A financial analyst encounters a client whose portfolio return has a mean yearly return of 24% and a standard deviation of 5%. Assuming a normal distribution, a 99% confidence interval for the expected return is\u00a0<em>closest <\/em>to:<\/p>\n<ol style=\"list-style-type: upper-alpha;\">\n<li>{0.08, 0.49}.<\/li>\n<li>{0.09, 0.39}.<\/li>\n<li>{0.09, 0.9}.<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <strong>B<\/strong>.<\/p>\n<p>For a 99% CI, approximately 99% of all the observations fall in the interval \\(\u03bc \u00b1 3\u03c3\\).<\/p>\n<p>Therefore,<\/p>\n<p>$$ \\begin{align*} \\text{Confidence interval at 99%} &amp; = \\left\\{ 0.24 \u2013 3 \\times 0.05, 0.24 + 3 \\times 0.05 \\right\\} \\\\<br \/>\n&amp; = \\left\\{ 0.09, 0.39 \\right\\} \\\\<br \/>\n\\end{align*} $$<\/p>\n<p><strong>B is incorrect<\/strong>. The calculation assumes a 95% CI:<\/p>\n<p>$$\\text{Confidence interval at 95%}=0.24\u20132\u00d70.05,0.24+2\u00d70.05=\\{0.14,0.34\\}$$<\/p>\n<p><strong>C is incorrect<\/strong>. The calculation assumes a 68% CI:<\/p>\n<p>$$\\text{Confidence interval at 68%}=0.24\u20130.05,0.24+0.05=\\{0.19,0.29\\}$$<\/p><\/blockquote>","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":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-29807","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>Normal Distribution &amp; Confidence Intervals | CFA Level 1<\/title>\n<meta name=\"description\" content=\"A confidence interval provides an estimated range for a population parameter, aiding in statistical inference under a normal distribution.\" \/>\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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