{"id":31315,"date":"2021-09-22T07:05:55","date_gmt":"2021-09-22T07:05:55","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=31315"},"modified":"2026-02-26T10:00:11","modified_gmt":"2026-02-26T10:00:11","slug":"confidence-intervals-2","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/confidence-intervals-2\/","title":{"rendered":"Confidence Intervals"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Sampling and Estimation (2025 Level I CFA\u00ae Exam \u2013 Quantitative Methods \u2013 Module 5)\",\n  \"description\": \"This CFA Level 1 Quantitative Methods video provides a comprehensive overview of Sampling and Estimation, including its theoretical and practical applications. Topics covered include probability vs. non-probability sampling, sampling errors, types of sampling methods (simple random, stratified, cluster, convenience, and judgmental), and key statistical concepts like the central limit theorem and confidence intervals. The session explains critical learning outcome statements (LOS), such as calculating and interpreting standard errors, identifying desirable estimator properties, and addressing biases like survivorship and time-period bias. Viewers will also learn resampling techniques, such as bootstrapping and jackknifing, for enhancing estimation accuracy. This in-depth tutorial equips CFA candidates with tools to master sampling techniques and their role in making population inferences, ensuring exam success.\",\n  \"uploadDate\": \"2021-11-11T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/QDmc4Pa92bs\/maxresdefault.jpg\",\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=QDmc4Pa92bs\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/QDmc4Pa92bs\",\n  \"duration\": \"PT37M40S\"\n}\n<\/script>\n\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\": \"How do you calculate a two-tailed 99% confidence interval for the population mean?\",\n    \"answerCount\": 1,\n    \"upvoteCount\": 0,\n    \"dateCreated\": \"2025-07-01T00:00:00+00:00\",\n    \"author\": {\n      \"@type\": \"Organization\",\n      \"name\": \"AnalystPrep\"\n    },\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"A two-tailed 99% confidence interval for the population mean is calculated as x\u0304 \u00b1 t\u03b1\/2 \u00d7 (S \/ \u221an). Using the given data, the correct confidence interval is (125.3, 127.91). A higher confidence level results in a wider interval.\",\n      \"dateCreated\": \"2025-07-01T00:00:00+00:00\",\n      \"upvoteCount\": 0,\n      \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/confidence-intervals-2\/\",\n      \"author\": {\n        \"@type\": \"Organization\",\n        \"name\": \"AnalystPrep\"\n      }\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-159.jpg\",\n  \"caption\": \"Confidence intervals and statistical estimation\",\n  \"width\": 1647,\n  \"height\": 972,\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<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\/2021\/09\/cfa-level-1-t-score.png\",\n  \"caption\": \"t-score and confidence interval calculation\",\n  \"width\": 750,\n  \"height\": 720,\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<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/QDmc4Pa92bs\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\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 is different from 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 a candidate lies in being able 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, 95% of the intervals would contain the true parameter value and just 5% would not contain it, hence, the phrase \u201cconfidence\u201d interval.<\/p>\n<div style=\"margin:22px 0;\">\n  <div style=\"text-align:center; margin:20px 0 24px;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"\n     style=\"display:inline-flex; align-items:center; justify-content:center; padding:10px 18px; border:2px solid #1a73e8; border-radius:999px; background:#f5f9ff; color:#1a73e8; text-decoration:none; font-weight:600; font-size:14px;\">\n    Practice confidence interval problems with CFA-style examples\n  <\/a>\n<\/div>\n\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>$$ \\text{CI} =\\text{Point estimate} \\pm \\text{Reliability factor} \u00d7 \\text{Standard error} $$<\/p>\n<p>Where:<\/p>\n<p>The 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>The Standard error is the standard error of the point estimate.<\/p>\n<h2><strong>Different Scenarios<\/strong><\/h2>\n<h3><strong style=\"color: revert; font-size: revert;\">1. Normal Distribution With a Known Variance<\/strong><\/h3>\n<p>We can calculate the confidence interval for the mean as:<\/p>\n<p>$$ x \\pm z_{\\alpha\/2} \u00d7 \\frac {\\sigma}{\\sqrt n} $$<\/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 table below 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\" 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<h3><strong>2. Normal Distribution With Unknown Variance<\/strong><\/h3>\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. Therefore,<\/p>\n<p>$$ CI = x \\pm t_{\\alpha\/2} \u00d7 \\frac {S}{\\sqrt n} $$<\/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<h3><strong>3. <\/strong><strong>Any Distribution<\/strong><strong> When Variance is Unknown, and the Sample Size is Large Enough<\/strong><\/h3>\n<p>Thanks to the Central Limit Theorem, we can approximate just about any non-normal distribution the same way we do a normal one, provided the sample size is large (n \u2265 30). Furthermore, we can use the relevant z-score when constructing a confidence interval for the population mean.<\/p>\n<p>However, some analysts may advocate for the use of a t-distribution in scenarios where the distribution is non-normal, and the population variance is unknown, even if n \u2265 30. Regardless of such arguments, the use of the z statistic would still be justified under such circumstances, provided the central limit theorem is applied correctly.<\/p>\n<h4><strong>Example: Confidence Interval<\/strong><\/h4>\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 12-year-olds.<\/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. As such, we will calculate the confidence interval for the mean as follows:<\/p>\n<p>$$ CI = x \\pm t_{\\alpha\/2} \u00d7 \\frac {S}{\\sqrt n} $$<\/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 loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-38554\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/cfa-level-1-t-score.png\" alt=\"\" width=\"750\" height=\"720\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/cfa-level-1-t-score.png 750w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/cfa-level-1-t-score-300x288.png 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/cfa-level-1-t-score-400x384.png 400w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/cfa-level-1-t-score-24x24.png 24w\" sizes=\"auto, (max-width: 750px) 100vw, 750px\" \/><\/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>Therefore, our confidence interval for \u03bc is (123.36, 129.84).<\/p>\n<blockquote>\n<h2><strong>Question<\/strong><\/h2>\n<p>Use the data from the example above to calculate a two-tailed 99% confidence interval for the population mean.<\/p>\n<ol style=\"list-style-type: upper-alpha;\">\n<li>(125.3, 127.91)<\/li>\n<li>(117.9, 135.3)<\/li>\n<li>(116.6, 136.6)<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <strong>A<\/strong>.<\/p>\n<p>$$ CI = x \\pm t_{\\alpha\/2} \u00d7 \\frac {S}{\\sqrt n} $$<\/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>$$ \\begin{align*}<br>CI &amp; = 126.6 \\pm 4.604\u00d7  \\frac {\\sqrt 6.8}{\\sqrt 5} \\\\<br>&amp; = 126.6 \\pm 5.4391 \\\\<br>\\end{align*} $$<\/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\n\n\n<div style=\"margin: 40px 0; padding: 30px; text-align: center; background-color: #f5f8fc; border-radius: 10px;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\"\n     target=\"_blank\"\n     class=\"ap-cta\"\n     data-cta-text=\"Start Free Trial\"\n     data-cta-type=\"button\"\n     data-cta-location=\"bottom_content\"\n     data-page-type=\"study_note\"\n     style=\"\n       display: inline-block;\n       padding: 14px 26px;\n       font-size: 18px;\n       font-weight: 700;\n       color: #ffffff;\n       background-color: #0b5ed7;\n       border-radius: 8px;\n       text-decoration: none;\n     \">\n    Start Free Trial \u2192\n  <\/a>\n  <p style=\"margin-top: 12px; font-size: 15px; color: #333;\">\n    Practice confidence interval &#038; statistical-methods problems with full solutions.\n  <\/p>\n<\/div>\n\n\n\n<p><\/p>\n","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 is different from from a point estimate which is a single, specific numerical value.<\/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-31315","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 Intervals Explained | CFA Level 1<\/title>\n<meta name=\"description\" content=\"A confidence interval shows the range where a true parameter likely falls, such as in a 95% confidence level, unlike a single-point estimate. 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