{"id":509,"date":"2023-02-27T17:29:59","date_gmt":"2023-02-27T17:29:59","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/?post_type=chapter&#038;p=509"},"modified":"2025-08-31T10:03:02","modified_gmt":"2025-08-31T10:03:02","slug":"sampling-methods-and-bias-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/sampling-methods-and-bias-learn-it-3\/","title":{"raw":"Sampling Methods and Bias: Learn It 1","rendered":"Sampling Methods and Bias: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Select a simple random sample from a finite population<\/li>\r\n\t<li>Understand and apply different sampling methods<\/li>\r\n\t<li>Determine and explain bias in a sampling method<\/li>\r\n<\/ul>\r\n<\/section>\r\n<section class=\"textbox recall\">\r\n<ul>\r\n\t<li>The <strong>population<\/strong> is an entire group of people, objects, or animals; usually a large group.<\/li>\r\n\t<li>A <strong>sample <\/strong>is a selected subset or subgroup of a population.<\/li>\r\n\t<li>For statistics, we want our sample to be representative of the population so we can make accurate inferences.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Sampling Methods<\/h2>\r\n<p class=\"para\" style=\"margin: 6.0pt 0in 6.0pt 0in;\">There are five common sampling methods used in research. These sampling methods include:<\/p>\r\n<ol>\r\n\t<li class=\"para\">simple random sampling<\/li>\r\n\t<li class=\"para\">systematic sampling<\/li>\r\n\t<li class=\"para\">stratified sampling<\/li>\r\n\t<li class=\"para\">cluster sampling<\/li>\r\n\t<li class=\"para\">convenience sampling<\/li>\r\n<\/ol>\r\n<h2>Simple Random Samples<\/h2>\r\n<p>In <strong>simple random sampling<\/strong>, every sample of a given size has the same chance of being selected. This results in every individual or entity of the population having an equal chance of being selected. There are different ways to collect a simple random sample, such as using a random number generator, placing equal-sized slips of paper in a hat and drawing a sample, rolling a six-sided die, or any other event where each outcome is assigned a subject and each subject has an equally likely chance of being selected.<\/p>\r\n<section class=\"textbox example\">In the following figure, a random number generator selected numbers 24, 22, 27, 25, and 13, resulting in the highlighted individuals being selected for the sample.\r\n[caption id=\"attachment_5723\" align=\"alignnone\" width=\"878\"]<img class=\"wp-image-5723\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/10\/03171045\/2.2.L-simple-random-sampling-1.png\" alt=\"Appropriate alternative text can be found in the description above.\" width=\"878\" height=\"357\" \/> Figure 1. A simple random sample selects individuals purely by chance, giving each person an equal opportunity to be included.[\/caption]\r\n<\/section>\r\n<h3>Random Number Generator<\/h3>\r\n<section class=\"textbox interact\">\r\n<p>A simple random sample relies on a random mechanism to choose a sample, without replacement, from the population so that every sample is equally likely to be chosen. Consider choosing 10 states randomly from the 50 United States.<\/p>\r\n<p style=\"padding-left: 40px;\"><strong><br \/>\r\nSTEP 1:<\/strong> Under \u201cChoose Minimum,\u201d select \u201c1.\u201d<\/p>\r\n<p style=\"padding-left: 40px;\"><strong style=\"font-size: 1rem; text-align: initial; background-color: initial;\">STEP 2:<\/strong><span style=\"font-size: 1rem; text-align: initial; background-color: initial;\"> Under \u201cChoose Maximum,\u201d select \u201c50.\u201d<\/span><\/p>\r\n<p style=\"padding-left: 40px;\"><strong style=\"font-size: 1rem; text-align: initial; background-color: initial;\">STEP 3:<\/strong><span style=\"font-size: 1rem; text-align: initial; background-color: initial;\"> Under \u201cHow many numbers do you want to generate,\u201d select \u201c10.\u201d<\/span><\/p>\r\n<p style=\"padding-left: 40px;\"><strong style=\"font-size: 1rem; text-align: initial; background-color: initial;\">STEP 4:<\/strong><span style=\"font-size: 1rem; text-align: initial; background-color: initial;\"> Under \u201cSample with Replacement,\u201d select \u201cNo.\u201d<\/span><\/p>\r\n<p style=\"padding-left: 40px;\"><strong style=\"font-size: 1rem; text-align: initial; background-color: initial;\">STEP 5:<\/strong><span style=\"font-size: 1rem; text-align: initial; background-color: initial;\"> Click \u201cGenerate.\u201d This will generate ten random numbers between 1 and 50.<\/span><\/p>\r\n<p><strong><span style=\"font-size: 1rem; text-align: initial; background-color: initial;\">Save these 10 numbers for the next question.<\/span><\/strong><\/p>\r\n<\/section>\r\n<p><br \/>\r\n<iframe src=\"https:\/\/lumen-learning.shinyapps.io\/randomnumbers\/\" width=\"100%\" height=\"850\"><\/iframe> [<a href=\"https:\/\/lumen-learning.shinyapps.io\/randomnumbers\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]3162[\/ohm2_question]<\/section>\r\n<section class=\"textbox example\">\r\n<p>Consider a small college with only 200 students, where 60% of these students were eligible for financial aid.<br \/>\r\n<br \/>\r\nWe can use this simplified situation to investigate how random samples relate to the population. This is the first step in creating a probability model that will be useful in inference. We know the <em data-start=\"577\" data-end=\"584\">truth<\/em> about the whole population (60% get aid). But in real life, we usually <strong data-start=\"656\" data-end=\"691\">don\u2019t know the whole population<\/strong>; we only see a <strong data-start=\"706\" data-end=\"717\">sample.<\/strong> So the question we want to ask is:<\/p>\r\n<p style=\"text-align: center;\"><em>How accurate are random samples at predicting this population proportion of 0.60?\u00a0<\/em><\/p>\r\n<p>[reveal-answer q=\"580589\"]Click to see why the proportion is written as 0.60.[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"580589\"]In statistics, we usually write percentages as decimals, since it makes calculations easier. 60% means <em data-start=\"304\" data-end=\"320\">60 out of 100.<\/em><\/p>\r\n<p style=\"text-align: center;\">[latex]60\\% = \\frac{60}{100} =60\u00f7100= 0.60[\/latex][\/hidden-answer]<\/p>\r\n<p>To answer this question, we randomly select 8 students and determine the proportion who are eligible for financial aid. We repeat this process several times. Here are the results for 3 random samples:<\/p>\r\n\r\n[caption id=\"attachment_3593\" align=\"aligncenter\" width=\"781\"]<img class=\"wp-image-3593 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/10\/05174016\/2.1.L.Diagram2-1.png\" alt=\"Financial aid eligibility: 3 random samples of students consisting of 8 students each (out of a total population of 200 students). The proportion eligible for financial aid in the population is .60. In the random samples, each student is assigned a number and then categorized as elibigle for financial aid or not. The sample proportions are as follows: Sample 1 has 6 students eligible for aid and six divided by 8 is 0.75. Sample 2 has 5 students eligible for aid and five divided by 8 is 0.625. Sample 3 has 3 students eligible for aid and three divided by 8 is 0.375. When you average the sample proportions and round to the tens place you get a proportion of .60. \" width=\"781\" height=\"566\" \/> Figure 2. When taking more than one random sample from the same population, random sample proportions may vary.[\/caption]\r\n\r\n<p>[reveal-answer q=\"257212\"]More about these random samples.[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"257212\"]<\/p>\r\n<p>Notice the following about these random samples:<\/p>\r\n<ul>\r\n\t<li>Each random sample comes from the same population, where the true proportion is 0.60.<\/li>\r\n\t<li>But the sample proportions vary\u2014each group of 8 gives a different result.<\/li>\r\n\t<li>Some samples are above 0.60, some are below. Some are good estimates (like 0.625), while others are not (like 0.375).<\/li>\r\n\t<li>Small samples (like just 8 students) are <strong data-start=\"1557\" data-end=\"1576\">highly variable<\/strong>\u2014they can be inaccurate. That\u2019s why a sample proportion doesn\u2019t always match the population proportion exactly. Later, we investigate the effect of increasing the size of the sample.<\/li>\r\n\t<li>The variability we see in proportions from random samples is due to chance.[\/hidden-answer]<\/li>\r\n<\/ul>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Select a simple random sample from a finite population<\/li>\n<li>Understand and apply different sampling methods<\/li>\n<li>Determine and explain bias in a sampling method<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox recall\">\n<ul>\n<li>The <strong>population<\/strong> is an entire group of people, objects, or animals; usually a large group.<\/li>\n<li>A <strong>sample <\/strong>is a selected subset or subgroup of a population.<\/li>\n<li>For statistics, we want our sample to be representative of the population so we can make accurate inferences.<\/li>\n<\/ul>\n<\/section>\n<h2>Sampling Methods<\/h2>\n<p class=\"para\" style=\"margin: 6.0pt 0in 6.0pt 0in;\">There are five common sampling methods used in research. These sampling methods include:<\/p>\n<ol>\n<li class=\"para\">simple random sampling<\/li>\n<li class=\"para\">systematic sampling<\/li>\n<li class=\"para\">stratified sampling<\/li>\n<li class=\"para\">cluster sampling<\/li>\n<li class=\"para\">convenience sampling<\/li>\n<\/ol>\n<h2>Simple Random Samples<\/h2>\n<p>In <strong>simple random sampling<\/strong>, every sample of a given size has the same chance of being selected. This results in every individual or entity of the population having an equal chance of being selected. There are different ways to collect a simple random sample, such as using a random number generator, placing equal-sized slips of paper in a hat and drawing a sample, rolling a six-sided die, or any other event where each outcome is assigned a subject and each subject has an equally likely chance of being selected.<\/p>\n<section class=\"textbox example\">In the following figure, a random number generator selected numbers 24, 22, 27, 25, and 13, resulting in the highlighted individuals being selected for the sample.<\/p>\n<figure id=\"attachment_5723\" aria-describedby=\"caption-attachment-5723\" style=\"width: 878px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-5723\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/10\/03171045\/2.2.L-simple-random-sampling-1.png\" alt=\"Appropriate alternative text can be found in the description above.\" width=\"878\" height=\"357\" \/><figcaption id=\"caption-attachment-5723\" class=\"wp-caption-text\">Figure 1. A simple random sample selects individuals purely by chance, giving each person an equal opportunity to be included.<\/figcaption><\/figure>\n<\/section>\n<h3>Random Number Generator<\/h3>\n<section class=\"textbox interact\">\n<p>A simple random sample relies on a random mechanism to choose a sample, without replacement, from the population so that every sample is equally likely to be chosen. Consider choosing 10 states randomly from the 50 United States.<\/p>\n<p style=\"padding-left: 40px;\"><strong><br \/>\nSTEP 1:<\/strong> Under \u201cChoose Minimum,\u201d select \u201c1.\u201d<\/p>\n<p style=\"padding-left: 40px;\"><strong style=\"font-size: 1rem; text-align: initial; background-color: initial;\">STEP 2:<\/strong><span style=\"font-size: 1rem; text-align: initial; background-color: initial;\"> Under \u201cChoose Maximum,\u201d select \u201c50.\u201d<\/span><\/p>\n<p style=\"padding-left: 40px;\"><strong style=\"font-size: 1rem; text-align: initial; background-color: initial;\">STEP 3:<\/strong><span style=\"font-size: 1rem; text-align: initial; background-color: initial;\"> Under \u201cHow many numbers do you want to generate,\u201d select \u201c10.\u201d<\/span><\/p>\n<p style=\"padding-left: 40px;\"><strong style=\"font-size: 1rem; text-align: initial; background-color: initial;\">STEP 4:<\/strong><span style=\"font-size: 1rem; text-align: initial; background-color: initial;\"> Under \u201cSample with Replacement,\u201d select \u201cNo.\u201d<\/span><\/p>\n<p style=\"padding-left: 40px;\"><strong style=\"font-size: 1rem; text-align: initial; background-color: initial;\">STEP 5:<\/strong><span style=\"font-size: 1rem; text-align: initial; background-color: initial;\"> Click \u201cGenerate.\u201d This will generate ten random numbers between 1 and 50.<\/span><\/p>\n<p><strong><span style=\"font-size: 1rem; text-align: initial; background-color: initial;\">Save these 10 numbers for the next question.<\/span><\/strong><\/p>\n<\/section>\n<p>\n<iframe loading=\"lazy\" src=\"https:\/\/lumen-learning.shinyapps.io\/randomnumbers\/\" width=\"100%\" height=\"850\"><\/iframe> [<a href=\"https:\/\/lumen-learning.shinyapps.io\/randomnumbers\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm3162\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=3162&theme=lumen&iframe_resize_id=ohm3162&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox example\">\n<p>Consider a small college with only 200 students, where 60% of these students were eligible for financial aid.<\/p>\n<p>We can use this simplified situation to investigate how random samples relate to the population. This is the first step in creating a probability model that will be useful in inference. We know the <em data-start=\"577\" data-end=\"584\">truth<\/em> about the whole population (60% get aid). But in real life, we usually <strong data-start=\"656\" data-end=\"691\">don\u2019t know the whole population<\/strong>; we only see a <strong data-start=\"706\" data-end=\"717\">sample.<\/strong> So the question we want to ask is:<\/p>\n<p style=\"text-align: center;\"><em>How accurate are random samples at predicting this population proportion of 0.60?\u00a0<\/em><\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q580589\">Click to see why the proportion is written as 0.60.<\/button><\/p>\n<div id=\"q580589\" class=\"hidden-answer\" style=\"display: none\">In statistics, we usually write percentages as decimals, since it makes calculations easier. 60% means <em data-start=\"304\" data-end=\"320\">60 out of 100.<\/em><\/p>\n<p style=\"text-align: center;\">[latex]60\\% = \\frac{60}{100} =60\u00f7100= 0.60[\/latex]<\/div>\n<\/div>\n<p>To answer this question, we randomly select 8 students and determine the proportion who are eligible for financial aid. We repeat this process several times. Here are the results for 3 random samples:<\/p>\n<figure id=\"attachment_3593\" aria-describedby=\"caption-attachment-3593\" style=\"width: 781px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3593 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/10\/05174016\/2.1.L.Diagram2-1.png\" alt=\"Financial aid eligibility: 3 random samples of students consisting of 8 students each (out of a total population of 200 students). The proportion eligible for financial aid in the population is .60. In the random samples, each student is assigned a number and then categorized as elibigle for financial aid or not. The sample proportions are as follows: Sample 1 has 6 students eligible for aid and six divided by 8 is 0.75. Sample 2 has 5 students eligible for aid and five divided by 8 is 0.625. Sample 3 has 3 students eligible for aid and three divided by 8 is 0.375. When you average the sample proportions and round to the tens place you get a proportion of .60.\" width=\"781\" height=\"566\" \/><figcaption id=\"caption-attachment-3593\" class=\"wp-caption-text\">Figure 2. When taking more than one random sample from the same population, random sample proportions may vary.<\/figcaption><\/figure>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q257212\">More about these random samples.<\/button><\/p>\n<div id=\"q257212\" class=\"hidden-answer\" style=\"display: none\">\n<p>Notice the following about these random samples:<\/p>\n<ul>\n<li>Each random sample comes from the same population, where the true proportion is 0.60.<\/li>\n<li>But the sample proportions vary\u2014each group of 8 gives a different result.<\/li>\n<li>Some samples are above 0.60, some are below. Some are good estimates (like 0.625), while others are not (like 0.375).<\/li>\n<li>Small samples (like just 8 students) are <strong data-start=\"1557\" data-end=\"1576\">highly variable<\/strong>\u2014they can be inaccurate. That\u2019s why a sample proportion doesn\u2019t always match the population proportion exactly. Later, we investigate the effect of increasing the size of the sample.<\/li>\n<li>The variability we see in proportions from random samples is due to chance.<\/div>\n<\/div>\n<\/li>\n<\/ul>\n<\/section>\n","protected":false},"author":13,"menu_order":20,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":375,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/509"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/users\/13"}],"version-history":[{"count":39,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/509\/revisions"}],"predecessor-version":[{"id":7008,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/509\/revisions\/7008"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/375"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/509\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=509"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=509"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=509"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=509"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}