{"id":220085,"date":"2025-05-27T13:18:04","date_gmt":"2025-05-27T13:18:04","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=220085"},"modified":"2025-05-27T13:18:06","modified_gmt":"2025-05-27T13:18:06","slug":"a-student-was-asked-to-find-a-95-confidence-interval-for-the-population-proportion-of-students-who-take-notes-using-data-from-a-random-sample-of-size-n87","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/27\/a-student-was-asked-to-find-a-95-confidence-interval-for-the-population-proportion-of-students-who-take-notes-using-data-from-a-random-sample-of-size-n87\/","title":{"rendered":"A student was asked to find a 95% confidence interval for the population proportion of students who take notes using data from a random sample of size n=87"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">A student was asked to find a 95% confidence interval for the population proportion of students who take notes using data from a random sample of size n=87. Which of the following is a correct interpretation of the interval 0.15, 0.25? Check all that are correct.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">With 95% confidence, a randomly selected student takes notes in a proportion of their classes that is between 0.15 and 0.25.<br>With 95% confidence, the proportion of all students who take notes is between 0.15 and 0.25.<br>There is a 95% chance that the proportion of the population is between 0.15 and 0.25.<br>There is a 95% chance that the proportion of notetakers in a sample of 87 students will be between 0.15 and 0.25.<br>The proportion of all students who take notes is between 0.15 and 0.25, 95% of the time.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u2705 Correct Interpretation:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>With 95% confidence, the proportion of all students who take notes is between 0.15 and 0.25.<\/strong><\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">\u274c Incorrect Interpretations:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>With 95% confidence, a randomly selected student takes notes in a proportion of their classes that is between 0.15 and 0.25.<\/strong><br><em>Incorrect:<\/em> This interpretation confuses the population proportion with an individual&#8217;s behavior. Confidence intervals refer to population parameters, not individuals.<\/li>\n\n\n\n<li><strong>There is a 95% chance that the proportion of the population is between 0.15 and 0.25.<\/strong><br><em>Incorrect:<\/em> The population proportion is a fixed (but unknown) value. The confidence interval either contains it or it doesn&#8217;t\u2014there&#8217;s no &#8220;chance&#8221; involved after the interval is calculated.<\/li>\n\n\n\n<li><strong>There is a 95% chance that the proportion of notetakers in a sample of 87 students will be between 0.15 and 0.25.<\/strong><br><em>Incorrect:<\/em> This misstates what the confidence interval means. It is not about the proportion in future samples but about estimating the population parameter.<\/li>\n\n\n\n<li><strong>The proportion of all students who take notes is between 0.15 and 0.25, 95% of the time.<\/strong><br><em>Incorrect:<\/em> This implies a probability applied to the fixed population parameter, which is a misunderstanding of the confidence interval concept.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">A <strong>confidence interval<\/strong> is a range of values, derived from sample data, that is likely to contain the true value of an unknown population parameter. In this case, the parameter is the <strong>population proportion of students who take notes<\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The interval given\u2014(0.15, 0.25)\u2014is a <strong>95% confidence interval<\/strong>. This means that if we were to take <strong>many random samples of size 87<\/strong>, and for each sample calculate a confidence interval in the same way, then <strong>95% of those intervals would contain the true population proportion<\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The correct interpretation focuses on estimating a population characteristic, not an individual\u2019s behavior or future samples. The statement <strong>&#8220;With 95% confidence, the proportion of all students who take notes is between 0.15 and 0.25&#8221;<\/strong> correctly expresses this. It reflects our degree of confidence in the <strong>procedure<\/strong>, not the probability of the interval after the fact.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The misconception that there is a &#8220;95% chance the population proportion is in this interval&#8221; wrongly treats the population proportion as variable. In fact, it&#8217;s fixed\u2014we&#8217;re uncertain about it, but it&#8217;s not changing. It&#8217;s the interval that varies from sample to sample.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Understanding this helps prevent common misinterpretations and reinforces the idea that <strong>confidence intervals are about populations, not individuals or probabilities after calculation<\/strong>.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/05\/learnexams-banner9-35.jpeg\" alt=\"\" class=\"wp-image-220086\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>A student was asked to find a 95% confidence interval for the population proportion of students who take notes using data from a random sample of size n=87. Which of the following is a correct interpretation of the interval 0.15, 0.25? Check all that are correct. With 95% confidence, a randomly selected student takes notes [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-220085","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/220085","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=220085"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/220085\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=220085"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=220085"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=220085"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}