[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-268685-105":59,"doc-detail-268685-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","confidence-intervals-and-the-central-limit-theorem","Confidence Intervals and the Central Limit Theorem","","A focused set of notes on constructing confidence intervals by leveraging the central limit theorem and bootstrap-based ideas. The material starts from sampling variability and explains how to form point estimates, then uses resampling to approximate the sampling distribution. It introduces the 2-SE approach and the percentile method for interval construction, generalizes normal-based interval formulas to any confidence level, and reviews the Normal distribution, its parameters, and the 95% rule with short practice prompts using common normal variants.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/confidence-intervals-and-the-central-limit-theorem/268685/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/confidence-intervals-and-the-central-limit-theorem/268685.png","ImageObject",300,407,{"name":92,"@type":93},"Mabel","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-06","2026-09-15",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",13,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What is the basic procedure for building confidence intervals discussed in the document?","Question",{"text":113,"@type":114},"The notes outline finding a point estimate from the sample, bootstrapping the data to mimic sampling variability, and then constructing a confidence interval using either the 2-SE method or percentile method from the bootstrap distribution.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How does the 2-SE method determine a confidence interval?",{"text":118,"@type":114},"It uses the bootstrap distribution to estimate the standard error of the point estimate (2-SE). The interval center is the point estimate, and the interval width is controlled by the standard error and a multiplier tied to the desired confidence level.",{"name":120,"@type":111,"acceptedAnswer":121},"Why does the document emphasize the Normal distribution and the 95% rule?",{"text":122,"@type":114},"Because normality enables interpreting areas under the curve as probabilities and supports generalizing confidence interval formulas. The 95% rule provides practical intuition for the middle mass of a Normal curve, and practice questions use standard and shifted/scaled Normal distributions.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},268685,1789434735,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":140,"language":141,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":142,"faqs":143,"seo_title":144,"seo_description":67,"update_tm":130,"read_time":145,"preview_page_count":4},7971474920318,"https://ap-avatar.wpscdn.com/avatar/74000ee537e6b0ca360?x-image-process=image/resize,m_fixed,w_180,h_180&k=1788430171632181674","Confidence Intervals and the Central Limit Theorem  \nRyan Miller  \n1/30  \nOutline  \n1. The Normal distribution  \n2. Central Limit Theorem  \n3. Confidence intervals using the Central Limit theorem  \nIntroduction  \nLately we’ve been focused on addressing the issue of sampling variability using confidence intervals. The general procedure looks like:  \n1) Use the sample data to find a point estimate of the population parameter of interest  \nIntroduction  \nLately we’ve been focused on addressing the issue of sampling variability using confidence intervals. The general procedure looks like:  \n1) Use the sample data to find a point estimate of the population parameter of interest  \n2) Bootstrap the sample data to mimic the process of sampling from the population (ie: sampling variability)  \nIntroduction  \nLately we’ve been focused on addressing the issue of sampling variability using confidence intervals. The general procedure looks like:  \n1) Use the sample data to find a point estimate of the population parameter of interest  \n2) Bootstrap the sample data to mimic the process of sampling from the population (ie: sampling variability)  \n3) Construct a confidence interval by using the bootstrap distribution to estimate the point estimate’s standard error (2-SE method) or by finding percentiles among the bootstrapped estimates (percentile method)  \nGeneralizing the 2-SE method  \nI The 2-SE method works when the bootstrap distribution is bell-shaped (due to the 68-95-99)  \nI Using the Normal Distribution, this approach can be generalized to confidence intervals (of any confidence level):  \nPoint Estimate ± c ∗ SE  \nI Within the interval’s MOE, the multiplier, c, can be adjusted to achieve any desired confidence level  \nThe Normal distribution  \nThe Normal curve, or Normal probability function, is a mathematical function that yields a bell-shaped distribution:  \n1  (X −µ)2  \nf (X ) = ~~  ~~ e − 2σ2  \n√2πσ2  \nI The parameter µ is a constant that defines the center of the bell-curve  \nI The parameter σ is a constant that defines the standard deviation of the bell-curve (how peaked or flat it is)  \n5/30  \nThe Normal distribution  \nThe Normal curve, or Normal probability function, is a mathematical function that yields a bell-shaped distribution:  \n1 − (X −µ)2  \nf (X ) = ~~ ~~ e 2σ2  \n√2πσ2  \nI  \nI  \nI  \nThe parameter µ is a constant that defines the center of the bell-curve  \nThe parameter σ is a constant that defines the standard deviation of the bell-curve (how peaked or flat it is)  \nThere infinitely many different Normal curves, one for each combination of µ and σ  \nI We will describe them using the notation: N (µ, σ)  \nThe Normal distribution  \nIf data follow a Normal distribution, the area under the  \ncurve describes the likelihood you see a value within a particular range:  \nThe 95% Rule  \nBecause the Normal probability function doesn’t have a closed-form integral, we must use software to find these areas  \nPractice  \nThe Theoretical Distributions section of StatKey allows us to find areas under various Normal curves:  \n1) Consider the Standard Normal distribution, or N(0,1), what values define the middle 90% of this distribution?  \n2) Consider a N(10,5) distribution, what proportion of this distribution is larger than 16?","cbCainrSpFDVyMUP","https://ap.wps.com/l/cbCainrSpFDVyMUP","pdf",356663,35,"English","# Outline\n## The Normal distribution\n## Central Limit Theorem\n## Confidence intervals using the Central Limit theorem\n## Generalizing the 2-SE method\n## The 95% Rule\n## Practice","[{\"question\":\"What is the basic procedure for building confidence intervals discussed in the document?\",\"answer\":\"The notes outline finding a point estimate from the sample, bootstrapping the data to mimic sampling variability, and then constructing a confidence interval using either the 2-SE method or percentile method from the bootstrap distribution.\"},{\"question\":\"How does the 2-SE method determine a confidence interval?\",\"answer\":\"It uses the bootstrap distribution to estimate the standard error of the point estimate (2-SE). The interval center is the point estimate, and the interval width is controlled by the standard error and a multiplier tied to the desired confidence level.\"},{\"question\":\"Why does the document emphasize the Normal distribution and the 95% rule?\",\"answer\":\"Because normality enables interpreting areas under the curve as probabilities and supports generalizing confidence interval formulas. The 95% rule provides practical intuition for the middle mass of a Normal curve, and practice questions use standard and shifted/scaled Normal distributions.\"}]","Confidence Intervals and the Central Limit Theorem | PDF",88]