[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-572083-105":3,"detail-sidebar-cat-1-en-105":81,"doc-detail-572083-en":127},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":74,"head_meta":76,"extra_data":78,"updated_unix":80},105,"en","ap-statistics-summary-of-confidence-intervals-and-hypothesis-tests","AP Statistics - Summary of Confidence Intervals and Hypothesis Tests","","AP Statistics summary table consolidating confidence interval and hypothesis test procedures across common one-sample, two-sample, and categorical settings. Covers mean and proportion intervals/tests under known and unknown sigma, includes calculator-driven confidence level and p-value options, and specifies key conditions such as SRS assumptions, normality or CLT/t robustness, and success-failure count requirements. Extends to two-sample mean and proportion comparisons using t or normal approximations, chi-square goodness-of-fit and independence/homogeneity tests, and regression slope inference with CI and t tests.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/template/","Template",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/template/general/","General",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/template/ap-statistics-summary-of-confidence-intervals-and-hypothesis-tests/572083/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/ap-statistics-summary-of-confidence-intervals-and-hypothesis-tests/572083.png","ImageObject",442,249,{"name":42,"@type":43},"OmBimo","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-21","2026-10-13",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",6,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"What conditions justify using the normal (z) procedures for one-sample proportions?","Question",{"text":63,"@type":64},"Use SRS, require the population to be at least 10 times the sample size, and ensure both success and failure counts are at least 10 so the normal approximation is valid.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"When is a t procedure used for one-sample or two-sample means?",{"text":68,"@type":64},"For means with σ unknown, the table uses t with df = n - 1 (one sample) or uses calculator df (two samples). It relies on normality or sufficiently large combined sample sizes (and robustness conditions) as listed.",{"name":70,"@type":61,"acceptedAnswer":71},"How are chi-square tests configured and what are the main requirement constraints?",{"text":72,"@type":64},"The chi-square section specifies GOF for one sample with 1 variable, independence for one sample with 2 variables, and homogeneity for two samples with 2 variables. It requires all expected counts to be at least 1 and no more than 20% of expected counts to be less than 5; p-values can be found via x2cdf.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},572083,1792105230,{"code":4,"msg":82,"data":83},"success",[84,89,94,99,104,109,114,119,124],{"id":85,"doc_module":22,"doc_module_name":25,"category_name":86,"show_sort_weight":87,"slug":88},11,"Presentations",90,"presentations",{"id":90,"doc_module":22,"doc_module_name":25,"category_name":91,"show_sort_weight":92,"slug":93},12,"Resumes",80,"resumes",{"id":95,"doc_module":22,"doc_module_name":25,"category_name":96,"show_sort_weight":97,"slug":98},14,"Invoices",70,"invoices",{"id":100,"doc_module":22,"doc_module_name":25,"category_name":101,"show_sort_weight":102,"slug":103},15,"Posters",60,"posters",{"id":105,"doc_module":22,"doc_module_name":25,"category_name":106,"show_sort_weight":107,"slug":108},16,"Social Media",50,"social-media",{"id":110,"doc_module":22,"doc_module_name":25,"category_name":111,"show_sort_weight":112,"slug":113},17,"Forms",40,"forms",{"id":115,"doc_module":22,"doc_module_name":25,"category_name":116,"show_sort_weight":117,"slug":118},18,"Letters",30,"letters",{"id":120,"doc_module":22,"doc_module_name":25,"category_name":121,"show_sort_weight":122,"slug":123},21,"Paper Templates",5,"papers-templates",{"id":125,"doc_module":22,"doc_module_name":25,"category_name":29,"show_sort_weight":4,"slug":126},158,"general-158",{"code":4,"msg":82,"data":128},{"doc_id":79,"user_id":129,"nickname":42,"user_avatar":130,"doc_module":22,"category_id":125,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":131,"file_id":132,"file_url":133,"file_type":134,"file_size":135,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":122,"language":136,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":137,"faqs":138,"seo_title":139,"seo_description":12,"update_tm":140,"read_time":26},962090893581,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","AP Statistics – Summary of Confidence Intervals and Hypothesis Tests  \n\n| Procedure | Formula | Conditions | Calculator Options |\n| --- | --- | --- | --- |\n| One Sample – Mean and Proportion |  |  |  |\n| Confidence Interval for mean µμ when σ is known | ! ± \" !~~ ~~!\\# | 1. SRS\u003Cbr>2. Given value of population standard deviation σ\u003Cbr>3. Population distribution is normal (if not stated, use CLT as long as n > 30) | |\n| Hypothesis Test for mean µμ when σ is known\u003Cbr>(Ho: µμ = µμo) | \" ! µ \\#\u003Cbr>! = ~~ ~~\u003Cbr>! | SAME AS ABOVE CI | \u003Cbr>*Can also find p-­‐‑value using 2nd-­‐‑Distr normalcdf(lower, upper, mean, sd) |\n| CI for mean µμ when σ is\u003Cbr>unknown | ! ± \" !  \\# \u003Cbr>$\u003Cbr>with df = n -­‐‑ 1\u003Cbr>| 1. SRS\u003Cbr>2. Using value of sample standard deviation s to estimate σ\u003Cbr>3. Population distribution is given as normal OR n > 30 (meaning t procedures are robust even if skewness and outliers exist) OR\u003Cbr>15 \u003C n \u003C 30 with normal probability plot showing little skewness and no extreme outliers OR n \u003C 15 with npp showing no outliers and noskewness | \u003Cbr>|\n\nAP Statistics – Summary of Confidence Intervals and Hypothesis Tests  \n\n| Procedure | Formula |  | Conditions | Calculator Options |\n| --- | --- | --- | --- | --- |\n| One Sample – Mean and Proportion – Continued |  |  |  |  |\n| Test for mean µμ when σ is unknown\u003Cbr>(Ho: µμ = µμo) | \" ! µ\u003Cbr>! =  \\#\u003Cbr>$\u003Cbr>with df = n -­‐‑ 1 |  | SAME AS ABOVE CI | \u003Cbr>*Can also find p-­‐‑value using 2nd-­‐‑Distr tcdf(lower, upper, df) |\n| CI for proportion p | ˆp± z * | | 1. SRS\u003Cbr>2. Population is at least 10 times n\u003Cbr>3. Counts of success nˆp and failures n(1-­‐ ˆp) are both at least\u003Cbr>10 (these counts verify the use of the normal approximation) | |\n| Test for\u003Cbr>proportion p\u003Cbr>(Ho: p = po) | \u003Cbr>z =\u003Cbr>ˆp -­‐ popo (1-­‐ po )\u003Cbr>n |  | 1. SRS\u003Cbr>2. Population is at least 10 times n\u003Cbr>3. Counts of success npo and failures n(1-­‐ po ) are both at least\u003Cbr>10 (these counts verify the use of the normal approximation) | \u003Cbr>*Can also find p-­‐‑value using 2nd-­‐‑Distr normalcdf(lower, upper, mean, sd) |\n\nAP Statistics – Summary of Confidence Intervals and Hypothesis Tests  \n\n| Procedure | Formula |  | Conditions | Calculator Options |\n| --- | --- | --- | --- | --- |\n| Two Samples – Means and Proportions |  |  |  |  |\n| CI for mean\u003Cbr>µμ1-­‐‑µμ2 when σ is\u003Cbr>unknown | (x1 -­‐ x2 ) ± t * | | 1. Populations are independent\u003Cbr>2. Both samples are from SRSs\u003Cbr>3. Using value of sample standard deviation s to estimate σ\u003Cbr>4. Population distributions are given as normal OR n1 + n2 > 30 (meaning t procedures are robust even if skewness and outliers exist) OR 15 \u003C n1 + n2 \u003C 30 with normal probability plots showing little skewness and no extreme outliers OR n1 + n2 \u003C 15 with npps showing no outliers and noskewness | \u003Cbr>|\n|  | with df read from calculator or use conservative estimate that df = n – 1 where n is the smaller of n1 or n2 |  |  |  |\n| Test for mean\u003Cbr>µμ 1-­‐‑ µμ2 when σ is\u003Cbr>unknown\u003Cbr>(Ho: µμ1 = µμ2) | \u003Cbr>t =\u003Cbr>(x1 -­‐ x2 )\u003Cbr>2 s1\u003Cbr>n1\u003Cbr>+\u003Cbr>2 s2\u003Cbr>n2\u003Cbr>with df read from calculator |  | SAME AS ABOVE CI | \u003Cbr>*Can also find p-­‐‑value using 2nd-­‐‑Distr tcdf(lower, upper, df) where df is either conservative estimate or value using long formula that calculator does automatically! |\n\nAP Statistics – Summary of Confidence Intervals and Hypothesis Tests  \n\n| Procedure | Formula | Conditions | Calculator Options |\n| --- | --- | --- | --- |\n| Two Samples – Means and Proportions – Continued |  |  |  |\n| Test for\u003Cbr>proportion\u003Cbr>p1 – p2 | ! \"\"\\# ! \"\"$ %\u003Cbr>! = ~~ ~~\u003Cbr> \"\"\\# !\\#! \"\"\\# %  \\#  +  \\# %''\u003Cbr>\\#$\\# $$&''\u003Cbr>where ˆpc = xn11~~ ~~xn22 | 1-­‐‑3 are SAME AS ABOVE CI\u003Cbr>4. Counts of success n1ˆpc and n2 ˆpc and failures n1 (1-­‐ ˆpc ) and n2 (1-­‐ ˆpc ) are all at least 5 (these counts verify the use of the normal approximation) | \u003Cbr>*Can also find p-­‐‑value using 2nd-­‐‑Distr normalcdf(lower, upper, mean, sd) where mean and sd are values from numerator and denominator of the formula fo","cbCaimlxh8mIpDBy","https://ap.wps.com/l/cbCaimlxh8mIpDBy","pdf",645167,"English","# One Sample - Mean and Proportion\n## Mean CI when σ is known\n## Mean Hypothesis test when σ is known\n## Mean CI when σ is unknown\n## Mean test when σ is unknown\n## Proportion CI\n## Proportion test\n# Two Samples - Means and Proportions\n## Mean CI (σ unknown)\n## Mean test (σ unknown)\n## Proportion test p1 - p2\n# Categorical Distributions\n## Chi-Square Test (GOF, Independence, Homogeneity)\n# Slope - Regression\n## CI for β\n## Test for β\n# Variable Legend\n## Common variables and symbols\n# Matched Pairs","[{\"question\":\"What conditions justify using the normal (z) procedures for one-sample proportions?\",\"answer\":\"Use SRS, require the population to be at least 10 times the sample size, and ensure both success and failure counts are at least 10 so the normal approximation is valid.\"},{\"question\":\"When is a t procedure used for one-sample or two-sample means?\",\"answer\":\"For means with σ unknown, the table uses t with df = n - 1 (one sample) or uses calculator df (two samples). It relies on normality or sufficiently large combined sample sizes (and robustness conditions) as listed.\"},{\"question\":\"How are chi-square tests configured and what are the main requirement constraints?\",\"answer\":\"The chi-square section specifies GOF for one sample with 1 variable, independence for one sample with 2 variables, and homogeneity for two samples with 2 variables. It requires all expected counts to be at least 1 and no more than 20% of expected counts to be less than 5; p-values can be found via x2cdf.\"}]","AP Statistics - Summary of Confidence Intervals and Hypothesis Tests | PDF",1791925366]