[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-268748-105":59,"doc-detail-268748-en":130},{"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":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","clt-based-inference-hypothesis-testing","CLT-based inference - hypothesis testing","","Lecture content on CLT-based inference and how to use the Central Limit Theorem for hypothesis testing in introductory data science. Defines key properties of the sampling distribution of the sample mean, including its mean, standard error, and approximate normality for sufficiently large n. Lists practical conditions for CLT (independence, random sampling, sample-size rules, sufficient skewness/normality guidance, and categorical success/failure counts). Connects these assumptions to constructing Z statistics, standardizing, and the hypothesis testing framework. Explains decision outcomes, including type I and type II errors.",{"@graph":69,"@context":122},[70,84,105],{"@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":30,"@type":76,"position":81},"https://docshare.wps.com/document/technology/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/clt-based-inference-hypothesis-testing/268748/",{"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/clt-based-inference-hypothesis-testing/268748.png","ImageObject",300,407,{"name":92,"@type":93},"Ethan Miller","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-17","2026-09-15",true,{"@type":102,"interactionType":103,"userInteractionCount":44},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What does the Central Limit Theorem say about the sampling distribution of the sample mean?","Question",{"text":112,"@type":113},"It states that the sampling distribution of the sample mean has mean equal to the population mean and standard deviation σ/√n, and for sufficiently large n its shape is approximately normal.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What conditions must hold for the CLT to be a good approximation?",{"text":117,"@type":113},"Observations should be independent, the sample should be random, and sample size should be large enough (with additional guidance for without-replacement sampling and for skewed or already-normal populations). For categorical data, common guidance is at least 10 successes and 10 failures.",{"name":119,"@type":110,"acceptedAnswer":120},"How are type I and type II errors related to the hypothesis testing decisions?",{"text":121,"@type":113},"With a given null hypothesis, the test can either fail to reject H0 or reject H0. Rejecting H0 when H0 is true corresponds to a type I error, while failing to reject H0 when H0 is false corresponds to a type II error.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},268748,1790485917,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":29,"category_name":30,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":44,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":145},687207017582,"https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d","CLT-based inferencehypothesis testing Intro to Data Science  \nShawn Santo  \n1 / 13  \nAnnouncements  \n Homework 04 out today  \n Hard deadline for Lab 06 is today  \n Keep thinking about what data you want to work with for your project  \nToday's agenda  \n Finish-up con¦dence intervals from last time  \n CLT-based inference for hypothesis testing  \nRecall  \n4 / 13  \nThe Central Limit Theorem  \nFor a population with a well-de¦ned mean μ and standard deviation σ, these three properties hold for the distribution of sample average ¯X, assuming certain conditions hold:  \n1. The mean of the sampling distribution is identical to the population mean μ,  \n2. The standard deviation of the distribution of the sample averages is σ/√n, orthe  (SE) of the mean, and  \n3. For n large enough (in the limit, as n → ∞), the shape of the sampling distribution of means is approximately normal (Gaussian) .  \nConditions  \nWhat are the conditions we need for the CLT to hold?  \n  The sampled observations must be independent. This is di¨cult to check, but the following are useful guidelines:  \n the sample must be random  \n if sampling without replacement, sample size must be less than 10% of the population size  \nif data are numerical, usually n > 30 is considered a large enough sample, but if the underlying population distribution is extremely skewed, more might be needed  \n if we know for sure that the underlying data are normal, then the distribution of sample averages will also be exactly normal, regardless of the sample size  \n if data are categorical, at least 10 successes and 10 failures.  \nCLT results: ¯X , ^p  \nAssuming the conditions for the CLT hold, ¯X approximately has the following distribution:  \nNormal (μ, σ/√n)  \nEquivalently, we can de¦ne the quantity Z, such that Z = μ~~n~~ , where Z has the following distribution:  \nNormal (0, 1)  \nAssuming the conditions for the CLT hold, ^p approximately has the following distribution:  \nNormal (p, √ ~~ ~~p~~ ~~(1p)~~ ~~ )  \nWe can standardize this in a similar way and de¦ne a quantity Z that is normally distributed with a mean of 0 and a standard deviation of 1.  \nThe hypothesis testing framework  \n1. Start with two hypotheses about the population: the null hypothesis and the alternative hypothesis.  \n2. Choose a (representative) sample, collect data, and analyze the data.  \n3. Figure out how likely it is to see data like what we observed,  the null hypothesis were in fact true.  \n4. If our data would have been extremely unlikely if the null claim were true, then we reject it and deem the alternative claim worthy of further study. Otherwise, we cannot reject the null claim.  \nThe \"errors\"  \nSuppose we test a certain null hypothesis, which can be either true or false (we never know for sure!) . We make one of two decisions given our data: either reject or fail to reject H0.  \nWe have the following four scenarios:  \n\n| | H0  | H0  |\n| --- | --- | --- |\n| Fail to reject H0 | Correct decision | Type II Error |\n| Reject H0 | Type I Error | Correct decision |\n\nIt is important to weigh the consequences of making each type of error.  \nCLT-based testing  \n10 / 13","cbCaii9hfjtQ5oao","https://ap.wps.com/l/cbCaii9hfjtQ5oao","pdf",216273,13,"English","# CLT-based inference for hypothesis testing\n## The Central Limit Theorem\n## Conditions for the CLT\n## Hypothesis testing framework\n## Errors: type I and type II\n## CLT-based testing","[{\"question\":\"What does the Central Limit Theorem say about the sampling distribution of the sample mean?\",\"answer\":\"It states that the sampling distribution of the sample mean has mean equal to the population mean and standard deviation σ/√n, and for sufficiently large n its shape is approximately normal.\"},{\"question\":\"What conditions must hold for the CLT to be a good approximation?\",\"answer\":\"Observations should be independent, the sample should be random, and sample size should be large enough (with additional guidance for without-replacement sampling and for skewed or already-normal populations). For categorical data, common guidance is at least 10 successes and 10 failures.\"},{\"question\":\"How are type I and type II errors related to the hypothesis testing decisions?\",\"answer\":\"With a given null hypothesis, the test can either fail to reject H0 or reject H0. Rejecting H0 when H0 is true corresponds to a type I error, while failing to reject H0 when H0 is false corresponds to a type II error.\"}]","CLT-based inference - hypothesis testing | PDF",1789436334,33]