{"id":215777,"date":"2025-05-17T18:34:08","date_gmt":"2025-05-17T18:34:08","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=215777"},"modified":"2025-05-17T18:34:10","modified_gmt":"2025-05-17T18:34:10","slug":"the-following-data-represent-the-age-in-weeks-at-which-babies-first-crawl-based-on-a-survey-of-12-mothers-3","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/17\/the-following-data-represent-the-age-in-weeks-at-which-babies-first-crawl-based-on-a-survey-of-12-mothers-3\/","title":{"rendered":"The following data represent the age (in weeks) at which babies first crawl based on a survey of 12 mothers."},"content":{"rendered":"\n<p>The following data represent the age (in weeks) at which babies first crawl based on a survey of 12 mothers. The data are normally distributed and s = 9.950 weeks. Construct and interpret a 90% confidence interval for the population standard deviation of the age (in weeks) at which babies first crawl. Click the icon to view the table of critical values of the chi-square distribution. Select the correct choice below and fill in the answer boxes to complete your choice. (Use ascending order. Round to three decimal places as needed.)<\/p>\n\n\n\n<p>A. There is a 90% probability that the true population standard deviation is between and<br>B. There is 90% confidence that the population standard deviation is between and<br>C. If repeated samples are taken, 90% of them will have the sample standard deviation between and<\/p>\n\n\n\n<p><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<p>We are asked to construct a <strong>90% confidence interval for the population standard deviation<\/strong> using sample data. Here&#8217;s how to do it step-by-step.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Given Information:<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Sample size ($n$) = 12<\/li>\n\n\n\n<li>Sample standard deviation ($s$) = 9.950 weeks<\/li>\n\n\n\n<li>The data is <strong>normally distributed<\/strong><\/li>\n\n\n\n<li>Confidence level = <strong>90%<\/strong><\/li>\n\n\n\n<li>Distribution used: <strong>Chi-square distribution<\/strong><\/li>\n\n\n\n<li>Degrees of freedom ($df$) = $n &#8211; 1 = 11$<\/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\"><strong>Step 1: Use Chi-Square Formula for Confidence Interval<\/strong><\/h3>\n\n\n\n<p>To find the confidence interval for the population standard deviation $\\sigma$, use the formula:<\/p>\n\n\n\n<p>$$<br>\\left( \\sqrt{\\frac{(n &#8211; 1) s^2}{\\chi^2_{\\alpha\/2}}}, \\sqrt{\\frac{(n &#8211; 1) s^2}{\\chi^2_{1 &#8211; \\alpha\/2}}} \\right)<br>$$<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$\\chi^2_{\\alpha\/2}$ and $\\chi^2_{1 &#8211; \\alpha\/2}$ are critical values from the chi-square distribution<\/li>\n\n\n\n<li>$\\alpha = 1 &#8211; 0.90 = 0.10 \\Rightarrow \\alpha\/2 = 0.05$<\/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\"><strong>Step 2: Find Chi-Square Critical Values<\/strong><\/h3>\n\n\n\n<p>From the chi-square table for $df = 11$:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$\\chi^2_{0.05} = 19.675$<\/li>\n\n\n\n<li>$\\chi^2_{0.95} = 4.575$<\/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\"><strong>Step 3: Plug into the Formula<\/strong><\/h3>\n\n\n\n<p>$$<br>\\text{Lower bound} = \\sqrt{\\frac{(11)(9.950)^2}{19.675}} = \\sqrt{\\frac{(11)(99.00)}{19.675}} = \\sqrt{\\frac{1089.96}{19.675}} = \\sqrt{55.39} \\approx 7.441<br>$$<\/p>\n\n\n\n<p>$$<br>\\text{Upper bound} = \\sqrt{\\frac{(11)(9.950)^2}{4.575}} = \\sqrt{\\frac{1089.96}{4.575}} = \\sqrt{238.16} \\approx 15.433<br>$$<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Answer:<\/strong><\/h3>\n\n\n\n<p>$$<br>\\boxed{(7.441,\\ 15.433)}<br>$$<\/p>\n\n\n\n<p>So the correct choice is:<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>B. There is 90% confidence that the population standard deviation is between 7.441 and 15.433 weeks.<\/strong><\/h3>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Explanation (300+ words):<\/strong><\/h3>\n\n\n\n<p>In inferential statistics, confidence intervals allow us to estimate a population parameter (like the standard deviation) based on a sample. In this case, we aim to estimate the <strong>population standard deviation<\/strong> of the age at which babies first crawl using a <strong>90% confidence interval<\/strong>.<\/p>\n\n\n\n<p>We start with a <strong>sample standard deviation (s = 9.950 weeks)<\/strong> taken from 12 babies. Since the sample size is small and the population is normally distributed, we use the <strong>chi-square distribution<\/strong> to construct the confidence interval for the population standard deviation.<\/p>\n\n\n\n<p>The formula for the confidence interval of a population <strong>standard deviation<\/strong> relies on chi-square critical values and the sample variance. We determine the degrees of freedom (df = n &#8211; 1 = 11) and look up the chi-square values corresponding to the <strong>5th and 95th percentiles<\/strong> (because it\u2019s a 90% confidence interval, and the remaining 10% is split between the two tails of the distribution).<\/p>\n\n\n\n<p>Using these chi-square values (4.575 and 19.675), we plug them into the standard deviation formula to get the interval:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Lower limit \u2248 7.441 weeks<\/li>\n\n\n\n<li>Upper limit \u2248 15.433 weeks<\/li>\n<\/ul>\n\n\n\n<p>This tells us that we can be <strong>90% confident<\/strong> the true population standard deviation of the age when babies first crawl lies within this range. This does <strong>not<\/strong> mean there&#8217;s a 90% chance the true value is in this interval\u2014it means that if we repeated this sampling many times, <strong>90% of the calculated intervals would contain<\/strong> the true standard deviation.<\/p>\n\n\n\n<p>Option B is the correct interpretation, while A misuses the word \u201cprobability\u201d (which applies before sampling), and C incorrectly describes the behavior of <strong>sample standard deviations<\/strong>, not the <strong>confidence interval for the population standard deviation<\/strong>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\u2705 Final Answer:<\/h3>\n\n\n\n<p><strong>B. There is 90% confidence that the population standard deviation is between 7.441 and 15.433 weeks.<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"<p>The following data represent the age (in weeks) at which babies first crawl based on a survey of 12 mothers. The data are normally distributed and s = 9.950 weeks. Construct and interpret a 90% confidence interval for the population standard deviation of the age (in weeks) at which babies first crawl. Click the icon [&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-215777","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/215777","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=215777"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/215777\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=215777"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=215777"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=215777"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}