[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-314880-105":59,"doc-detail-314880-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","2024-ap-statistics-free-response-questions-formulas-and-section-ii","2024 AP® Statistics - Free-Response Questions - Formulas and Section II","","2024 AP® Statistics free-response materials compile key formulas and reference tables for descriptive statistics, probability and distributions, and sampling distributions used in inferential statistics. The page includes expressions for mean, standard deviation, confidence intervals, standardized test statistics, chi-square calculations, and sampling distributions for proportions, means, and simple linear regression. It also provides AP exam section instructions and begins QUESTION 1 and QUESTION 2, which require completing appropriate inference procedures based on sample data and relative frequencies.",{"@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":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/2024-ap-statistics-free-response-questions-formulas-and-section-ii/314880/",{"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/2024-ap-statistics-free-response-questions-formulas-and-section-ii/314880.png","ImageObject",300,407,{"name":92,"@type":93},"Berry Peter","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-05","2026-09-20",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",16,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What types of formulas are provided in the AP Statistics reference section?","Question",{"text":113,"@type":114},"The material lists formulas for descriptive statistics, probability and distributions, and inferential statistics including confidence intervals and sampling distributions.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How is a confidence interval expressed in the provided formulas?",{"text":118,"@type":114},"A confidence interval is written as statistic ± (critical value)(standard error of statistic).",{"name":120,"@type":111,"acceptedAnswer":121},"What do QUESTION 1 and QUESTION 2 require students to do?",{"text":122,"@type":114},"QUESTION 1 asks for an appropriate inference procedure to test whether proportions differ by age group at a specified significance level. QUESTION 2 asks students to complete segmented bar graphs and address parts of a fund-raiser distribution problem.","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},314880,1789902056,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":19,"category_name":20,"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},1374402524268,"https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2","2024  \nAP® Statistics  \nFree-Response Questions  \n© 2024 College Board. College Board, Advanced Placement, AP, AP Central, and the acorn logo are registered trademarks of College Board. Visit College Board on the web: [collegeboard.org](collegeboard.org).  \n[AP Central is the official online home for the AP Program: apcentral.collegeboard.org](AP Central is the official online home for the AP Program: apcentral.collegeboard.org).  \nFormulas for AP Statistics  \nI. Descriptive Statistics  \nx~~ ~~ = 1n~~ ~~ ∑ xi = ∑nxi~~ ~~  \n = a + bx  \nr = n~~ ~~~~ ~~1 ∑ 􀂩􀂨􀂧 ~~ ~~xi~~ ~~s−x~~ ~~x~~  ~~ 􀂷􀂹􀂸 􀂩􀂨􀂧 ~~ ~~yi~~ ~~s−y~~ ~~y~~ ~~􀂷􀂹􀂸 II. Probability and Distributions  \nP ( A ∪ B) = P ( A) + P( B ) − P( A ∩ B)  \nsx = ~~ ~~n~~ ~~~~ ~~1 ∑ ( xi − x~~ ~~ )2 = y = a + bx  \nb = r ~~ ~~ssy~~ ~~  \nx  \nP ( A | B ) = P~~ ~~(AP~~(~~∩B~~)~~B~~ ~~)  \n| Probability Distribution | Mean | Standard Deviation |\n| --- | --- | --- |\n| Discrete random variable, X | µX = E ( X ) = ∑ xiP ( xi ) | σX = ∑ ( xi − µX )2 P ( xi ) |\n| If 􀜺 has a binomial distribution with parameters n and p, then:\u003Cbr>P ( X = x ) = 􀂧􀂩􀂨 nx􀂷􀂹􀂸 px ( 1 − p )n − x where x = 0, 1, 2, 3, 􀀡 , n | µX = np | σX = np ( 1 − p ) |\n| If 􀜺 has a geometric distribution with parameter p, then:\u003Cbr>P ( X = x ) = ( 1 − p )x −1 p where x = 1, 2, 3, 􀀡 | µX = 1p |  1 − p σ =\u003Cbr>X p |\n\nIII. Sampling Distributions and Inferential Statistics  \n\n| Standardized test statistic: ~~ ~~stanstatidardstic~~ ~~−~~ ~~paerror~~ ~~ofrtameterhe~~ ~~statistic |\n| --- |\n| Confidence interval: statistic ± ( critical value )( standard error of statistic) |\n\nChi-square statistic: χ 2 = ∑ ~~ ~~(~~ ~~observedexp−~~ ~~expectedected~~ ~~)2~~ ~~  \nIII. Sampling Distributions and Inferential Statistics (continued) Sampling distributions for proportions:  \n\n| Random\u003Cbr>Variable | Parameters of\u003Cbr>Sampling Distribution |  |  |  | Standard Error* of Sample Statistic |  |  |\n| --- | --- | --- | --- | --- | --- | --- | --- |\n| For one population:\u003Cbr>ˆ\u003Cbr>p | µ = p | σ = |  | | s = |  | |\n| For two populations:\u003Cbr>1 − 2 | µ1 − 2 = p1 − p2 | σ1 − 2 = | \u003Cbr>\u003Cbr>p1 ( 1 − p1 ) p2 ( 1 − p2 )+\u003Cbr>n n\u003Cbr>1 2 |  | s1 − 2 = | |  |\n|  |  |  |  |  | When p1 = p2 is assumed:\u003Cbr>s1 − 2 = c ( 1 − c ) 􀂧􀂩􀂨 1n1 + ~~ ~~1n2~~ ~~􀂷􀂹􀂸 where c = Xn11~~ ~~~~ ~~Xn22 |  |  |\n\nSampling distributions for means:  \n\n| Random\u003Cbr>Variable | Parameters of Sampling Distribution |  |  | Standard Error* of Sample Statistic |  |\n| --- | --- | --- | --- | --- | --- |\n| For one population:\u003Cbr>X | µX = µ | σ   =  σ \u003Cbr>X\u003Cbr>~~ ~~ n\u003Cbr>|  | s   =  s  X\u003Cbr>~~ ~~ n\u003Cbr>|  |\n| For two populations:\u003Cbr>X − X\u003Cbr>1 2 | µX − X = µ1 − µ2 1 2 | σ ~~ ~~ ~~ ~~ = X − X\u003Cbr>1 2 | | s ~~ ~~ ~~ ~~ = X − X\u003Cbr>1 2 | |\n\nSampling distributions for simple linear regression:  \n\n| Random\u003Cbr>Variable | Parameters of Sampling Distribution |  | Standard Error* of Sample Statistic |  |\n| --- | --- | --- | --- | --- |\n| For slope: b | µb = β | σb = σ~~ ~~σ~~ ~~n~~  ~~ , x\u003Cbr>where σ =\u003Cbr>x\u003Cbr>\u003Cbr>| \u003Cbr>s\u003Cbr>sb = ,\u003Cbr>s n − 1\u003Cbr>x\u003Cbr>where s =\u003Cbr>∑ ( yi − i )2 n − 2 |  |\n|  |  |  | and s = x | |\n\n*Standard deviation is a measurement of variability from the theoretical population. Standard error is the estimate of the standard deviation. If the standard deviation of the statistic is assumed to be known, then the standard deviation should be used instead of the standard error.  \nBegin your response to QUESTION 1 on this page.  \nSTATISTICS  \nSECTION II  \nTotal Time—1 hour and 30 minutes  \n6 Questions  \nSECTION II, Part A  \nSuggested Time—1 hour and 5 minutes  \n5 Questions  \nDirections: Show all your work. Indicate clearly the methods you use, because you will be scored on the correctness of your methods as well as on the accuracy and completeness of your results and explanations.  \n1. A large exercise center has several thousand members from age 18 to 55 years and several thousand members age  \n56 and older. The manager of the center is considering offering online fitness classes. The manager is investigating whet","cbCaikAlmWt0LBhl","https://ap.wps.com/l/cbCaikAlmWt0LBhl","pdf",3244567,21,"English","# Formulas for AP Statistics\n## I. Descriptive Statistics\n## II. Probability and Distributions\n## III. Sampling Distributions and Inferential Statistics\n# Exam Instructions and Questions\n## QUESTION 1\n## QUESTION 2","[{\"question\":\"What types of formulas are provided in the AP Statistics reference section?\",\"answer\":\"The material lists formulas for descriptive statistics, probability and distributions, and inferential statistics including confidence intervals and sampling distributions.\"},{\"question\":\"How is a confidence interval expressed in the provided formulas?\",\"answer\":\"A confidence interval is written as statistic ± (critical value)(standard error of statistic).\"},{\"question\":\"What do QUESTION 1 and QUESTION 2 require students to do?\",\"answer\":\"QUESTION 1 asks for an appropriate inference procedure to test whether proportions differ by age group at a specified significance level. QUESTION 2 asks students to complete segmented bar graphs and address parts of a fund-raiser distribution problem.\"}]","2024 AP® Statistics - Free-Response Questions - Formulas and Section II | PDF",53]