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.DocSearch-Hit-text,.DocSearch-Hit[aria-selected=true] .DocSearch-Hit-title,.DocSearch-Hit[aria-selected=true] mark{color:var(--docsearch-hit-active-color)!important}.DocSearch-ErrorScreen,.DocSearch-NoResults,.DocSearch-StartScreen{font-size:.9em;margin:0 auto;padding:36px 0;text-align:center;width:80%}.DocSearch-Screen-Icon{color:var(--docsearch-muted-color);padding-bottom:12px}.DocSearch-NoResults-Prefill-List{display:inline-block;padding-bottom:24px;text-align:left}.DocSearch-NoResults-Prefill-List ul{display:inline-block;padding:8px 0 0}.DocSearch-NoResults-Prefill-List li{list-style-position:inside;list-style-type:"» 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*{box-sizing:border-box}.toggleButton_gllP,html{-webkit-tap-highlight-color:transparent}:root{--ifm-color-scheme:light;--ifm-dark-value:10%;--ifm-darker-value:15%;--ifm-darkest-value:30%;--ifm-light-value:15%;--ifm-lighter-value:30%;--ifm-lightest-value:50%;--ifm-contrast-background-value:90%;--ifm-contrast-foreground-value:70%;--ifm-contrast-background-dark-value:70%;--ifm-contrast-foreground-dark-value:90%;--ifm-color-primary:#3578e5;--ifm-color-secondary:#ebedf0;--ifm-color-success:#00a400;--ifm-color-info:#54c7ec;--ifm-color-warning:#ffba00;--ifm-color-danger:#fa383e;--ifm-color-primary-dark:#306cce;--ifm-color-primary-darker:#2d66c3;--ifm-color-primary-darkest:#2554a0;--ifm-color-primary-light:#538ce9;--ifm-color-primary-lighter:#72a1ed;--ifm-color-primary-lightest:#9abcf2;--ifm-color-primary-contrast-background:#ebf2fc;--ifm-color-primary-contrast-foreground:#102445;--ifm-color-secondary-dark:#d4d5d8;--ifm-color-secondary-darker:#c8c9cc;--ifm-color-secondary-darkest:#a4a6a8;--ifm-color-secondary-light:#eef0f2;--ifm-color-secondary-lighter:#f1f2f5;--ifm-color-secondary-lightest:#f5f6f8;--ifm-color-secondary-contrast-background:#fdfdfe;--ifm-color-secondary-contrast-foreground:#474748;--ifm-color-success-dark:#009400;--ifm-color-success-darker:#008b00;--ifm-color-success-darkest:#007300;--ifm-color-success-light:#26b226;--ifm-color-success-lighter:#4dbf4d;--ifm-color-success-lightest:#80d280;--ifm-color-success-contrast-background:#e6f6e6;--ifm-color-success-contrast-foreground:#003100;--ifm-color-info-dark:#4cb3d4;--ifm-color-info-darker:#47a9c9;--ifm-color-info-darkest:#3b8ba5;--ifm-color-info-light:#6ecfef;--ifm-color-info-lighter:#87d8f2;--ifm-color-info-lightest:#aae3f6;--ifm-color-info-contrast-background:#eef9fd;--ifm-color-info-contrast-foreground:#193c47;--ifm-color-warning-dark:#e6a700;--ifm-color-warning-darker:#d99e00;--ifm-color-warning-darkest:#b38200;--ifm-color-warning-light:#ffc426;--ifm-color-warning-lighter:#ffcf4d;--ifm-color-warning-lightest:#ffdd80;--ifm-color-warning-contrast-background:#fff8e6;--ifm-color-warning-contrast-foreground:#4d3800;--ifm-color-danger-dark:#e13238;--ifm-color-danger-darker:#d53035;--ifm-color-danger-darkest:#af272b;--ifm-color-danger-light:#fb565b;--ifm-color-danger-lighter:#fb7478;--ifm-color-danger-lightest:#fd9c9f;--ifm-color-danger-contrast-background:#ffebec;--ifm-color-danger-contrast-foreground:#4b1113;--ifm-color-white:#fff;--ifm-color-black:#000;--ifm-color-gray-0:var(--ifm-color-white);--ifm-color-gray-100:#f5f6f7;--ifm-color-gray-200:#ebedf0;--ifm-color-gray-300:#dadde1;--ifm-color-gray-400:#ccd0d5;--ifm-color-gray-500:#bec3c9;--ifm-color-gray-600:#8d949e;--ifm-color-gray-700:#606770;--ifm-color-gray-800:#444950;--ifm-color-gray-900:#1c1e21;--ifm-color-gray-1000:var(--ifm-color-black);--ifm-color-emphasis-0:var(--ifm-color-gray-0);--ifm-color-emphasis-100:var(--ifm-color-gray-100);--ifm-color-emphasis-200:var(--ifm-color-gray-200);--ifm-color-emphasis-300:var(--ifm-color-gray-300);--ifm-color-emphasis-400:var(--ifm-color-gray-400);--ifm-color-emphasis-600:var(--ifm-color-gray-600);--ifm-color-emphasis-700:var(--ifm-color-gray-700);--ifm-color-emphasis-800:var(--ifm-color-gray-800);--ifm-color-emphasis-900:var(--ifm-color-gray-900);--ifm-color-emphasis-1000:var(--ifm-color-gray-1000);--ifm-color-content:var(--ifm-color-emphasis-900);--ifm-color-content-inverse:var(--ifm-color-emphasis-0);--ifm-color-content-secondary:#525860;--ifm-background-color:#0000;--ifm-background-surface-color:var(--ifm-color-content-inverse);--ifm-global-border-width:1px;--ifm-global-radius:0.4rem;--ifm-hover-overlay:#0000000d;--ifm-font-color-base:var(--ifm-color-content);--ifm-font-color-base-inverse:var(--ifm-color-content-inverse);--ifm-font-color-secondary:var(--ifm-color-content-secondary);--ifm-font-family-base:system-ui,-apple-system,Segoe 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#0000001a;--ifm-z-index-dropdown:100;--ifm-z-index-fixed:200;--ifm-z-index-overlay:400;--ifm-container-width:1140px;--ifm-container-width-xl:1320px;--ifm-code-background:#f6f7f8;--ifm-code-border-radius:var(--ifm-global-radius);--ifm-code-font-size:90%;--ifm-code-padding-horizontal:0.1rem;--ifm-code-padding-vertical:0.1rem;--ifm-pre-background:var(--ifm-code-background);--ifm-pre-border-radius:var(--ifm-code-border-radius);--ifm-pre-color:inherit;--ifm-pre-line-height:1.45;--ifm-pre-padding:1rem;--ifm-heading-color:inherit;--ifm-heading-margin-top:0;--ifm-heading-margin-bottom:var(--ifm-spacing-vertical);--ifm-heading-font-family:var(--ifm-font-family-base);--ifm-heading-font-weight:var(--ifm-font-weight-bold);--ifm-heading-line-height:1.25;--ifm-h1-font-size:2rem;--ifm-h2-font-size:1.5rem;--ifm-h3-font-size:1.25rem;--ifm-h4-font-size:1rem;--ifm-h5-font-size:0.875rem;--ifm-h6-font-size:0.85rem;--ifm-image-alignment-padding:1.25rem;--ifm-leading-desktop:1.25;--ifm-leading:calc(var(--ifm-leading-desktop)*1rem);--ifm-list-left-padding:2rem;--ifm-list-margin:1rem;--ifm-list-item-margin:0.25rem;--ifm-list-paragraph-margin:1rem;--ifm-table-cell-padding:0.75rem;--ifm-table-background:#0000;--ifm-table-stripe-background:#00000008;--ifm-table-border-width:1px;--ifm-table-border-color:var(--ifm-color-emphasis-300);--ifm-table-head-background:inherit;--ifm-table-head-color:inherit;--ifm-table-head-font-weight:var(--ifm-font-weight-bold);--ifm-table-cell-color:inherit;--ifm-link-color:var(--ifm-color-primary);--ifm-link-decoration:none;--ifm-link-hover-color:var(--ifm-link-color);--ifm-link-hover-decoration:underline;--ifm-paragraph-margin-bottom:var(--ifm-leading);--ifm-blockquote-font-size:var(--ifm-font-size-base);--ifm-blockquote-border-left-width:2px;--ifm-blockquote-padding-horizontal:var(--ifm-spacing-horizontal);--ifm-blockquote-padding-vertical:0;--ifm-blockquote-shadow:none;--ifm-blockquote-color:var(--ifm-color-emphasis-800);--ifm-blockquote-border-color:var(--ifm-color-emphasis-300);--ifm-hr-background-color:var(--ifm-color-emphasis-500);--ifm-hr-height:1px;--ifm-hr-margin-vertical:1.5rem;--ifm-scrollbar-size:7px;--ifm-scrollbar-track-background-color:#f1f1f1;--ifm-scrollbar-thumb-background-color:silver;--ifm-scrollbar-thumb-hover-background-color:#a7a7a7;--ifm-alert-background-color:inherit;--ifm-alert-border-color:inherit;--ifm-alert-border-radius:var(--ifm-global-radius);--ifm-alert-border-width:0px;--ifm-alert-border-left-width:5px;--ifm-alert-color:var(--ifm-font-color-base);--ifm-alert-padding-horizontal:var(--ifm-spacing-horizontal);--ifm-alert-padding-vertical:var(--ifm-spacing-vertical);--ifm-alert-shadow:var(--ifm-global-shadow-lw);--ifm-avatar-intro-margin:1rem;--ifm-avatar-intro-alignment:inherit;--ifm-avatar-photo-size:3rem;--ifm-badge-background-color:inherit;--ifm-badge-border-color:inherit;--ifm-badge-border-radius:var(--ifm-global-radius);--ifm-badge-border-width:var(--ifm-global-border-width);--ifm-badge-color:var(--ifm-color-white);--ifm-badge-padding-horizontal:calc(var(--ifm-spacing-horizontal)*0.5);--ifm-badge-padding-vertical:calc(var(--ifm-spacing-vertical)*0.25);--ifm-breadcrumb-border-radius:1.5rem;--ifm-breadcrumb-spacing:0.5rem;--ifm-breadcrumb-color-active:var(--ifm-color-primary);--ifm-breadcrumb-item-background-active:var(--ifm-hover-overlay);--ifm-breadcrumb-padding-horizontal:0.8rem;--ifm-breadcrumb-padding-vertical:0.4rem;--ifm-breadcrumb-size-multiplier:1;--ifm-breadcrumb-separator:url('data:image/svg+xml;utf8,');--ifm-breadcrumb-separator-filter:none;--ifm-breadcrumb-separator-size:0.5rem;--ifm-breadcrumb-separator-size-multiplier:1.25;--ifm-button-background-color:inherit;--ifm-button-border-color:var(--ifm-button-background-color);--ifm-button-border-width:var(--ifm-global-border-width);--ifm-button-font-weight:var(--ifm-font-weight-bold);--ifm-button-padding-horizontal:1.5rem;--ifm-button-padding-vertical:0.375rem;--ifm-button-size-multiplier:1;--ifm-button-transition-duration:var(--ifm-transition-fast);--ifm-button-border-radius:calc(var(--ifm-global-radius)*var(--ifm-button-size-multiplier));--ifm-button-group-spacing:2px;--ifm-card-background-color:var(--ifm-background-surface-color);--ifm-card-border-radius:calc(var(--ifm-global-radius)*2);--ifm-card-horizontal-spacing:var(--ifm-global-spacing);--ifm-card-vertical-spacing:var(--ifm-global-spacing);--ifm-toc-border-color:var(--ifm-color-emphasis-300);--ifm-toc-link-color:var(--ifm-color-content-secondary);--ifm-toc-padding-vertical:0.5rem;--ifm-toc-padding-horizontal:0.5rem;--ifm-dropdown-background-color:var(--ifm-background-surface-color);--ifm-dropdown-font-weight:var(--ifm-font-weight-semibold);--ifm-dropdown-link-color:var(--ifm-font-color-base);--ifm-dropdown-hover-background-color:var(--ifm-hover-overlay);--ifm-footer-background-color:var(--ifm-color-emphasis-100);--ifm-footer-color:inherit;--ifm-footer-link-color:var(--ifm-color-emphasis-700);--ifm-footer-link-hover-color:var(--ifm-color-primary);--ifm-foot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It tests whether two samples come from the same population distribution. To conduct a permutation test:"]}),(0,s.jsxs)(t.ol,{children:["\n",(0,s.jsx)(t.li,{children:"Shuffle the group labels (i.e. the Trues and Falses) to generate two new samples under the null."}),"\n",(0,s.jsx)(t.li,{children:"Compute the test statistic (i.e. the difference in group means)."}),"\n",(0,s.jsx)(t.li,{children:"Repeat steps 1 and 2 to generate an empirical distribution of the test statistics (i.e. the difference in group means)."}),"\n",(0,s.jsx)(t.li,{children:"See where the observed statistic lies in the empirical distribution. 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The observed differences in our samples cannot be explained by random chance alone."]}),"\n"]})]}),"\n",(0,s.jsxs)(t.p,{children:['Since the alternative hypothesis is of the form "A and B are different," the ',(0,s.jsx)(t.strong,{children:"test statistic"})," should measure distance and use an absolute value."]}),"\n",(0,s.jsxs)(t.p,{children:[(0,s.jsxs)(t.span,{className:"katex",children:[(0,s.jsx)(t.span,{className:"katex-mathml",children:(0,s.jsx)(t.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,s.jsxs)(t.semantics,{children:[(0,s.jsx)(t.mrow,{children:(0,s.jsx)(t.mo,{children:"\u2234"})}),(0,s.jsx)(t.annotation,{encoding:"application/x-tex",children:"\\therefore"})]})})}),(0,s.jsx)(t.span,{className:"katex-html","aria-hidden":"true",children:(0,s.jsxs)(t.span,{className:"base",children:[(0,s.jsx)(t.span,{className:"strut",style:{height:"0.6922em"}}),(0,s.jsx)(t.span,{className:"mrel amsrm",children:"\u2234"})]})})]})," We use ",(0,s.jsx)(t.strong,{children:"absolute difference in group means"})," as the test statistic."]}),"\n",(0,s.jsx)(t.hr,{}),"\n",(0,s.jsx)(t.h3,{id:"2-query-the-dataframe",children:"2. Query the DataFrame"}),"\n",(0,s.jsx)(t.p,{children:"Since we want to compare the distributions of only cats and dogs, we need to make sure to only include the relevant pieces of data (e.g., cats and dogs weights)."}),"\n",(0,s.jsx)(t.pre,{children:(0,s.jsx)(t.code,{className:"language-python",children:"# Create a boolean Series that references which rows fulfill either condition.\nquerying_condition = (full_pets.get('Species') == 'dog') | (full_pets.get('Species') == 'cat')\n# Query.\ncats_dogs = full_pets[querying_condition]\n# Display the first 5 rows only.\ncats_dogs.take(np.arange(5)) \n"})}),"\n",(0,s.jsx)(a.A,{data:'{"columns":["ID","Species","Color","Weight","Age","Is_Cat","Owner_Comment"],"index":[0,1,2,3,4],"data":[["dog_001","dog","black",40.0,5.0,false," There are no bad dogs, only bad owners."],["cat_001","cat","golden",1.5,0.2,true,"My best birthday present ever!!!"],["cat_002","cat","black",15.0,9.0,true,"****All you need is love and a cat.****"],["dog_002","dog","white",80.0,2.0,false,"Love is a wet nose and a wagging tail."],["dog_003","dog","black",25.0,0.5,false,"Be the person your dog thinks you are."]]}'}),"\n",(0,s.jsx)(t.hr,{}),"\n",(0,s.jsx)(t.h3,{id:"3-create-a-function-to-calculate-test-statistic",children:"3. Create a function to calculate test statistic"}),"\n",(0,s.jsx)(t.p,{children:"Since our hypotheses depend on the test statistic, create a function to be able to calculate it during every trial of our permutation test."}),"\n",(0,s.jsx)(t.pre,{children:(0,s.jsx)(t.code,{className:"language-python",children:"def difference_in_means(cats_dogs):\n \"\"\"\n Calculate the absolute difference in the mean weight of dogs and cats.\n ---\n Input:\n cats_dogs: a DataFrame containing the columns 'Species' and 'Weight'.\n ---\n Output:\n The absolute difference in the mean weight of dogs and cats.\n \"\"\"\n means = cats_dogs.groupby('Species').mean()\n return np.abs((means.get('Weight').loc['dog'] - means.get('Weight').loc['cat']))\n"})}),"\n",(0,s.jsx)(t.p,{children:(0,s.jsx)(t.em,{children:"Note: Although we can simply do this in the for-loop shown in the next step, we can practice good coding habits by separating our code into readable bits!"})}),"\n",(0,s.jsx)(t.hr,{}),"\n",(0,s.jsx)(t.h3,{id:"4-simulate-the-permutation-test",children:"4. Simulate the permutation test"}),"\n",(0,s.jsx)(t.pre,{children:(0,s.jsx)(t.code,{className:"language-python",children:"n = 500 # Number of simulations.\nstatistics = np.array([]) # Array to keep track of the difference in means for each iteration.\nfor i in np.arange(n): # Run the simulation `n` number of times\n # 1. Shuffle the species.\n shuffled = cats_dogs.assign(Species=np.random.permutation(cats_dogs.get('Species')))\n\n # 2. Compute the test statistic.\n statistic = difference_in_means(shuffled)\n\n # 3. Save the result.\n statistics = np.append(statistics, statistic)\n"})}),"\n",(0,s.jsxs)(t.p,{children:["This code will run the permutation test 500 times, but a different reasonable number can be used instead. It is ",(0,s.jsx)(t.strong,{children:"crucial"})," to keep track of the difference in means each time our for-loop runs so that the number of simulated values can be displayed."]}),"\n",(0,s.jsx)(t.hr,{}),"\n",(0,s.jsx)(t.h3,{id:"5-conclusion",children:"5. Conclusion"}),"\n",(0,s.jsx)(t.pre,{children:(0,s.jsx)(t.code,{className:"language-python",children:'observed = difference_in_means(cats_dogs)\np_value = np.count_nonzero(statistics >= observed) / n\n\nprint("The observed value of the test statistic is:", observed)\nprint("The p-value is:", p_value)\n'})}),"\n",(0,s.jsxs)(t.p,{children:[(0,s.jsx)(t.strong,{children:"The observed value of the test statistic is: 30.361111111111107"})," ",(0,s.jsx)("br",{}),"\n",(0,s.jsx)(t.strong,{children:"The p-value is: 0.004"})]}),"\n",(0,s.jsx)(t.p,{children:"Using a significance level of 0.05:"}),"\n",(0,s.jsx)(t.admonition,{title:"Conclusion",type:"info",children:(0,s.jsxs)(t.ul,{children:["\n",(0,s.jsx)(t.li,{children:"Under the null hypothesis, we rarely see a difference greater than the observed value."}),"\n",(0,s.jsxs)(t.li,{children:["Therefore, we ",(0,s.jsx)(t.strong,{children:"reject"})," the null hypothesis: the evidence implies that the two groups do not come from the same distribution."]}),"\n",(0,s.jsx)(t.li,{children:"Still, we cannot conclude that species causes a different weight because there may be other confounding variables."}),"\n"]})}),"\n",(0,s.jsx)(t.hr,{}),"\n",(0,s.jsx)(t.h3,{id:"6-extra",children:"6. Extra"}),"\n",(0,s.jsx)(t.p,{children:"Let's see how our observed statistic compares to the overall simulated values!"}),"\n",(0,s.jsx)(t.pre,{children:(0,s.jsx)(t.code,{className:"language-python",children:"# Create the histogram.\nbpd.DataFrame().assign(statistics=statistics).plot(kind='hist', bins=20, density=True, ec='w')\n# Don't worry about this line - you won't need to know it for this course!\nplt.axvline(x=observed, c='black', linewidth=4, label='population difference in means')\n"})}),"\n",(0,s.jsx)(t.p,{children:(0,s.jsx)(t.img,{alt:"Distribution",src:n(5771).A+"",width:"576",height:"413"})}),"\n",(0,s.jsxs)(t.p,{children:["From this graph, we can tell that there is barely any data to the ",(0,s.jsx)(t.strong,{children:"right"})," of the black vertical line (our observed test statistic), meaning we have a very low p-value!"]})]})}function p(e={}){const{wrapper:t}={...(0,i.R)(),...e.components};return t?(0,s.jsx)(t,{...e,children:(0,s.jsx)(u,{...e})}):u(e)}},8597:(e,t,n)=>{n.d(t,{A:()=>i});n(6540);var s=n(4848);const i=e=>{let{data:t}=e,n=[],i=[],a="",r=[];if(t){const e=JSON.parse(t);n=e.columns||[],i=e.data||[],a=e.indexName||"Index",r=e.index||[]}return(0,s.jsx)("div",{className:"dataframe-container",children:(0,s.jsxs)("table",{className:"dataframe-table",children:[(0,s.jsx)("thead",{children:(0,s.jsxs)("tr",{children:[(0,s.jsx)("th",{children:a}),n.map(((e,t)=>(0,s.jsx)("th",{children:e},t)))]})}),(0,s.jsx)("tbody",{children:i.map(((e,t)=>(0,s.jsxs)("tr",{children:[(0,s.jsx)("td",{className:"index-value",children:r[t]}),e.map(((e,t)=>{return(0,s.jsx)("td",{children:(n=e,"boolean"==typeof n?n.toString().charAt(0).toUpperCase()+n.toString().slice(1):n)},t);var n}))]},t)))})]})})}},5158:(e,t,n)=>{n.d(t,{A:()=>a});var s=n(6540),i=n(4848);const a=function(e){let{src:t,sourceLink:n,width:a="100%",height:r="100%",paddingBottom:o="56.25%"}=e;const[l,c]=(0,s.useState)(0);return(0,i.jsxs)(i.Fragment,{children:[(0,i.jsx)("div",{style:{position:"relative",height:"0",paddingBottom:o,overflow:"hidden",maxWidth:"100%"},children:(0,i.jsx)("iframe",{src:t,style:{position:"absolute",top:0,left:0,width:a,height:r,frameBorder:"0",allowFullScreen:!0}},l)}),(0,i.jsx)("div",{style:{display:"flex",justifyContent:"center",margin:"20px 0"},children:(0,i.jsx)("a",{href:n,target:"_blank",rel:"noopener noreferrer",children:"(Source)"})}),(0,i.jsx)("div",{style:{display:"flex",justifyContent:"center",margin:"10px 0"},children:(0,i.jsx)("button",{onClick:()=>{c((e=>e+1))},style:{backgroundColor:"#4CAF50",color:"white",padding:"10px 20px",border:"none",borderRadius:"5px",cursor:"pointer",outline:"none",fontSize:"16px",fontWeight:"bold",boxShadow:"0 4px 8px rgba(0,0,0,0.1)",transition:"background-color 0.3s"},onMouseOver:e=>e.currentTarget.style.backgroundColor="#45a049",onMouseOut:e=>e.currentTarget.style.backgroundColor="#4CAF50",children:"Restart Diagram"})})]})}},5771:(e,t,n)=>{n.d(t,{A:()=>s});const s=n.p+"assets/images/permutation-test-fae4df12733e667b05989b96bf08d9a9.png"},8453:(e,t,n)=>{n.d(t,{R:()=>r,x:()=>o});var s=n(6540);const i={},a=s.createContext(i);function r(e){const t=s.useContext(a);return s.useMemo((function(){return"function"==typeof e?e(t):{...t,...e}}),[t,e])}function o(e){let t;return t=e.disableParentContext?"function"==typeof e.components?e.components(i):e.components||i:r(e.components),s.createElement(a.Provider,{value:t},e.children)}}}]); \ No newline at end of file +"use strict";(self.webpackChunkbpd_reference=self.webpackChunkbpd_reference||[]).push([[9894],{6171:(e,t,n)=>{n.r(t),n.d(t,{assets:()=>d,contentTitle:()=>l,default:()=>p,frontMatter:()=>o,metadata:()=>c,toc:()=>h});var s=n(4848),i=n(8453),a=n(8597),r=n(5158);const o={sidebar_position:7,sidebar_label:"Permutation Test",title:"Permutation Test",tags:["querying","functions","for-loop","histogram"]},l=void 0,c={id:"statistical-inference/permutation_test",title:"Permutation Test",description:"Concept",source:"@site/docs/statistical-inference/permutation_test.md",sourceDirName:"statistical-inference",slug:"/statistical-inference/permutation_test",permalink:"/bpd-reference/docs/statistical-inference/permutation_test",draft:!1,unlisted:!1,editUrl:"https://github.com/dsc-courses/bpd-reference/tree/main/docs/statistical-inference/permutation_test.md",tags:[{label:"querying",permalink:"/bpd-reference/docs/tags/querying"},{label:"functions",permalink:"/bpd-reference/docs/tags/functions"},{label:"for-loop",permalink:"/bpd-reference/docs/tags/for-loop"},{label:"histogram",permalink:"/bpd-reference/docs/tags/histogram"}],version:"current",sidebarPosition:7,frontMatter:{sidebar_position:7,sidebar_label:"Permutation Test",title:"Permutation Test",tags:["querying","functions","for-loop","histogram"]},sidebar:"statsSidebar",previous:{title:"Hypothesis Test",permalink:"/bpd-reference/docs/statistical-inference/hypothesis_test"},next:{title:"The Standard Normal Distribution",permalink:"/bpd-reference/docs/statistical-inference/standard_normal_distribution"}},d={},h=[{value:"Concept",id:"concept",level:2},{value:"Code Example",id:"code-example",level:2},{value:"1. State the question/hypothesis",id:"1-state-the-questionhypothesis",level:3},{value:"2. Query the DataFrame",id:"2-query-the-dataframe",level:3},{value:"3. Create a function to calculate test statistic",id:"3-create-a-function-to-calculate-test-statistic",level:3},{value:"4. Simulate the permutation test",id:"4-simulate-the-permutation-test",level:3},{value:"5. Conclusion",id:"5-conclusion",level:3},{value:"6. Extra",id:"6-extra",level:3}];function u(e){const t={a:"a",admonition:"admonition",annotation:"annotation",code:"code",em:"em",h2:"h2",h3:"h3",hr:"hr",img:"img",li:"li",math:"math",mo:"mo",mrow:"mrow",ol:"ol",p:"p",pre:"pre",semantics:"semantics",span:"span",strong:"strong",ul:"ul",...(0,i.R)(),...e.components};return(0,s.jsxs)(s.Fragment,{children:[(0,s.jsx)(t.h2,{id:"concept",children:"Concept"}),"\n",(0,s.jsx)(t.admonition,{title:"Key Idea",type:"tip",children:(0,s.jsx)(t.p,{children:"We use a permutation test to test whether two samples are drawn from the same population."})}),"\n",(0,s.jsxs)(t.admonition,{title:"Terminology",type:"info",children:[(0,s.jsxs)(t.p,{children:[(0,s.jsx)(t.strong,{children:"Permutation"}),": A permutation of a sequence is a rearrangement of the elements in that sequence."]}),(0,s.jsxs)(t.ul,{children:["\n",(0,s.jsx)(t.li,{children:"Example: 'BAC' and 'CAB' are both permutations of the string 'ABC'."}),"\n",(0,s.jsx)(t.li,{children:"We create permutations by shuffling."}),"\n"]}),(0,s.jsxs)(t.p,{children:[(0,s.jsx)(t.strong,{children:"Permutation tests:"})," A permutation test is a type of A/B test (and a type of hypothesis test). It tests whether two samples come from the same population distribution. To conduct a permutation test:"]}),(0,s.jsxs)(t.ol,{children:["\n",(0,s.jsx)(t.li,{children:"Shuffle the group labels (i.e. the Trues and Falses) to generate two new samples under the null."}),"\n",(0,s.jsx)(t.li,{children:"Compute the test statistic (i.e. the difference in group means)."}),"\n",(0,s.jsx)(t.li,{children:"Repeat steps 1 and 2 to generate an empirical distribution of the test statistics (i.e. the difference in group means)."}),"\n",(0,s.jsx)(t.li,{children:"See where the observed statistic lies in the empirical distribution. If, in our simulations, we rarely saw a difference in group means as extreme as the observed difference in group means, we have evidence to reject the null."}),"\n"]})]}),"\n",(0,s.jsx)(t.admonition,{type:"note",children:(0,s.jsx)(t.p,{children:"While shuffling, the size of the two new samples should be the same as the observed samples."})}),"\n",(0,s.jsxs)(t.p,{children:["The diagram below provides an overview of conducting a permutation test, although it references a different dataset. ",(0,s.jsx)(t.a,{href:"https://www.jwilber.me/permutationtest/",children:"Here"})," is another useful visualization. For additional helpful visual guides, please visit the ",(0,s.jsx)(t.a,{href:"https://dsc10.com/diagrams/",children:"Diagrams"})," site."]}),"\n",(0,s.jsx)(r.A,{src:"https://docs.google.com/presentation/d/e/2PACX-1vSovXDonR6EmjrT45h4pY1mwmcKFMWVSdgpbKHC5HNTm9sbG7dojvvCDEQCjuk2dk1oA4gmwMogr8ZL/embed?start=true&loop=false&delayms=3000&rm=minimal",sourceLink:"https://docs.google.com/presentation/d/1TTWoQ7FU4tzBGc7EskF6WYccIn1BWQbTSizAiFc2p2o/edit?usp=sharing"}),"\n",(0,s.jsx)(t.hr,{}),"\n",(0,s.jsx)(t.h2,{id:"code-example",children:"Code Example"}),"\n",(0,s.jsx)(t.h3,{id:"1-state-the-questionhypothesis",children:"1. State the question/hypothesis"}),"\n",(0,s.jsxs)(t.admonition,{title:"Hypotheses",type:"info",children:[(0,s.jsx)(t.p,{children:"We will test the following pair of hypotheses at the standard p = 0.05 significance level:"}),(0,s.jsxs)(t.ul,{children:["\n",(0,s.jsxs)(t.li,{children:[(0,s.jsx)(t.strong,{children:"Null Hypothesis:"})," In the population, weights of dogs and cats have the ",(0,s.jsx)(t.strong,{children:"same"})," distribution, and the observed differences in our samples are due to random chance."]}),"\n",(0,s.jsxs)(t.li,{children:[(0,s.jsx)(t.strong,{children:"Alternative Hypothesis:"})," In the population, dogs and cats have ",(0,s.jsx)(t.strong,{children:"different"})," weights, on average. The observed differences in our samples cannot be explained by random chance alone."]}),"\n"]})]}),"\n",(0,s.jsxs)(t.p,{children:['Since the alternative hypothesis is of the form "A and B are different," the ',(0,s.jsx)(t.strong,{children:"test statistic"})," should measure distance and use an absolute value."]}),"\n",(0,s.jsxs)(t.p,{children:[(0,s.jsxs)(t.span,{className:"katex",children:[(0,s.jsx)(t.span,{className:"katex-mathml",children:(0,s.jsx)(t.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,s.jsxs)(t.semantics,{children:[(0,s.jsx)(t.mrow,{children:(0,s.jsx)(t.mo,{children:"\u2234"})}),(0,s.jsx)(t.annotation,{encoding:"application/x-tex",children:"\\therefore"})]})})}),(0,s.jsx)(t.span,{className:"katex-html","aria-hidden":"true",children:(0,s.jsxs)(t.span,{className:"base",children:[(0,s.jsx)(t.span,{className:"strut",style:{height:"0.6922em"}}),(0,s.jsx)(t.span,{className:"mrel amsrm",children:"\u2234"})]})})]})," We use ",(0,s.jsx)(t.strong,{children:"absolute difference in group means"})," as the test statistic."]}),"\n",(0,s.jsx)(t.hr,{}),"\n",(0,s.jsx)(t.h3,{id:"2-query-the-dataframe",children:"2. Query the DataFrame"}),"\n",(0,s.jsx)(t.p,{children:"Since we want to compare the distributions of only cats and dogs, we need to make sure to only include the relevant pieces of data (e.g., cats and dogs weights)."}),"\n",(0,s.jsx)(t.pre,{children:(0,s.jsx)(t.code,{className:"language-python",children:"# Create a boolean Series that references which rows fulfill either condition.\nquerying_condition = (full_pets.get('Species') == 'dog') | (full_pets.get('Species') == 'cat')\n# Query.\ncats_dogs = full_pets[querying_condition]\n# Display the first 5 rows only.\ncats_dogs.take(np.arange(5)) \n"})}),"\n",(0,s.jsx)(a.A,{data:'{"columns":["ID","Species","Color","Weight","Age","Is_Cat","Owner_Comment"],"index":[0,1,2,3,4],"data":[["dog_001","dog","black",40.0,5.0,false," There are no bad dogs, only bad owners."],["cat_001","cat","golden",1.5,0.2,true,"My best birthday present ever!!!"],["cat_002","cat","black",15.0,9.0,true,"****All you need is love and a cat.****"],["dog_002","dog","white",80.0,2.0,false,"Love is a wet nose and a wagging tail."],["dog_003","dog","black",25.0,0.5,false,"Be the person your dog thinks you are."]]}'}),"\n",(0,s.jsx)(t.hr,{}),"\n",(0,s.jsx)(t.h3,{id:"3-create-a-function-to-calculate-test-statistic",children:"3. Create a function to calculate test statistic"}),"\n",(0,s.jsx)(t.p,{children:"Since our hypotheses depend on the test statistic, create a function to be able to calculate it during every trial of our permutation test."}),"\n",(0,s.jsx)(t.pre,{children:(0,s.jsx)(t.code,{className:"language-python",children:"def difference_in_means(cats_dogs):\n \"\"\"\n Calculate the absolute difference in the mean weight of dogs and cats.\n ---\n Input:\n cats_dogs: a DataFrame containing the columns 'Species' and 'Weight'.\n ---\n Output:\n The absolute difference in the mean weight of dogs and cats.\n \"\"\"\n means = cats_dogs.groupby('Species').mean()\n return np.abs((means.get('Weight').loc['dog'] - means.get('Weight').loc['cat']))\n"})}),"\n",(0,s.jsx)(t.p,{children:(0,s.jsx)(t.em,{children:"Note: Although we can simply do this in the for-loop shown in the next step, we can practice good coding habits by separating our code into readable bits!"})}),"\n",(0,s.jsx)(t.hr,{}),"\n",(0,s.jsx)(t.h3,{id:"4-simulate-the-permutation-test",children:"4. Simulate the permutation test"}),"\n",(0,s.jsx)(t.pre,{children:(0,s.jsx)(t.code,{className:"language-python",children:"n = 500 # Number of simulations.\nstatistics = np.array([]) # Array to keep track of the difference in means for each iteration.\nfor i in np.arange(n): # Run the simulation `n` number of times\n # 1. Shuffle the species.\n shuffled = cats_dogs.assign(Species=np.random.permutation(cats_dogs.get('Species')))\n\n # 2. Compute the test statistic.\n statistic = difference_in_means(shuffled)\n\n # 3. Save the result.\n statistics = np.append(statistics, statistic)\n"})}),"\n",(0,s.jsxs)(t.p,{children:["This code will run the permutation test 500 times, but a different reasonable number can be used instead. It is ",(0,s.jsx)(t.strong,{children:"crucial"})," to keep track of the difference in means each time our for-loop runs so that the number of simulated values can be displayed."]}),"\n",(0,s.jsx)(t.hr,{}),"\n",(0,s.jsx)(t.h3,{id:"5-conclusion",children:"5. Conclusion"}),"\n",(0,s.jsx)(t.pre,{children:(0,s.jsx)(t.code,{className:"language-python",children:'observed = difference_in_means(cats_dogs)\np_value = np.count_nonzero(statistics >= observed) / n\n\nprint("The observed value of the test statistic is:", observed)\nprint("The p-value is:", p_value)\n'})}),"\n",(0,s.jsxs)(t.p,{children:[(0,s.jsx)(t.strong,{children:"The observed value of the test statistic is: 30.361111111111107"})," ",(0,s.jsx)("br",{}),"\n",(0,s.jsx)(t.strong,{children:"The p-value is: 0.004"})]}),"\n",(0,s.jsx)(t.p,{children:"Using a significance level of 0.05:"}),"\n",(0,s.jsx)(t.admonition,{title:"Conclusion",type:"info",children:(0,s.jsxs)(t.ul,{children:["\n",(0,s.jsx)(t.li,{children:"Under the null hypothesis, we rarely see a difference greater than the observed value."}),"\n",(0,s.jsxs)(t.li,{children:["Therefore, we ",(0,s.jsx)(t.strong,{children:"reject"})," the null hypothesis: the evidence implies that the two groups do not come from the same distribution."]}),"\n",(0,s.jsx)(t.li,{children:"Still, we cannot conclude that species causes a different weight because there may be other confounding variables."}),"\n"]})}),"\n",(0,s.jsx)(t.hr,{}),"\n",(0,s.jsx)(t.h3,{id:"6-extra",children:"6. Extra"}),"\n",(0,s.jsx)(t.p,{children:"Let's see how our observed statistic compares to the overall simulated values!"}),"\n",(0,s.jsx)(t.pre,{children:(0,s.jsx)(t.code,{className:"language-python",children:"# Create the histogram.\nbpd.DataFrame().assign(statistics=statistics).plot(kind='hist', bins=20, density=True, ec='w')\n# Don't worry about this line - you won't need to know it for this course!\nplt.axvline(x=observed, c='black', linewidth=4, label='population difference in means')\n"})}),"\n",(0,s.jsx)(t.p,{children:(0,s.jsx)(t.img,{alt:"Distribution",src:n(5771).A+"",width:"576",height:"413"})}),"\n",(0,s.jsxs)(t.p,{children:["From this graph, we can tell that there is barely any data to the ",(0,s.jsx)(t.strong,{children:"right"})," of the black vertical line (our observed test statistic), meaning we have a very low p-value!"]})]})}function p(e={}){const{wrapper:t}={...(0,i.R)(),...e.components};return 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Extra",id:"6-extra",level:3}];function c(e){const t={admonition:"admonition",annotation:"annotation",code:"code",em:"em",h2:"h2",h3:"h3",hr:"hr",img:"img",li:"li",math:"math",mo:"mo",mrow:"mrow",ol:"ol",p:"p",pre:"pre",semantics:"semantics",span:"span",strong:"strong",ul:"ul",...(0,o.R)(),...e.components};return(0,n.jsxs)(n.Fragment,{children:[(0,n.jsx)(t.h2,{id:"concept",children:"Concept"}),"\n",(0,n.jsx)(t.admonition,{title:"Key Idea",type:"tip",children:(0,n.jsx)(t.p,{children:"We use a hypothesis test to test whether a sample was drawn from a population."})}),"\n",(0,n.jsxs)(t.admonition,{title:"Terminology",type:"info",children:[(0,n.jsxs)(t.p,{children:[(0,n.jsx)(t.strong,{children:"Hypothesis Test"}),": A process that test whether a sample was drawn from a population, given a population distribution and a sample."]}),(0,n.jsx)(t.p,{children:"To conduct a hypothesis test:"}),(0,n.jsxs)(t.ol,{children:["\n",(0,n.jsx)(t.li,{children:"Simulate the experiment many times under the assumption that the null hypothesis is true."}),"\n",(0,n.jsx)(t.li,{children:"Compute a test statistic on each of the simulated samples, as well as on the observed sample."}),"\n",(0,n.jsx)(t.li,{children:"Look at the resulting empirical distribution of test statistics and see where the observed test statistic falls. 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State the question/hypothesis"}),"\n",(0,n.jsxs)(t.admonition,{title:"Hypotheses",type:"info",children:[(0,n.jsx)(t.p,{children:"We will test the following pair of hypotheses at the standard p = 0.05 significance level."}),(0,n.jsxs)(t.ul,{children:["\n",(0,n.jsxs)(t.li,{children:[(0,n.jsx)(t.strong,{children:"Null Hypothesis:"}),' The proportion of "white" dogs among dogs is ',(0,n.jsx)(t.strong,{children:"equal"}),' to the proportion of "white" pets in the overall population.']}),"\n",(0,n.jsxs)(t.li,{children:[(0,n.jsx)(t.strong,{children:"Alternative Hypothesis:"}),' The distribution of "white" dogs among dogs is ',(0,n.jsx)(t.strong,{children:"greater than"}),' the proportion of "white" pets in the overall population.']}),"\n"]})]}),"\n",(0,n.jsxs)(t.p,{children:['Since the alternative hypothesis is of the form "A is ',(0,n.jsx)(t.strong,{children:(0,n.jsx)(t.em,{children:"greater than"})}),' B," the ',(0,n.jsx)(t.strong,{children:"test statistic"})," should measure ",(0,n.jsx)(t.strong,{children:"signed difference"})," rather than absolute difference."]}),"\n",(0,n.jsxs)(t.p,{children:[(0,n.jsxs)(t.span,{className:"katex",children:[(0,n.jsx)(t.span,{className:"katex-mathml",children:(0,n.jsx)(t.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(t.semantics,{children:[(0,n.jsx)(t.mrow,{children:(0,n.jsx)(t.mo,{children:"\u2234"})}),(0,n.jsx)(t.annotation,{encoding:"application/x-tex",children:"\\therefore"})]})})}),(0,n.jsx)(t.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(t.span,{className:"base",children:[(0,n.jsx)(t.span,{className:"strut",style:{height:"0.6922em"}}),(0,n.jsx)(t.span,{className:"mrel amsrm",children:"\u2234"})]})})]})," We use ",(0,n.jsx)(t.strong,{children:"the difference in proportions"})," of white dogs among dogs and white pets in the overall poplulation as the test statistic."]}),"\n",(0,n.jsx)(t.hr,{}),"\n",(0,n.jsx)(t.h3,{id:"2-calculate-the-overall-proportion",children:"2. Calculate the overall proportion"}),"\n",(0,n.jsx)(t.p,{children:"We first compute the proportion of white pets in the overall population, which will be used for calculating the test statistic."}),"\n",(0,n.jsx)(t.pre,{children:(0,n.jsx)(t.code,{className:"language-python",children:"#Method 1: Obtain a series of boolean values (1s and 0s), and calculate the mean\noverall_prop = (full_pets.get('Color')=='white').mean()\n\n#Method 2: Query the DataFrame and obtain the proportion by using .shape[0]\n#overall_prop = full_pets[full_pets.get('Color')=='white'].shape[0]/full_pets.shape[0]\n\noverall_prop\n"})}),"\n",(0,n.jsx)(t.p,{children:"0.2631578947368421"}),"\n",(0,n.jsx)(t.hr,{}),"\n",(0,n.jsx)(t.h3,{id:"3-create-a-function-to-calculate-test-statistic",children:"3. Create a function to calculate test statistic"}),"\n",(0,n.jsx)(t.p,{children:"Create a function to calculate the test statistic during every trial of our hypothesis test."}),"\n",(0,n.jsx)(t.pre,{children:(0,n.jsx)(t.code,{className:"language-python",children:'def difference_in_proportions(dogs_prop, overall_prop):\n """\n Calculate the difference in proportions of white dogs among dogs and white pets in the overall population.\n ---\n Input:\n dogs_prop: the proportion of white dogs among all dogs.\n overall_prop: the proportion of white pets in the overall population.\n ---\n Output:\n The difference in proportions of white dogs among dogs and white pets in the overall population.\n """\n return dogs_prop - overall_prop\n'})}),"\n",(0,n.jsx)(t.p,{children:(0,n.jsx)(t.em,{children:"Note: Although we can simply do this in the for-loop shown in the next step, we can practice good coding habits by separating our code into readable bits!"})}),"\n",(0,n.jsx)(t.hr,{}),"\n",(0,n.jsx)(t.h3,{id:"4-compute-the-observed-statistic",children:"4. Compute the observed statistic"}),"\n",(0,n.jsx)(t.pre,{children:(0,n.jsx)(t.code,{className:"language-python",children:"# Query the DataFrame to only include dogs\ndogs_df = full_pets[full_pets.get('Species') == 'dog']\n# Compute the observed proportion of white dogs among all dogs\nobs_dogs_prop = (dogs_df.get('Color') == 'white').mean()\n# Calculate the observed statistic\nobserved_stat = difference_in_proportions(obs_dogs_prop, overall_prop)\nobserved_stat\n"})}),"\n",(0,n.jsx)(t.p,{children:"0.18128654970760233"}),"\n",(0,n.jsx)(t.hr,{}),"\n",(0,n.jsx)(t.h3,{id:"4-simulate-the-hypothesis-test-under-the-null-hypothesis",children:"4. Simulate the hypothesis test under the null hypothesis"}),"\n",(0,n.jsx)(t.pre,{children:(0,n.jsx)(t.code,{className:"language-python",children:'n = 500 # Number of simulations.\ntest_statistics = np.array([]) # Array to keep track of the test statistic for each iteration.\nfor i in np.arange(n): # Run the simulation `n` number of times\n # 1. Simulate the dogs distribution\n \n # Calculte the smaple size (number of dogs)\n sample_size = dogs_df.shape[0]\n \n # Under the null hypothesis, proportion of "white" dogs among dogs is equal to the proportion of white pets among the overall population\n model_proportions = np.array([obs_dogs_prop, 1-obs_dogs_prop]) \n \n # Calculate the the proportion of white dogs among all dogs.\n simulation_proportion = np.random.multinomial(sample_size, model_proportions)[0] / sample_size\n \n # 2. Compute the test statistic\n test_stat = difference_in_proportions(simulation_proportion, overall_prop)\n\n # 3. Save the result\n test_statistics = np.append(test_statistics, test_stat)\n'})}),"\n",(0,n.jsxs)(t.p,{children:["This code will run the permutation test 500 times, but a different reasonable number can be used instead. It is ",(0,n.jsx)(t.strong,{children:"crucial"})," to keep track of the test statistic each time our for-loop runs so that the number of simulated values can be stored."]}),"\n",(0,n.jsx)(t.hr,{}),"\n",(0,n.jsx)(t.h3,{id:"5-conclusion",children:"5. 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\ No newline at end of file +"use strict";(self.webpackChunkbpd_reference=self.webpackChunkbpd_reference||[]).push([[972],{3007:(e,t,s)=>{s.r(t),s.d(t,{assets:()=>l,contentTitle:()=>r,default:()=>p,frontMatter:()=>o,metadata:()=>a,toc:()=>h});var n=s(4848),i=s(8453);s(8597),s(5158);const o={sidebar_position:6,sidebar_label:"Hypothesis Test",title:"Hypothesis Test",tags:["querying","functions","for-loop","histogram"]},r=void 0,a={id:"statistical-inference/hypothesis_test",title:"Hypothesis Test",description:"Concept",source:"@site/docs/statistical-inference/hypothesis_test.md",sourceDirName:"statistical-inference",slug:"/statistical-inference/hypothesis_test",permalink:"/bpd-reference/docs/statistical-inference/hypothesis_test",draft:!1,unlisted:!1,editUrl:"https://github.com/dsc-courses/bpd-reference/tree/main/docs/statistical-inference/hypothesis_test.md",tags:[{label:"querying",permalink:"/bpd-reference/docs/tags/querying"},{label:"functions",permalink:"/bpd-reference/docs/tags/functions"},{label:"for-loop",permalink:"/bpd-reference/docs/tags/for-loop"},{label:"histogram",permalink:"/bpd-reference/docs/tags/histogram"}],version:"current",sidebarPosition:6,frontMatter:{sidebar_position:6,sidebar_label:"Hypothesis Test",title:"Hypothesis Test",tags:["querying","functions","for-loop","histogram"]},sidebar:"statsSidebar",previous:{title:"P-value",permalink:"/bpd-reference/docs/statistical-inference/p_value"},next:{title:"Permutation Test",permalink:"/bpd-reference/docs/statistical-inference/permutation_test"}},l={},h=[{value:"Concept",id:"concept",level:2},{value:"Code Example",id:"code-example",level:2},{value:"1. 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Extra",id:"6-extra",level:3}];function c(e){const t={admonition:"admonition",annotation:"annotation",code:"code",em:"em",h2:"h2",h3:"h3",hr:"hr",img:"img",li:"li",math:"math",mo:"mo",mrow:"mrow",ol:"ol",p:"p",pre:"pre",semantics:"semantics",span:"span",strong:"strong",ul:"ul",...(0,i.R)(),...e.components};return(0,n.jsxs)(n.Fragment,{children:[(0,n.jsx)(t.h2,{id:"concept",children:"Concept"}),"\n",(0,n.jsx)(t.admonition,{title:"Key Idea",type:"tip",children:(0,n.jsx)(t.p,{children:"We use a hypothesis test to test whether a sample was drawn from a population."})}),"\n",(0,n.jsxs)(t.admonition,{title:"Terminology",type:"info",children:[(0,n.jsxs)(t.p,{children:[(0,n.jsx)(t.strong,{children:"Hypothesis Test"}),": A process that test whether a sample was drawn from a population, given a population distribution and a sample."]}),(0,n.jsx)(t.p,{children:"To conduct a hypothesis test:"}),(0,n.jsxs)(t.ol,{children:["\n",(0,n.jsx)(t.li,{children:"Simulate the experiment many times under the assumption that the null hypothesis is true."}),"\n",(0,n.jsx)(t.li,{children:"Compute a test statistic on each of the simulated samples, as well as on the observed sample."}),"\n",(0,n.jsx)(t.li,{children:"Look at the resulting empirical distribution of test statistics and see where the observed test statistic falls. 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State the question/hypothesis"}),"\n",(0,n.jsxs)(t.admonition,{title:"Hypotheses",type:"info",children:[(0,n.jsx)(t.p,{children:"We will test the following pair of hypotheses at the standard p = 0.05 significance level."}),(0,n.jsxs)(t.ul,{children:["\n",(0,n.jsxs)(t.li,{children:[(0,n.jsx)(t.strong,{children:"Null Hypothesis:"}),' The proportion of "white" dogs among dogs is ',(0,n.jsx)(t.strong,{children:"equal"}),' to the proportion of "white" pets in the overall population.']}),"\n",(0,n.jsxs)(t.li,{children:[(0,n.jsx)(t.strong,{children:"Alternative Hypothesis:"}),' The distribution of "white" dogs among dogs is ',(0,n.jsx)(t.strong,{children:"greater than"}),' the proportion of "white" pets in the overall population.']}),"\n"]})]}),"\n",(0,n.jsxs)(t.p,{children:['Since the alternative hypothesis is of the form "A is ',(0,n.jsx)(t.strong,{children:(0,n.jsx)(t.em,{children:"greater than"})}),' B," the ',(0,n.jsx)(t.strong,{children:"test statistic"})," should measure ",(0,n.jsx)(t.strong,{children:"signed difference"})," rather than absolute difference."]}),"\n",(0,n.jsxs)(t.p,{children:[(0,n.jsxs)(t.span,{className:"katex",children:[(0,n.jsx)(t.span,{className:"katex-mathml",children:(0,n.jsx)(t.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(t.semantics,{children:[(0,n.jsx)(t.mrow,{children:(0,n.jsx)(t.mo,{children:"\u2234"})}),(0,n.jsx)(t.annotation,{encoding:"application/x-tex",children:"\\therefore"})]})})}),(0,n.jsx)(t.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(t.span,{className:"base",children:[(0,n.jsx)(t.span,{className:"strut",style:{height:"0.6922em"}}),(0,n.jsx)(t.span,{className:"mrel amsrm",children:"\u2234"})]})})]})," We use ",(0,n.jsx)(t.strong,{children:"the difference in proportions"})," of white dogs among dogs and white pets in the overall poplulation as the test statistic."]}),"\n",(0,n.jsx)(t.hr,{}),"\n",(0,n.jsx)(t.h3,{id:"2-calculate-the-overall-proportion",children:"2. Calculate the overall proportion"}),"\n",(0,n.jsx)(t.p,{children:"We first compute the proportion of white pets in the overall population, which will be used for calculating the test statistic."}),"\n",(0,n.jsx)(t.pre,{children:(0,n.jsx)(t.code,{className:"language-python",children:"#Method 1: Obtain a series of boolean values (1s and 0s), and calculate the mean\noverall_prop = (full_pets.get('Color')=='white').mean()\n\n#Method 2: Query the DataFrame and obtain the proportion by using .shape[0]\n#overall_prop = full_pets[full_pets.get('Color')=='white'].shape[0]/full_pets.shape[0]\n\noverall_prop\n"})}),"\n",(0,n.jsx)(t.p,{children:"0.2631578947368421"}),"\n",(0,n.jsx)(t.hr,{}),"\n",(0,n.jsx)(t.h3,{id:"3-create-a-function-to-calculate-test-statistic",children:"3. 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Simulate the dogs distribution\n \n # Calculte the smaple size (number of dogs)\n sample_size = dogs_df.shape[0]\n \n # Under the null hypothesis, proportion of "white" dogs among dogs is equal to the proportion of white pets among the overall population\n model_proportions = np.array([obs_dogs_prop, 1-obs_dogs_prop]) \n \n # Calculate the the proportion of white dogs among all dogs.\n simulation_proportion = np.random.multinomial(sample_size, model_proportions)[0] / sample_size\n \n # 2. Compute the test statistic\n test_stat = difference_in_proportions(simulation_proportion, overall_prop)\n\n # 3. Save the result\n test_statistics = np.append(test_statistics, test_stat)\n'})}),"\n",(0,n.jsxs)(t.p,{children:["This code will run the permutation test 500 times, but a different reasonable number can be used instead. 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\ No newline at end of file diff --git a/assets/js/f095f4cd.62435f21.js b/assets/js/f095f4cd.62435f21.js deleted file mode 100644 index f2bb4073..00000000 --- a/assets/js/f095f4cd.62435f21.js +++ /dev/null @@ -1 +0,0 @@ -"use strict";(self.webpackChunkbpd_reference=self.webpackChunkbpd_reference||[]).push([[28],{8406:(e,A,t)=>{t.r(A),t.d(A,{assets:()=>d,contentTitle:()=>o,default:()=>h,frontMatter:()=>r,metadata:()=>l,toc:()=>c});var n=t(4848),a=t(8453),s=t(8597),i=t(5158);const r={sidebar_position:8,sidebar_label:"Boostrapping and Confidence Intervals",title:"Bootstrapping and Confidence Intervals",tags:["for-loop","histogram"]},o=void 0,l={id:"statistical-inference/bootstrapping",title:"Bootstrapping and Confidence Intervals",description:"Concept",source:"@site/docs/statistical-inference/bootstrapping.md",sourceDirName:"statistical-inference",slug:"/statistical-inference/bootstrapping",permalink:"/bpd-reference/docs/statistical-inference/bootstrapping",draft:!1,unlisted:!1,editUrl:"https://github.com/dsc-courses/bpd-reference/tree/main/docs/statistical-inference/bootstrapping.md",tags:[{label:"for-loop",permalink:"/bpd-reference/docs/tags/for-loop"},{label:"histogram",permalink:"/bpd-reference/docs/tags/histogram"}],version:"current",sidebarPosition:8,frontMatter:{sidebar_position:8,sidebar_label:"Boostrapping and Confidence Intervals",title:"Bootstrapping and Confidence Intervals",tags:["for-loop","histogram"]},sidebar:"statsSidebar",previous:{title:"Standard Units, Correlation, Regression",permalink:"/bpd-reference/docs/statistical-inference/standard_units"},next:{title:"Hypothesis Test vs. Permutation Test",permalink:"/bpd-reference/docs/statistical-inference/concept check/hypothesis_vs_permutation"}},d={},c=[{value:"Concept",id:"concept",level:2},{value:"Code Example",id:"code-example",level:2},{value:"1. Take a random sample of size 12 from the full_pets DataFrame.",id:"1-take-a-random-sample-of-size-12-from-the-full_pets-dataframe",level:3},{value:"2. Find the observed parameter",id:"2-find-the-observed-parameter",level:3},{value:"3. Bootstrap the sample 10,000 times with replacement",id:"3-bootstrap-the-sample-10000-times-with-replacement",level:3},{value:"4. Create a 95% confidence interval",id:"4-create-a-95-confidence-interval",level:3},{value:"5. Conclusion",id:"5-conclusion",level:3},{value:"6. Extra",id:"6-extra",level:3}];function p(e){const A={a:"a",admonition:"admonition",code:"code",h2:"h2",h3:"h3",hr:"hr",img:"img",li:"li",ol:"ol",p:"p",pre:"pre",strong:"strong",ul:"ul",...(0,a.R)(),...e.components};return(0,n.jsxs)(n.Fragment,{children:[(0,n.jsx)(A.h2,{id:"concept",children:"Concept"}),"\n",(0,n.jsx)(A.admonition,{title:"Key Idea",type:"tip",children:(0,n.jsx)(A.p,{children:"We use a bootstrapping to estimate the distribution of the sample statistic to see how different our it could have been. We use confidence intervals to define a range that captures most of the bootstrapped distribution of the sample statistic."})}),"\n",(0,n.jsxs)(A.admonition,{title:"Terminology",type:"info",children:[(0,n.jsxs)(A.p,{children:[(0,n.jsx)(A.strong,{children:"Bootstrapping"}),": Bootstrapping is a type of hypothesis test that involves resampling from a single sample to estimate the\ndistribution of the sample statistic. It answers the question of how different the sample statistic could have been if given a different sample. To conduct bootstrapping:"]}),(0,n.jsxs)(A.ol,{children:["\n",(0,n.jsx)(A.li,{children:"Resample from the original sample with replacement."}),"\n",(0,n.jsx)(A.li,{children:"Calculate the sample statistic on the bootstrapped resample."}),"\n",(0,n.jsx)(A.li,{children:"Save the results into an array."}),"\n",(0,n.jsx)(A.li,{children:"Repeat steps 1 through 3 to generate an empirical distribution of the test statistic."}),"\n",(0,n.jsx)(A.li,{children:"Calculate the confidence interval and see if the observed statistic lies in it. If the observed statistic is not\nin our confidence interval, we have evidence to reject the null."}),"\n"]}),(0,n.jsxs)(A.p,{children:[(0,n.jsx)(A.strong,{children:"Confidence Intervals:"})," A confidence interval is a range that captures most of the distribution of the bootstrapped sample statistic in\nthe hopes of also containing the true population parameter within it. If\nwe were to construct a 95% confidence interval, we aren't saying that there is a 95% chance that the true population parameter lies in the\ninterval as the interval either contains it or it doesn't. Instead, we are saying that approximately 95% of the time, the intervals you\ncreate will contain the true population parameter. For example, if we generated 100 confidence intervals, about 95 of them will have the\ntrue population parameter."]})]}),"\n",(0,n.jsx)(A.admonition,{type:"note",children:(0,n.jsx)(A.p,{children:"When resampling, the size of the resample should be the same as the original sample with replacement."})}),"\n",(0,n.jsx)(A.p,{children:"The diagram below provides an overview of conducting bootstrapping, although it references a different dataset."}),"\n",(0,n.jsx)(i.A,{src:"https://docs.google.com/presentation/d/e/2PACX-1vS_iYHJYXSVMMZ-YQVFwMEFR6EFN3FDSAvaMyUm-YJfLQgRMTHm3vI-wWJJ5999eFJq70nWp2hyItZg/embed?start=true&loop=false&delayms=3000&rm=minimal",sourceLink:"https://docs.google.com/presentation/d/1oYakqMdI7z61BthvgWUJvbT21bKTXg01KmAibglepBI/edit?usp=sharing"}),"\n",(0,n.jsxs)(A.p,{children:["The diagram below provides an overview of creating confidence intervals, although it references a different situation. For additional helpful visual guides, please visit the ",(0,n.jsx)(A.a,{href:"https://dsc10.com/diagrams/",children:"Diagrams"})," site."]}),"\n",(0,n.jsx)(i.A,{src:"https://docs.google.com/presentation/d/e/2PACX-1vTaPZsueXI6fey_5cj2Y1TevkR1joBvpwaWVsZNvgBlnJSrw1EiBLHJywkFH_QNLU5Tdr6JZgDrhFxG/embed?start=true&loop=false&delayms=3000&rm=minimal",sourceLink:"https://docs.google.com/presentation/d/1l5GVyl60FrornEaZnGL6REV4gjEydRncNVi6k1hHUsA/edit?usp=sharing"}),"\n",(0,n.jsx)(A.hr,{}),"\n",(0,n.jsx)(A.h2,{id:"code-example",children:"Code Example"}),"\n",(0,n.jsxs)(A.h3,{id:"1-take-a-random-sample-of-size-12-from-the-full_pets-dataframe",children:["1. Take a random sample of size 12 from the ",(0,n.jsx)(A.code,{children:"full_pets"})," DataFrame."]}),"\n",(0,n.jsxs)(A.p,{children:["Let's say we didn't have access to all of the information in the ",(0,n.jsx)(A.code,{children:"full_pets"})," DataFrame and were only able to collect a sample of 12 pets."]}),"\n",(0,n.jsx)(A.pre,{children:(0,n.jsx)(A.code,{className:"language-python",children:"# Magic to ensure that we get the same results every time this code is run. \nnp.random.seed(42)\n\n# sample\npets_sample = full_pets.sample(12, replace=False)\npets_sample\n"})}),"\n",(0,n.jsx)(s.A,{data:'{"columns":["Unnamed: 0","ID","Species","Color","Weight","Age","Is_Cat","Owner_Comment"],"index":[18,14,4,13,10,7,6,3,2,15,17,8],"data":[[18,"cat_006","cat","black",3.0,0.5,true,"No, thank you!"],[14,"dog_007","dog","white",50.0,6.1,false,"No, thank you!"],[4,"dog_003","dog","black",25.0,0.5,false,"Be the person your dog thinks you are."],[13,"ham_003","hamster","black",0.5,0.1,false,"No, thank you!"],[10,"dog_006","dog","golden",35.0,4.0,false,"No, thank you!"],[7,"cat_003","cat","black",10.0,0.0,true,"No, thank you!"],[6,"ham_002","hamster","golden",0.25,0.2,false,"No, thank you!"],[3,"dog_002","dog","white",80.0,2.0,false,"Love is a wet nose and a wagging tail."],[2,"cat_002","cat","black",15.0,9.0,true,"****All you need is love and a cat.****"],[15,"ham_004","hamster","golden",0.25,0.2,false,"No, thank you!"],[17,"dog_009","dog","white",30.0,4.8,false,"No, thank you!"],[8,"dog_004","dog","black",45.0,6.7,false,"No, thank you!"]]}'}),"\n",(0,n.jsx)(A.hr,{}),"\n",(0,n.jsx)(A.h3,{id:"2-find-the-observed-parameter",children:"2. Find the observed parameter"}),"\n",(0,n.jsx)(A.p,{children:"In this case, we are interested in finding the median weight of the entire population."}),"\n",(0,n.jsx)(A.pre,{children:(0,n.jsx)(A.code,{className:"language-python",children:"pets_sample = full_pets.sample(12, replace=False)\nprint('Median of pets_sample weight:', pets_sample.get('Weight').median())\n"})}),"\n",(0,n.jsx)(A.p,{children:"Median of pets_sample weight: 20.0"}),"\n",(0,n.jsx)(A.hr,{}),"\n",(0,n.jsx)(A.h3,{id:"3-bootstrap-the-sample-10000-times-with-replacement",children:"3. Bootstrap the sample 10,000 times with replacement"}),"\n",(0,n.jsx)(A.p,{children:"Since we were only able to collect one random sample from the full population, we can't be sure if this singular guess predicts the\ntrue population parameter well. We can't go out and collect\nanother random sample, so we will resample from the original sample with replacement to simulate what could've been."}),"\n",(0,n.jsx)(A.pre,{children:(0,n.jsx)(A.code,{className:"language-python",children:"boot_medians = np.array([])\nfor i in np.arange(10000):\n # 1. resample the data\n resample = pets_sample.sample(pets_sample.shape[0], replace=True)\n\n # 2. calculate the median of the resample\n boot_median = resample.get('Weight').median()\n\n # 3. append the median to the array\n boot_medians = np.append(boot_medians, boot_median)\n"})}),"\n",(0,n.jsx)(A.p,{children:"This code will create 10,000 bootstrapped samples and calculate the median for each of them, but a different\nreasonable number can be used instead. Since these samples are all random, the information in each sample and\nmedian will be different from one another."}),"\n",(0,n.jsx)(A.hr,{}),"\n",(0,n.jsx)(A.h3,{id:"4-create-a-95-confidence-interval",children:"4. Create a 95% confidence interval"}),"\n",(0,n.jsx)(A.p,{children:"Instead of using a single number to estimate the true population parameter, we create a range of where we think it is."}),"\n",(0,n.jsx)(A.pre,{children:(0,n.jsx)(A.code,{className:"language-python",children:"# Get the 95% confidence interval\nleft = np.percentile(boot_medians, 2.5) # 2.5th percentile\nright = np.percentile(boot_medians, 97.5) # 97.5th percentile\n"})}),"\n",(0,n.jsx)(A.p,{children:"Remember that the 95% confidence interval does not mean we have a 95% chance of containing the true population parameter. It means\nthat about 95% of all intervals we create will contain the true population parameter."}),"\n",(0,n.jsx)(A.hr,{}),"\n",(0,n.jsx)(A.h3,{id:"5-conclusion",children:"5. Conclusion"}),"\n",(0,n.jsx)(A.pre,{children:(0,n.jsx)(A.code,{className:"language-python",children:"left, right\n"})}),"\n",(0,n.jsx)(A.p,{children:"(1.75, 40.0)"}),"\n",(0,n.jsx)(A.admonition,{title:"Conclusion",type:"info",children:(0,n.jsxs)(A.ul,{children:["\n",(0,n.jsx)(A.li,{children:"From this interval, we are 95% confident that the true population median lies somewhere between 1.75 and 40."}),"\n",(0,n.jsx)(A.li,{children:"We have no way of knowing where exactly in this interval does the true population median falls or even if it is contained at all."}),"\n",(0,n.jsx)(A.li,{children:"What we do know is that if we were to repeat the process and generate multiple confidence intervals, roughly 95% of them will\ncontain the true population median."}),"\n"]})}),"\n",(0,n.jsx)(A.hr,{}),"\n",(0,n.jsx)(A.h3,{id:"6-extra",children:"6. Extra"}),"\n",(0,n.jsx)(A.p,{children:"Let's look at the distribution of the bootstrapped medians!"}),"\n",(0,n.jsx)(A.pre,{children:(0,n.jsx)(A.code,{className:"language-python",children:"# Create the histogram.\n# Plot the histogram of boot_medians\nplt.hist(boot_medians, bins=20, density=True, ec = 'w')\n\nplt.show()\n"})}),"\n",(0,n.jsx)(A.p,{children:(0,n.jsx)(A.img,{alt:"Distribution",src:t(3698).A+"",width:"556",height:"413"})}),"\n",(0,n.jsxs)(A.p,{children:["A 95% confidence level means that ",(0,n.jsx)(A.strong,{children:"approximately 95% of the time, the intervals you create through this process will contain the true population parameter"}),"."]})]})}function h(e={}){const{wrapper:A}={...(0,a.R)(),...e.components};return A?(0,n.jsx)(A,{...e,children:(0,n.jsx)(p,{...e})}):p(e)}},8597:(e,A,t)=>{t.d(A,{A:()=>a});t(6540);var n=t(4848);const a=e=>{let{data:A}=e,t=[],a=[],s="",i=[];if(A){const 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0,l={id:"statistical-inference/bootstrapping",title:"Bootstrapping and Confidence Intervals",description:"Concept",source:"@site/docs/statistical-inference/bootstrapping.md",sourceDirName:"statistical-inference",slug:"/statistical-inference/bootstrapping",permalink:"/bpd-reference/docs/statistical-inference/bootstrapping",draft:!1,unlisted:!1,editUrl:"https://github.com/dsc-courses/bpd-reference/tree/main/docs/statistical-inference/bootstrapping.md",tags:[{label:"for-loop",permalink:"/bpd-reference/docs/tags/for-loop"},{label:"histogram",permalink:"/bpd-reference/docs/tags/histogram"}],version:"current",sidebarPosition:8,frontMatter:{sidebar_position:8,sidebar_label:"Boostrapping and Confidence Intervals",title:"Bootstrapping and Confidence Intervals",tags:["for-loop","histogram"]},sidebar:"statsSidebar",previous:{title:"Standard Units, Correlation, Regression",permalink:"/bpd-reference/docs/statistical-inference/standard_units"},next:{title:"Hypothesis Test vs. Permutation Test",permalink:"/bpd-reference/docs/statistical-inference/concept check/hypothesis_vs_permutation"}},d={},c=[{value:"Concept",id:"concept",level:2},{value:"Code Example",id:"code-example",level:2},{value:"1. Take a random sample of size 12 from the full_pets DataFrame.",id:"1-take-a-random-sample-of-size-12-from-the-full_pets-dataframe",level:3},{value:"2. Find the observed parameter",id:"2-find-the-observed-parameter",level:3},{value:"3. Bootstrap the sample 10,000 times with replacement",id:"3-bootstrap-the-sample-10000-times-with-replacement",level:3},{value:"4. Create a 95% confidence interval",id:"4-create-a-95-confidence-interval",level:3},{value:"5. Conclusion",id:"5-conclusion",level:3},{value:"6. Extra",id:"6-extra",level:3}];function p(e){const A={a:"a",admonition:"admonition",code:"code",h2:"h2",h3:"h3",hr:"hr",img:"img",li:"li",ol:"ol",p:"p",pre:"pre",strong:"strong",ul:"ul",...(0,a.R)(),...e.components};return(0,n.jsxs)(n.Fragment,{children:[(0,n.jsx)(A.h2,{id:"concept",children:"Concept"}),"\n",(0,n.jsx)(A.admonition,{title:"Key Idea",type:"tip",children:(0,n.jsx)(A.p,{children:"We use a bootstrapping to estimate the distribution of the sample statistic to see how different our it could have been. We use confidence intervals to define a range that captures most of the bootstrapped distribution of the sample statistic."})}),"\n",(0,n.jsxs)(A.admonition,{title:"Terminology",type:"info",children:[(0,n.jsxs)(A.p,{children:[(0,n.jsx)(A.strong,{children:"Bootstrapping"}),": Bootstrapping is a type of hypothesis test that involves resampling from a single sample to estimate the\ndistribution of the sample statistic. It answers the question of how different the sample statistic could have been if given a different sample. To conduct bootstrapping:"]}),(0,n.jsxs)(A.ol,{children:["\n",(0,n.jsx)(A.li,{children:"Resample from the original sample with replacement."}),"\n",(0,n.jsx)(A.li,{children:"Calculate the sample statistic on the bootstrapped resample."}),"\n",(0,n.jsx)(A.li,{children:"Save the results into an array."}),"\n",(0,n.jsx)(A.li,{children:"Repeat steps 1 through 3 to generate an empirical distribution of the test statistic."}),"\n",(0,n.jsx)(A.li,{children:"Calculate the confidence interval and see if the observed statistic lies in it. If the observed statistic is not\nin our confidence interval, we have evidence to reject the null."}),"\n"]}),(0,n.jsxs)(A.p,{children:[(0,n.jsx)(A.strong,{children:"Confidence Intervals:"})," A confidence interval is a range that captures most of the distribution of the bootstrapped sample statistic in\nthe hopes of also containing the true population parameter within it. If\nwe were to construct a 95% confidence interval, we aren't saying that there is a 95% chance that the true population parameter lies in the\ninterval as the interval either contains it or it doesn't. Instead, we are saying that approximately 95% of the time, the intervals you\ncreate will contain the true population parameter. For example, if we generated 100 confidence intervals, about 95 of them will have the\ntrue population parameter."]})]}),"\n",(0,n.jsx)(A.admonition,{type:"note",children:(0,n.jsx)(A.p,{children:"When resampling, the size of the resample should be the same as the original sample with replacement."})}),"\n",(0,n.jsx)(A.p,{children:"The diagram below provides an overview of conducting bootstrapping, although it references a different dataset."}),"\n",(0,n.jsx)(r.A,{src:"https://docs.google.com/presentation/d/e/2PACX-1vS_iYHJYXSVMMZ-YQVFwMEFR6EFN3FDSAvaMyUm-YJfLQgRMTHm3vI-wWJJ5999eFJq70nWp2hyItZg/embed?start=true&loop=false&delayms=3000&rm=minimal",sourceLink:"https://docs.google.com/presentation/d/1oYakqMdI7z61BthvgWUJvbT21bKTXg01KmAibglepBI/edit?usp=sharing"}),"\n",(0,n.jsxs)(A.p,{children:["The diagram below provides an overview of creating confidence intervals, although it references a different situation. For additional helpful visual guides, please visit the ",(0,n.jsx)(A.a,{href:"https://dsc10.com/diagrams/",children:"Diagrams"})," site."]}),"\n",(0,n.jsx)(r.A,{src:"https://docs.google.com/presentation/d/e/2PACX-1vTaPZsueXI6fey_5cj2Y1TevkR1joBvpwaWVsZNvgBlnJSrw1EiBLHJywkFH_QNLU5Tdr6JZgDrhFxG/embed?start=true&loop=false&delayms=3000&rm=minimal",sourceLink:"https://docs.google.com/presentation/d/1l5GVyl60FrornEaZnGL6REV4gjEydRncNVi6k1hHUsA/edit?usp=sharing"}),"\n",(0,n.jsx)(A.hr,{}),"\n",(0,n.jsx)(A.h2,{id:"code-example",children:"Code Example"}),"\n",(0,n.jsxs)(A.h3,{id:"1-take-a-random-sample-of-size-12-from-the-full_pets-dataframe",children:["1. Take a random sample of size 12 from the ",(0,n.jsx)(A.code,{children:"full_pets"})," DataFrame."]}),"\n",(0,n.jsxs)(A.p,{children:["Let's say we didn't have access to all of the information in the ",(0,n.jsx)(A.code,{children:"full_pets"})," DataFrame and were only able to collect a sample of 12 pets."]}),"\n",(0,n.jsx)(A.pre,{children:(0,n.jsx)(A.code,{className:"language-python",children:"# Magic to ensure that we get the same results every time this code is run. \nnp.random.seed(42)\n\n# sample\npets_sample = full_pets.sample(12, replace=False)\npets_sample\n"})}),"\n",(0,n.jsx)(s.A,{data:'{"columns":["Unnamed: 0","ID","Species","Color","Weight","Age","Is_Cat","Owner_Comment"],"index":[18,14,4,13,10,7,6,3,2,15,17,8],"data":[[18,"cat_006","cat","black",3.0,0.5,true,"No, thank you!"],[14,"dog_007","dog","white",50.0,6.1,false,"No, thank you!"],[4,"dog_003","dog","black",25.0,0.5,false,"Be the person your dog thinks you are."],[13,"ham_003","hamster","black",0.5,0.1,false,"No, thank you!"],[10,"dog_006","dog","golden",35.0,4.0,false,"No, thank you!"],[7,"cat_003","cat","black",10.0,0.0,true,"No, thank you!"],[6,"ham_002","hamster","golden",0.25,0.2,false,"No, thank you!"],[3,"dog_002","dog","white",80.0,2.0,false,"Love is a wet nose and a wagging tail."],[2,"cat_002","cat","black",15.0,9.0,true,"****All you need is love and a cat.****"],[15,"ham_004","hamster","golden",0.25,0.2,false,"No, thank you!"],[17,"dog_009","dog","white",30.0,4.8,false,"No, thank you!"],[8,"dog_004","dog","black",45.0,6.7,false,"No, thank you!"]]}'}),"\n",(0,n.jsx)(A.hr,{}),"\n",(0,n.jsx)(A.h3,{id:"2-find-the-observed-parameter",children:"2. Find the observed parameter"}),"\n",(0,n.jsx)(A.p,{children:"In this case, we are interested in finding the median weight of the entire population."}),"\n",(0,n.jsx)(A.pre,{children:(0,n.jsx)(A.code,{className:"language-python",children:"pets_sample = full_pets.sample(12, replace=False)\nprint('Median of pets_sample weight:', pets_sample.get('Weight').median())\n"})}),"\n",(0,n.jsx)(A.p,{children:"Median of pets_sample weight: 20.0"}),"\n",(0,n.jsx)(A.hr,{}),"\n",(0,n.jsx)(A.h3,{id:"3-bootstrap-the-sample-10000-times-with-replacement",children:"3. Bootstrap the sample 10,000 times with replacement"}),"\n",(0,n.jsx)(A.p,{children:"Since we were only able to collect one random sample from the full population, we can't be sure if this singular guess predicts the\ntrue population parameter well. We can't go out and collect\nanother random sample, so we will resample from the original sample with replacement to simulate what could've been."}),"\n",(0,n.jsx)(A.pre,{children:(0,n.jsx)(A.code,{className:"language-python",children:"boot_medians = np.array([])\nfor i in np.arange(10000):\n # 1. resample the data\n resample = pets_sample.sample(pets_sample.shape[0], replace=True)\n\n # 2. calculate the median of the resample\n boot_median = resample.get('Weight').median()\n\n # 3. append the median to the array\n boot_medians = np.append(boot_medians, boot_median)\n"})}),"\n",(0,n.jsx)(A.p,{children:"This code will create 10,000 bootstrapped samples and calculate the median for each of them, but a different\nreasonable number can be used instead. Since these samples are all random, the information in each sample and\nmedian will be different from one another."}),"\n",(0,n.jsx)(A.hr,{}),"\n",(0,n.jsx)(A.h3,{id:"4-create-a-95-confidence-interval",children:"4. Create a 95% confidence interval"}),"\n",(0,n.jsx)(A.p,{children:"Instead of using a single number to estimate the true population parameter, we create a range of where we think it is."}),"\n",(0,n.jsx)(A.pre,{children:(0,n.jsx)(A.code,{className:"language-python",children:"# Get the 95% confidence interval\nleft = np.percentile(boot_medians, 2.5) # 2.5th percentile\nright = np.percentile(boot_medians, 97.5) # 97.5th percentile\n"})}),"\n",(0,n.jsx)(A.p,{children:"Remember that the 95% confidence interval does not mean we have a 95% chance of containing the true population parameter. It means\nthat about 95% of all intervals we create will contain the true population parameter."}),"\n",(0,n.jsx)(A.hr,{}),"\n",(0,n.jsx)(A.h3,{id:"5-conclusion",children:"5. Conclusion"}),"\n",(0,n.jsx)(A.pre,{children:(0,n.jsx)(A.code,{className:"language-python",children:"left, right\n"})}),"\n",(0,n.jsx)(A.p,{children:"(1.75, 40.0)"}),"\n",(0,n.jsx)(A.admonition,{title:"Conclusion",type:"info",children:(0,n.jsxs)(A.ul,{children:["\n",(0,n.jsx)(A.li,{children:"From this interval, we are 95% confident that the true population median lies somewhere between 1.75 and 40."}),"\n",(0,n.jsx)(A.li,{children:"We have no way of knowing where exactly in this interval does the true population median falls or even if it is contained at all."}),"\n",(0,n.jsx)(A.li,{children:"What we do know is that if we were to repeat the process and generate multiple confidence intervals, roughly 95% of them will\ncontain the true population median."}),"\n"]})}),"\n",(0,n.jsx)(A.hr,{}),"\n",(0,n.jsx)(A.h3,{id:"6-extra",children:"6. Extra"}),"\n",(0,n.jsx)(A.p,{children:"Let's look at the distribution of the bootstrapped medians!"}),"\n",(0,n.jsx)(A.pre,{children:(0,n.jsx)(A.code,{className:"language-python",children:"# Create the histogram.\n# Plot the histogram of boot_medians\nplt.hist(boot_medians, bins=20, density=True, ec = 'w')\n\nplt.show()\n"})}),"\n",(0,n.jsx)(A.p,{children:(0,n.jsx)(A.img,{alt:"Distribution",src:t(3698).A+"",width:"556",height:"413"})}),"\n",(0,n.jsxs)(A.p,{children:["A 95% confidence level means that ",(0,n.jsx)(A.strong,{children:"approximately 95% of the time, the intervals you create through this process will contain the true population parameter"}),"."]})]})}function h(e={}){const{wrapper:A}={...(0,a.R)(),...e.components};return A?(0,n.jsx)(A,{...e,children:(0,n.jsx)(p,{...e})}):p(e)}},8597:(e,A,t)=>{t.d(A,{A:()=>a});t(6540);var n=t(4848);const a=e=>{let{data:A}=e,t=[],a=[],s="",r=[];if(A){const 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