{"id":2616,"date":"2024-07-12T10:34:34","date_gmt":"2024-07-12T14:34:34","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/?post_type=chapter&#038;p=2616"},"modified":"2024-07-24T01:00:38","modified_gmt":"2024-07-24T05:00:38","slug":"hypothesis-testing-for-proportions-examples","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/chapter\/hypothesis-testing-for-proportions-examples\/","title":{"raw":"Hypothesis Testing for One Proportion","rendered":"Hypothesis Testing for One Proportion"},"content":{"raw":"<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Learning Objectives<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">Learn how to perform left, right and two-tailed hypothesis tests for proportions from one sample.<\/div>\r\n<\/div>\r\nIn this section, we will step through several examples of how to perform hypothesis testing for sample proportions. First, let us recap the required formulas from the <a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/chapter\/steps-to-perform-a-hypothesis-testing\/\">Hypothesis Steps section<\/a>:\r\n<h2>Hypotheses<\/h2>\r\nH<sub>0<\/sub>: [latex]p = p_{original}[\/latex] (all tails)\r\n\r\nH<sub>A<\/sub>: [latex]p &lt; p_{original}[\/latex] (left-tailed), [latex]p &gt; p_{original}[\/latex] (right-tailed), [latex]p \\neq p_{original}[\/latex] (two-tailed)\r\n<h2>Test Statistic<\/h2>\r\nThe test statistic formula is the same, regardless of what tailed test we are performing: \\[z_{test} = \\frac{\\bar{p}-p}{\\sqrt{\\frac{p(1-p)}{n}}} \\]\r\n<h2>P-Values<\/h2>\r\nWe use Excel's NORM.S.DIST() function to calculate all p-values:\r\n\r\n\\[p_{\\text{ left-tailed}}=\\text{NORM.S.DIST}(z_{test},\\text{TRUE})\\]\r\n\r\n\\[p_{\\text{ two-tailed &amp; z &lt; 0}}=2\\times\\text{NORM.S.DIST}(z_{test},\\text{TRUE}) \\]\r\n\r\n\\[p_{\\text{ two-tailed &amp; z &gt; 0}}=2\\times(1-\\text{NORM.S.DIST}(z_{test},\\text{TRUE})) \u00a0\\]\r\n\r\n\\[p_{\\text{ right-tailed}}=1-\\text{NORM.S.DIST}(z_{test},\\text{TRUE}) \\]\r\n\r\n<strong>Remember: <\/strong>For two-tailed tests, we double the area outside of the z[latex]_{test}[\/latex] score to account for the fact that we are interested in either tail (the left or right tail). We double the area beyond the test statistic to account for this.\r\n<h1>Click-Through Rates Right-tailed Test (Excel Solutions VIDEO)<\/h1>\r\nIn this section, we will have an ad campaign being run on the <a href=\"https:\/\/supersede.media\/blog\/topic\/pay-per-click\/search-network-vs-display-network\">Google Ads Search Network<\/a>. We will ask if there has been a change in the Click-Through Rates based on a change to the ad. We will use a right-tailed hypothesis test to perform this test. Before we jump into the example, let us first discuss Click-Through Rates (CTR).\r\n<h2>Click-Through Rates (CTR)<\/h2>\r\nWhat is a Click-Through Rate (CTR)? It is the percentage of time the person clicks through to a website, whether they click on your ad, click on website brought up in a search, etc. It can be defined as the ratio of clicks to impressions.\r\n\r\n\\[CTR = \\frac{\\text{Number of Clicks}}{\\text{Number of Impressions}}\\times 100\\% \\]\r\n\r\n<strong>Note<\/strong>: An <a href=\"https:\/\/www.appsflyer.com\/glossary\/ad-impression\/#:~:text=An%20ad%20impression%2C%20also%20known,have%20potentially%20viewed%20your%20content.\">impression<\/a> is a 'view' of the ad, search suggestion or video in question.\r\n<h2>Example 58.1<\/h2>\r\n<strong><span style=\"color: #003366\">Problem Setup<\/span><\/strong>: You and your marketing team are running an ad on Google Ad's Search Network. The campaign has had a Click-Through Rate of 1.91% until recently. You recently enabled your ad to appear on Google's search partners in addition to searches on Google itself. You believe this change has created an increased Click-Through Rate.\r\n\r\n[caption id=\"attachment_2622\" align=\"aligncenter\" width=\"419\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/GoogleAds-scaled-e1720810262343.jpg\"><img class=\"wp-image-2622\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/GoogleAds-scaled-e1720810262343.jpg\" alt=\"Photo of Google's search page and &quot;Pexels' is typed in the search bar. Suggested searches show up below the typed word.\" width=\"419\" height=\"384\" \/><\/a> Figure 58.1 Google suggested results from search[\/caption]\r\n\r\nYour Google Ads account, since the change, shows 77 clicks-through on 3054 impressions.\r\n\r\n<span style=\"color: #003366\"><strong>Question:<\/strong><\/span> Is there sufficient evidence, at the 5% level of significance, to conclude that turning on the feature to allow your ad to also be seen via Google's search partners increased the Click-Through Rate?\r\n\r\n<span style=\"color: #003366\"><strong>Solutions<\/strong><\/span>: <a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example58-1_Solutions.xlsx\">Click here<\/a>\u00a0to download the Excel solutions and click on the sections below to reveal the written solutions shown in the above video.\r\n\r\n[embed]https:\/\/youtu.be\/jj3R_p-9weg[\/embed]\r\n<div>\r\n<h1>Assumptions<\/h1>\r\nFor proportions, there is a slightly different set of criteria for a large enough sample size. The sample size needs to be large enough such that:\r\n<ul>\r\n \t<li>[latex]n\\times p &gt; 5[\/latex]<\/li>\r\n \t<li>[latex]n\\times (1-p) &gt; 5[\/latex]<\/li>\r\n<\/ul>\r\nIf the above criteria are satisfied, the mean of the distribution of the sample proportions, denoted \u03bc<sub>p<\/sub> , equals the population proportion. In addition, the distribution of the sample proportion is approximately normal.\r\n<h1>Hypotheses<\/h1>\r\n<ul>\r\n \t<li>H<sub>0<\/sub>: p = 0.0191<\/li>\r\n \t<li>H<sub>A<\/sub>: p &gt; 0.0191<\/li>\r\n<\/ul>\r\n<h1>Rejection Region<\/h1>\r\n<span style=\"color: #000000\">We want to prove that the click-through rate (CTR) is now higher than 1.91%. For this reason, we need our sample result to be significantly higher than 1.91%. The rejection region would be the upper 5% right tail. The original proportion of 1.91% is what we are comparing to and is marked in the middle of the graph<\/span>.\r\n\r\n[caption id=\"attachment_2633\" align=\"aligncenter\" width=\"592\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example58-1.jpg\"><img class=\"wp-image-2633 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example58-1.jpg\" alt=\"Bell curve with right-most 5 percent highlighted and p equals to 0.0191\" width=\"592\" height=\"278\" \/><\/a> Figure 58.2 Rejection region[\/caption]\r\n<h1>Type of Test<\/h1>\r\nThis is a right-tailed test. In order to prove that the click-through rate (CTR) is now higher than 1.91%, we will need our sample result to be far enough above the original rate of 1.91% and be, therefore, in the upper right tail (upper rejection region).\r\n<h1>Test Statistic<\/h1>\r\nIn order to calculate the test statistic, we need to first calculate the sample proportion:\r\n\r\n[latex] \\bar{p}=\\frac{\\bar{x}}{n}=\\frac{77}{3054} =0.0252 = 2.52\\% [\/latex]\r\n\r\nWe can now use the sample proportion to calculate the test statistic:\r\n\r\n[latex]\r\n\r\n\\begin{align}\r\n\r\nz_{test} = \\frac{\\bar{p}-p}{\\sqrt{\\frac{p(1-p)}{n}}} &amp;= \\frac{0.0252-0.0191}{\\sqrt{\\frac{0.0191(1-0.0191)}{3054}}} \\\\\r\n\r\n&amp;= \\frac{0.006113}{0.002477}= 2.4680\r\n\r\n\\end{align}\r\n\r\n[\/latex]\r\n<h1>P-value<\/h1>\r\nBecause we are performing a right-tailed test on a mean, we will use Excel's <a href=\"https:\/\/support.microsoft.com\/en-us\/office\/norm-s-dist-function-1e787282-3832-4520-a9ae-bd2a8d99ba88\">NORM.S.DIST.(z<sub>test<\/sub>, 1)<\/a> function:\r\n\r\n[latex] p\\text{-value}=\\text{NORM.S.DIST}(2.4680, 1) = 0.0068 [\/latex]\r\n<h1>Decision<\/h1>\r\nWe can see, in the figure below, that the test result of 2.52% lies far within the rejection region:\r\n\r\n[caption id=\"attachment_2652\" align=\"aligncenter\" width=\"543\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example58-1_SOLUTION.jpg\"><img class=\"size-full wp-image-2652\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example58-1_SOLUTION.jpg\" alt=\"Bell curve with right 5 percent shaded. A line is marked where p-bar of 2 point 52 percent is noted. Above this region is written zero point 68 percent.\" width=\"543\" height=\"246\" \/><\/a> Figure 58.3 Rejection rejection with sample proportion indicated on graph[\/caption]\r\n\r\nThis means that the p-value (0.68%) is less than the level of significance of (5%) and we should reject H<sub>0<\/sub>.\r\n<h1>Conclusion<\/h1>\r\nBecause we rejected H<sub>0<\/sub>, there is sufficient evidence to conclude that turning on the feature to allow your ad to also be seen via Google's search partners increased the Click-Through Rate.\r\n\r\n<\/div>\r\n<h1>Quality COntrol Left-tailed Test (Written Solutions Video)<\/h1>\r\nIn this example, we will perform a left-tailed hypothesis test for a sample with one proportion. We will return to our example of quality control for <a href=\"https:\/\/fastenerandfixing.com\/application-technology\/space-fasteners-play-their-part\/\">space fasteners<\/a> that could be supplied to organizations and companies like <a href=\"https:\/\/standards.nasa.gov\/standard\/NASA\/NASA-STD-6008\">NASA<\/a>, <a href=\"https:\/\/www.spacex.com\/\">SpaceX<\/a> and other companies that build satellites and other machines used in space.\r\n<table class=\"no-lines\" style=\"border-collapse: collapse;width: 100%\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%\">\r\n\r\n[caption id=\"attachment_2345\" align=\"alignnone\" width=\"600\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Space_fasteners.jpg\"><img class=\"wp-image-2345 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Space_fasteners.jpg\" alt=\"Image of nuts, bolts and other fasteners that could be used in space.\" width=\"600\" height=\"397\" \/><\/a> Figure 58.4 Possible space fasteners[\/caption]<\/td>\r\n<td style=\"width: 50%\">\r\n\r\n[caption id=\"attachment_2346\" align=\"alignnone\" width=\"598\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Satellite_image.jpg\"><img class=\"wp-image-2346 \" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Satellite_image.jpg\" alt=\"Image of satellite orbiting over Earth.\" width=\"598\" height=\"396\" \/><\/a> Figure 58.5 Satellite orbiting over Earth[\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Example 58.2<\/h2>\r\n<span style=\"color: #003366\"><strong>Problem Setup:<\/strong><\/span> A company supplies Standard Hexagon Head Cap Screws. They have recently upgraded their machines and believe that they have decreased the percent of screws deemed as \"defective.\" Historically, they would have 1 out of 125 screws deemed as 'defective.'\r\n\r\n[caption id=\"attachment_2411\" align=\"aligncenter\" width=\"333\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/HexBoltScrew.jpg\"><img class=\" wp-image-2411\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/HexBoltScrew.jpg\" alt=\"Picture with two hex bolt screws\" width=\"333\" height=\"286\" \/><\/a> Figure 58.6 Military hexagon head cap screws[\/caption]\r\n\r\n<span style=\"color: #003366\"><strong>Question<\/strong><\/span>: Since replacing their machines, they sampled 1,000 screws and found that 6 were defective. At the 1% level of significance, is there sufficient evidence to conclude that the percent of defective screws has decreased since replacing their machines?\r\n\r\n<span style=\"color: #003366\"><strong>Solutions<\/strong><\/span>: <a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example38-2_Left-Tailed_Hypothesis-Testing-on-Proportions.pdf\">Click here to download the written solutions<\/a> shown in the video below. <a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example58-2_Solutions.xlsx\">Click here to download the Excel solutions<\/a>.\r\n\r\nhttps:\/\/youtu.be\/24cvLIP4kkQ\r\n\r\n<span style=\"color: #003366\"><strong>Conclusion<\/strong><\/span>: Because we failed to reject H<sub>0<\/sub>, there is not sufficient evidence to conclude that the percent of defective screws has decreased since replacing the machines in the production plant. Perhaps we can conclude that the money could have better been spent on improving other processes\/machinery in the plant?\r\n<div>\r\n<h1>Written Solutions (click to Reveal)<\/h1>\r\n<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example58-2_Written_SOLUTION.jpg\"><img class=\"alignnone size-full wp-image-2667\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example58-2_Written_SOLUTION.jpg\" alt=\"Handwritten solutions for example. Image only.\" width=\"922\" height=\"733\" \/><\/a>\r\n\r\n<\/div>\r\n<h1>Polling Votes Two-tailed Test (Exercise)<\/h1>\r\nIn this example, we will look at polling votes for an election example. We will perform a two-tailed proportion test.\r\n<h2>Example 58.3<\/h2>\r\n<strong><span style=\"color: #003366\">Problem Setup<\/span><\/strong>: <em>Abacus Pollsters<\/em> polls the electorate before the next election. Previous to the poll, it was forecasted that 28% of the electorate would vote for the <a href=\"https:\/\/en.wikipedia.org\/wiki\/The_Canada_Party\">Canada Party<\/a> in the upcoming election. They poll 1,200 people and find that 282 people responded that they planned to vote for the Canada Party in the upcoming election.\r\n\r\n[caption id=\"attachment_2673\" align=\"aligncenter\" width=\"402\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/CanadaParty2.jpg\" rel=\"https:\/\/pixabay.com\/illustrations\/crowd-lego-staff-choice-voter-1699137\/\"><img class=\"wp-image-2673 \" title=\"Photo by Matthias Wewering from Pixabay\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/CanadaParty2.jpg\" alt=\"Picture of arm with canada flag painted on it. Hand is giving a thumbs up.\" width=\"402\" height=\"263\" \/><\/a> Figure 58.7 Canada Party thumbs up.[\/caption]\r\n\r\n<strong><span style=\"color: #003366\">Question<\/span><\/strong>: Is there sufficient evidence, at the 5% level of significance, to conclude that the percent of people forecasted to vote for the Canada Party has changed?\r\n\r\n<span style=\"color: #003366\"><strong>You Try<\/strong><\/span>: Scroll through the questions below to solve the hypothesis test problem above.\r\n<div class=\"textbox textbox--exercises\">\r\n<div class=\"textbox__content\">\r\n\r\n[h5p id=\"184\"]\r\n\r\n<\/div>\r\n<\/div>\r\n<span style=\"color: #003366\"><strong>Solutions<\/strong><\/span>: <a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example58-3_Solutions.xlsx\">Click here<\/a> to download the Excel solutions to the above exercise.","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">Learn how to perform left, right and two-tailed hypothesis tests for proportions from one sample.<\/div>\n<\/div>\n<p>In this section, we will step through several examples of how to perform hypothesis testing for sample proportions. First, let us recap the required formulas from the <a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/chapter\/steps-to-perform-a-hypothesis-testing\/\">Hypothesis Steps section<\/a>:<\/p>\n<h2>Hypotheses<\/h2>\n<p>H<sub>0<\/sub>: [latex]p = p_{original}[\/latex] (all tails)<\/p>\n<p>H<sub>A<\/sub>: [latex]p < p_{original}[\/latex] (left-tailed), [latex]p > p_{original}[\/latex] (right-tailed), [latex]p \\neq p_{original}[\/latex] (two-tailed)<\/p>\n<h2>Test Statistic<\/h2>\n<p>The test statistic formula is the same, regardless of what tailed test we are performing: \\[z_{test} = \\frac{\\bar{p}-p}{\\sqrt{\\frac{p(1-p)}{n}}} \\]<\/p>\n<h2>P-Values<\/h2>\n<p>We use Excel&#8217;s NORM.S.DIST() function to calculate all p-values:<\/p>\n<p>\\[p_{\\text{ left-tailed}}=\\text{NORM.S.DIST}(z_{test},\\text{TRUE})\\]<\/p>\n<p>\\[p_{\\text{ two-tailed &amp; z &lt; 0}}=2\\times\\text{NORM.S.DIST}(z_{test},\\text{TRUE}) \\]<\/p>\n<p>\\[p_{\\text{ two-tailed &amp; z &gt; 0}}=2\\times(1-\\text{NORM.S.DIST}(z_{test},\\text{TRUE})) \u00a0\\]<\/p>\n<p>\\[p_{\\text{ right-tailed}}=1-\\text{NORM.S.DIST}(z_{test},\\text{TRUE}) \\]<\/p>\n<p><strong>Remember: <\/strong>For two-tailed tests, we double the area outside of the z[latex]_{test}[\/latex] score to account for the fact that we are interested in either tail (the left or right tail). We double the area beyond the test statistic to account for this.<\/p>\n<h1>Click-Through Rates Right-tailed Test (Excel Solutions VIDEO)<\/h1>\n<p>In this section, we will have an ad campaign being run on the <a href=\"https:\/\/supersede.media\/blog\/topic\/pay-per-click\/search-network-vs-display-network\">Google Ads Search Network<\/a>. We will ask if there has been a change in the Click-Through Rates based on a change to the ad. We will use a right-tailed hypothesis test to perform this test. Before we jump into the example, let us first discuss Click-Through Rates (CTR).<\/p>\n<h2>Click-Through Rates (CTR)<\/h2>\n<p>What is a Click-Through Rate (CTR)? It is the percentage of time the person clicks through to a website, whether they click on your ad, click on website brought up in a search, etc. It can be defined as the ratio of clicks to impressions.<\/p>\n<p>\\[CTR = \\frac{\\text{Number of Clicks}}{\\text{Number of Impressions}}\\times 100\\% \\]<\/p>\n<p><strong>Note<\/strong>: An <a href=\"https:\/\/www.appsflyer.com\/glossary\/ad-impression\/#:~:text=An%20ad%20impression%2C%20also%20known,have%20potentially%20viewed%20your%20content.\">impression<\/a> is a &#8216;view&#8217; of the ad, search suggestion or video in question.<\/p>\n<h2>Example 58.1<\/h2>\n<p><strong><span style=\"color: #003366\">Problem Setup<\/span><\/strong>: You and your marketing team are running an ad on Google Ad&#8217;s Search Network. The campaign has had a Click-Through Rate of 1.91% until recently. You recently enabled your ad to appear on Google&#8217;s search partners in addition to searches on Google itself. You believe this change has created an increased Click-Through Rate.<\/p>\n<figure id=\"attachment_2622\" aria-describedby=\"caption-attachment-2622\" style=\"width: 419px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/GoogleAds-scaled-e1720810262343.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2622\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/GoogleAds-scaled-e1720810262343.jpg\" alt=\"Photo of Google's search page and &quot;Pexels' is typed in the search bar. Suggested searches show up below the typed word.\" width=\"419\" height=\"384\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/GoogleAds-scaled-e1720810262343.jpg 1920w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/GoogleAds-scaled-e1720810262343-300x275.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/GoogleAds-scaled-e1720810262343-1024x940.jpg 1024w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/GoogleAds-scaled-e1720810262343-768x705.jpg 768w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/GoogleAds-scaled-e1720810262343-1536x1410.jpg 1536w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/GoogleAds-scaled-e1720810262343-65x60.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/GoogleAds-scaled-e1720810262343-225x206.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/GoogleAds-scaled-e1720810262343-350x321.jpg 350w\" sizes=\"auto, (max-width: 419px) 100vw, 419px\" \/><\/a><figcaption id=\"caption-attachment-2622\" class=\"wp-caption-text\">Figure 58.1 Google suggested results from search<\/figcaption><\/figure>\n<p>Your Google Ads account, since the change, shows 77 clicks-through on 3054 impressions.<\/p>\n<p><span style=\"color: #003366\"><strong>Question:<\/strong><\/span> Is there sufficient evidence, at the 5% level of significance, to conclude that turning on the feature to allow your ad to also be seen via Google&#8217;s search partners increased the Click-Through Rate?<\/p>\n<p><span style=\"color: #003366\"><strong>Solutions<\/strong><\/span>: <a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example58-1_Solutions.xlsx\">Click here<\/a>\u00a0to download the Excel solutions and click on the sections below to reveal the written solutions shown in the above video.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"How to determine if you have increased your click through rate with a Hypothesis Test\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/jj3R_p-9weg?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div>\n<h1>Assumptions<\/h1>\n<p>For proportions, there is a slightly different set of criteria for a large enough sample size. The sample size needs to be large enough such that:<\/p>\n<ul>\n<li>[latex]n\\times p > 5[\/latex]<\/li>\n<li>[latex]n\\times (1-p) > 5[\/latex]<\/li>\n<\/ul>\n<p>If the above criteria are satisfied, the mean of the distribution of the sample proportions, denoted \u03bc<sub>p<\/sub> , equals the population proportion. In addition, the distribution of the sample proportion is approximately normal.<\/p>\n<h1>Hypotheses<\/h1>\n<ul>\n<li>H<sub>0<\/sub>: p = 0.0191<\/li>\n<li>H<sub>A<\/sub>: p &gt; 0.0191<\/li>\n<\/ul>\n<h1>Rejection Region<\/h1>\n<p><span style=\"color: #000000\">We want to prove that the click-through rate (CTR) is now higher than 1.91%. For this reason, we need our sample result to be significantly higher than 1.91%. The rejection region would be the upper 5% right tail. The original proportion of 1.91% is what we are comparing to and is marked in the middle of the graph<\/span>.<\/p>\n<figure id=\"attachment_2633\" aria-describedby=\"caption-attachment-2633\" style=\"width: 592px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example58-1.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2633 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example58-1.jpg\" alt=\"Bell curve with right-most 5 percent highlighted and p equals to 0.0191\" width=\"592\" height=\"278\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example58-1.jpg 592w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example58-1-300x141.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example58-1-65x31.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example58-1-225x106.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example58-1-350x164.jpg 350w\" sizes=\"auto, (max-width: 592px) 100vw, 592px\" \/><\/a><figcaption id=\"caption-attachment-2633\" class=\"wp-caption-text\">Figure 58.2 Rejection region<\/figcaption><\/figure>\n<h1>Type of Test<\/h1>\n<p>This is a right-tailed test. In order to prove that the click-through rate (CTR) is now higher than 1.91%, we will need our sample result to be far enough above the original rate of 1.91% and be, therefore, in the upper right tail (upper rejection region).<\/p>\n<h1>Test Statistic<\/h1>\n<p>In order to calculate the test statistic, we need to first calculate the sample proportion:<\/p>\n<p>[latex]\\bar{p}=\\frac{\\bar{x}}{n}=\\frac{77}{3054} =0.0252 = 2.52\\%[\/latex]<\/p>\n<p>We can now use the sample proportion to calculate the test statistic:<\/p>\n<p>[latex]\\begin{align}    z_{test} = \\frac{\\bar{p}-p}{\\sqrt{\\frac{p(1-p)}{n}}} &= \\frac{0.0252-0.0191}{\\sqrt{\\frac{0.0191(1-0.0191)}{3054}}} \\\\    &= \\frac{0.006113}{0.002477}= 2.4680    \\end{align}[\/latex]<\/p>\n<h1>P-value<\/h1>\n<p>Because we are performing a right-tailed test on a mean, we will use Excel&#8217;s <a href=\"https:\/\/support.microsoft.com\/en-us\/office\/norm-s-dist-function-1e787282-3832-4520-a9ae-bd2a8d99ba88\">NORM.S.DIST.(z<sub>test<\/sub>, 1)<\/a> function:<\/p>\n<p>[latex]p\\text{-value}=\\text{NORM.S.DIST}(2.4680, 1) = 0.0068[\/latex]<\/p>\n<h1>Decision<\/h1>\n<p>We can see, in the figure below, that the test result of 2.52% lies far within the rejection region:<\/p>\n<figure id=\"attachment_2652\" aria-describedby=\"caption-attachment-2652\" style=\"width: 543px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example58-1_SOLUTION.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-2652\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example58-1_SOLUTION.jpg\" alt=\"Bell curve with right 5 percent shaded. A line is marked where p-bar of 2 point 52 percent is noted. Above this region is written zero point 68 percent.\" width=\"543\" height=\"246\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example58-1_SOLUTION.jpg 543w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example58-1_SOLUTION-300x136.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example58-1_SOLUTION-65x29.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example58-1_SOLUTION-225x102.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example58-1_SOLUTION-350x159.jpg 350w\" sizes=\"auto, (max-width: 543px) 100vw, 543px\" \/><\/a><figcaption id=\"caption-attachment-2652\" class=\"wp-caption-text\">Figure 58.3 Rejection rejection with sample proportion indicated on graph<\/figcaption><\/figure>\n<p>This means that the p-value (0.68%) is less than the level of significance of (5%) and we should reject H<sub>0<\/sub>.<\/p>\n<h1>Conclusion<\/h1>\n<p>Because we rejected H<sub>0<\/sub>, there is sufficient evidence to conclude that turning on the feature to allow your ad to also be seen via Google&#8217;s search partners increased the Click-Through Rate.<\/p>\n<\/div>\n<h1>Quality COntrol Left-tailed Test (Written Solutions Video)<\/h1>\n<p>In this example, we will perform a left-tailed hypothesis test for a sample with one proportion. We will return to our example of quality control for <a href=\"https:\/\/fastenerandfixing.com\/application-technology\/space-fasteners-play-their-part\/\">space fasteners<\/a> that could be supplied to organizations and companies like <a href=\"https:\/\/standards.nasa.gov\/standard\/NASA\/NASA-STD-6008\">NASA<\/a>, <a href=\"https:\/\/www.spacex.com\/\">SpaceX<\/a> and other companies that build satellites and other machines used in space.<\/p>\n<table class=\"no-lines\" style=\"border-collapse: collapse;width: 100%\">\n<tbody>\n<tr>\n<td style=\"width: 50%\">\n<figure id=\"attachment_2345\" aria-describedby=\"caption-attachment-2345\" style=\"width: 600px\" class=\"wp-caption alignnone\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Space_fasteners.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2345 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Space_fasteners.jpg\" alt=\"Image of nuts, bolts and other fasteners that could be used in space.\" width=\"600\" height=\"397\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Space_fasteners.jpg 600w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Space_fasteners-300x199.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Space_fasteners-65x43.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Space_fasteners-225x149.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Space_fasteners-350x232.jpg 350w\" sizes=\"auto, (max-width: 600px) 100vw, 600px\" \/><\/a><figcaption id=\"caption-attachment-2345\" class=\"wp-caption-text\">Figure 58.4 Possible space fasteners<\/figcaption><\/figure>\n<\/td>\n<td style=\"width: 50%\">\n<figure id=\"attachment_2346\" aria-describedby=\"caption-attachment-2346\" style=\"width: 598px\" class=\"wp-caption alignnone\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Satellite_image.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2346\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Satellite_image.jpg\" alt=\"Image of satellite orbiting over Earth.\" width=\"598\" height=\"396\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Satellite_image.jpg 600w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Satellite_image-300x199.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Satellite_image-65x43.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Satellite_image-225x149.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Satellite_image-350x232.jpg 350w\" sizes=\"auto, (max-width: 598px) 100vw, 598px\" \/><\/a><figcaption id=\"caption-attachment-2346\" class=\"wp-caption-text\">Figure 58.5 Satellite orbiting over Earth<\/figcaption><\/figure>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Example 58.2<\/h2>\n<p><span style=\"color: #003366\"><strong>Problem Setup:<\/strong><\/span> A company supplies Standard Hexagon Head Cap Screws. They have recently upgraded their machines and believe that they have decreased the percent of screws deemed as &#8220;defective.&#8221; Historically, they would have 1 out of 125 screws deemed as &#8216;defective.&#8217;<\/p>\n<figure id=\"attachment_2411\" aria-describedby=\"caption-attachment-2411\" style=\"width: 333px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/HexBoltScrew.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2411\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/HexBoltScrew.jpg\" alt=\"Picture with two hex bolt screws\" width=\"333\" height=\"286\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/HexBoltScrew.jpg 330w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/HexBoltScrew-300x257.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/HexBoltScrew-65x56.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/HexBoltScrew-225x193.jpg 225w\" sizes=\"auto, (max-width: 333px) 100vw, 333px\" \/><\/a><figcaption id=\"caption-attachment-2411\" class=\"wp-caption-text\">Figure 58.6 Military hexagon head cap screws<\/figcaption><\/figure>\n<p><span style=\"color: #003366\"><strong>Question<\/strong><\/span>: Since replacing their machines, they sampled 1,000 screws and found that 6 were defective. At the 1% level of significance, is there sufficient evidence to conclude that the percent of defective screws has decreased since replacing their machines?<\/p>\n<p><span style=\"color: #003366\"><strong>Solutions<\/strong><\/span>: <a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example38-2_Left-Tailed_Hypothesis-Testing-on-Proportions.pdf\">Click here to download the written solutions<\/a> shown in the video below. <a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example58-2_Solutions.xlsx\">Click here to download the Excel solutions<\/a>.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Hypothesis Test to Determine if Replacing Machines Decreases the Defective Rate\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/24cvLIP4kkQ?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><span style=\"color: #003366\"><strong>Conclusion<\/strong><\/span>: Because we failed to reject H<sub>0<\/sub>, there is not sufficient evidence to conclude that the percent of defective screws has decreased since replacing the machines in the production plant. Perhaps we can conclude that the money could have better been spent on improving other processes\/machinery in the plant?<\/p>\n<div>\n<h1>Written Solutions (click to Reveal)<\/h1>\n<p><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example58-2_Written_SOLUTION.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-2667\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example58-2_Written_SOLUTION.jpg\" alt=\"Handwritten solutions for example. Image only.\" width=\"922\" height=\"733\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example58-2_Written_SOLUTION.jpg 922w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example58-2_Written_SOLUTION-300x239.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example58-2_Written_SOLUTION-768x611.jpg 768w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example58-2_Written_SOLUTION-65x52.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example58-2_Written_SOLUTION-225x179.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example58-2_Written_SOLUTION-350x278.jpg 350w\" sizes=\"auto, (max-width: 922px) 100vw, 922px\" \/><\/a><\/p>\n<\/div>\n<h1>Polling Votes Two-tailed Test (Exercise)<\/h1>\n<p>In this example, we will look at polling votes for an election example. We will perform a two-tailed proportion test.<\/p>\n<h2>Example 58.3<\/h2>\n<p><strong><span style=\"color: #003366\">Problem Setup<\/span><\/strong>: <em>Abacus Pollsters<\/em> polls the electorate before the next election. Previous to the poll, it was forecasted that 28% of the electorate would vote for the <a href=\"https:\/\/en.wikipedia.org\/wiki\/The_Canada_Party\">Canada Party<\/a> in the upcoming election. They poll 1,200 people and find that 282 people responded that they planned to vote for the Canada Party in the upcoming election.<\/p>\n<figure id=\"attachment_2673\" aria-describedby=\"caption-attachment-2673\" style=\"width: 402px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/CanadaParty2.jpg\" rel=\"https:\/\/pixabay.com\/illustrations\/crowd-lego-staff-choice-voter-1699137\/\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2673\" title=\"Photo by Matthias Wewering from Pixabay\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/CanadaParty2.jpg\" alt=\"Picture of arm with canada flag painted on it. Hand is giving a thumbs up.\" width=\"402\" height=\"263\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/CanadaParty2.jpg 1280w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/CanadaParty2-300x196.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/CanadaParty2-1024x670.jpg 1024w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/CanadaParty2-768x502.jpg 768w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/CanadaParty2-65x43.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/CanadaParty2-225x147.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/CanadaParty2-350x229.jpg 350w\" sizes=\"auto, (max-width: 402px) 100vw, 402px\" \/><\/a><figcaption id=\"caption-attachment-2673\" class=\"wp-caption-text\">Figure 58.7 Canada Party thumbs up.<\/figcaption><\/figure>\n<p><strong><span style=\"color: #003366\">Question<\/span><\/strong>: Is there sufficient evidence, at the 5% level of significance, to conclude that the percent of people forecasted to vote for the Canada Party has changed?<\/p>\n<p><span style=\"color: #003366\"><strong>You Try<\/strong><\/span>: Scroll through the questions below to solve the hypothesis test problem above.<\/p>\n<div class=\"textbox textbox--exercises\">\n<div class=\"textbox__content\">\n<div id=\"h5p-184\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-184\" class=\"h5p-iframe\" data-content-id=\"184\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Example 58.3 Two-Tailed Hypothesis Testing One Proportion\"><\/iframe><\/div>\n<\/div>\n<\/div>\n<\/div>\n<p><span style=\"color: #003366\"><strong>Solutions<\/strong><\/span>: <a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example58-3_Solutions.xlsx\">Click here<\/a> to download the Excel solutions to the above exercise.<\/p>\n","protected":false},"author":865,"menu_order":4,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2616","chapter","type-chapter","status-publish","hentry"],"part":2505,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/2616","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/users\/865"}],"version-history":[{"count":25,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/2616\/revisions"}],"predecessor-version":[{"id":2678,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/2616\/revisions\/2678"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/parts\/2505"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/2616\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/media?parent=2616"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=2616"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/contributor?post=2616"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/license?post=2616"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}