{"id":1128,"date":"2023-06-22T01:56:32","date_gmt":"2023-06-22T01:56:32","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/module-9-cheat-sheet\/"},"modified":"2025-02-11T00:03:10","modified_gmt":"2025-02-11T00:03:10","slug":"module-9-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/module-9-cheat-sheet\/","title":{"raw":"Module 8: Cheat Sheet","rendered":"Module 8: Cheat Sheet"},"content":{"raw":"<p style=\"text-align: right;\"><span style=\"font-size: 14pt;\"><a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Statistics+Exemplar\/Cheat+Sheets\/Module+8_+Cheat+Sheet.pdf\" target=\"_blank\" rel=\"noopener\">Download a PDF of this page here.<\/a><\/span><\/p>\r\n<h4 style=\"text-align: right;\"><a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Statistics+Exemplar\/Cheat+Sheets\/Spanish\/Mo%CC%81dulo+9_+Hoja+de+trucos+-+Espan%CC%83ol.pdf\" target=\"_blank\" rel=\"noopener\">Download the Spanish version here<\/a><\/h4>\r\n<h2>Essential Concepts<\/h2>\r\n<ul>\r\n\t<li style=\"font-weight: 400;\">One of the goals of statistical inference is to draw a conclusion about a population on the basis of a random sample from the population. Random samples vary, so we need to understand how much they vary and how they relate to the population. Our ultimate goal is to create a probability model that describes the long-run behavior of sample measurements. We use this model to make inferences about the population.<\/li>\r\n\t<li style=\"font-weight: 400;\">When we want to describe the characteristics of a sample, we call the values statistics. However, when we want to describe the characteristics of a population, we call those values parameters.<\/li>\r\n\t<li style=\"font-weight: 400;\">We can use mathematical theory to derive expressions for the mean and standard deviation of the sampling distribution of the sample proportion. When taking many random samples of size [latex]n[\/latex] from a population distribution with population proportion [latex]p[\/latex]:<\/li>\r\n<\/ul>\r\n<ul>\r\n\t<li style=\"list-style-type: none;\">\r\n<ul>\r\n\t<li style=\"font-weight: 400;\">The mean of the distribution of sample proportions is [latex]p[\/latex].<\/li>\r\n\t<li style=\"font-weight: 400;\">The standard deviation of the distribution of sample proportions is [latex]\\sqrt{\\frac{p(1-p)}{n}}[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<ul>\r\n\t<li style=\"font-weight: 400;\">In order to get a sense of the pattern of variation in sample proportions, we need to generate more than five samples. The distribution showing how sample proportions vary from sample to sample is called a sampling distribution of the sample proportion.<\/li>\r\n\t<li>The\u00a0Central Limit Theorem\u00a0states that, as the sample size gets larger, the distribution of the sample proportion will become closer to a normal distribution.<\/li>\r\n\t<li style=\"font-weight: 400;\">When taking many random samples of size [latex]n[\/latex] from a population distribution with population proportion [latex]p[\/latex]:<\/li>\r\n<\/ul>\r\n<ul>\r\n\t<li style=\"list-style-type: none;\">\r\n<ul>\r\n\t<li style=\"font-weight: 400;\">The mean of the distribution of sample proportions is [latex]p[\/latex].<\/li>\r\n\t<li style=\"font-weight: 400;\">The standard deviation of the distribution of sample proportions is [latex]\\sqrt{\\frac{p(1-p)}{n}}[\/latex].<\/li>\r\n\t<li style=\"font-weight: 400;\">If [latex]np\\geq 10[\/latex] and [latex]n(1-p) \\geq 10[\/latex], then the Central Limit Theorem states that the distribution of the sample proportions follows an approximate normal distribution with mean [latex]p[\/latex] and standard deviation [latex]\\sqrt{\\frac{p(1-p)}{n}}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li style=\"font-weight: 400;\">In practice, we do not know the population proportion, nor do we have the luxury of taking thousands of random samples. Instead, we observe a single random sample. In this case, we need to estimate the mean and standard deviation of the sample proportion:<\/li>\r\n<\/ul>\r\n<ul>\r\n\t<li style=\"list-style-type: none;\">\r\n<ul>\r\n\t<li style=\"font-weight: 400;\">The estimated mean of the distribution of sample proportions is [latex]\\hat{p}[\/latex].<\/li>\r\n\t<li style=\"font-weight: 400;\">To distinguish it from the true standard deviation of sample proportions, we call the estimated standard deviation of sample proportions the standard error of [latex]\\hat{p}[\/latex]:<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>If the desired standard deviation is known, then we can calculate the sample size needed by working backwards from the standard deviation formula.<\/li>\r\n<\/ul>\r\n<h2>Key Equations<\/h2>\r\n<p><strong>sample size<\/strong><\/p>\r\n<p style=\"padding-left: 40px;\">[latex]n = \\frac{p(1-p)}{(\\sigma_{\\hat{p}})^2}[\/latex]<\/p>\r\n<p><strong>standard error:<\/strong><\/p>\r\n<p style=\"padding-left: 40px;\">[latex]SE = \\sqrt{\\frac{\\hat{p}(1-\\hat{p})}{n}}[\/latex]<\/p>\r\n<p><strong>standard deviation<\/strong><\/p>\r\n<p style=\"padding-left: 40px;\">[latex]\\sigma_\\hat{p}=\\sqrt{\\frac{p(1-p)}{n}}[\/latex]<\/p>\r\n<h2>Glossary<\/h2>\r\n<p><strong>Central Limit Theorem<\/strong><\/p>\r\n<p style=\"padding-left: 40px;\">If [latex]np\\geq 10[\/latex] and [latex]n(1-p) \\geq 10[\/latex], then the Central Limit Theorem states that the distribution of the sample proportions follows an approximate normal distribution with mean p and standard deviation [latex]\\sqrt{\\frac{p(1-p)}{n}}[\/latex]<\/p>\r\n<p><strong>parameters<\/strong><\/p>\r\n<p style=\"padding-left: 40px;\">numbers that describe a population<\/p>\r\n<p><strong>population<\/strong><\/p>\r\n<p style=\"padding-left: 40px;\">the population is the entire collection of individuals or objects that you want to learn about<\/p>\r\n<p><strong>sample<\/strong><\/p>\r\n<p style=\"padding-left: 40px;\">a sample is a part of the population that is selected for study<\/p>\r\n<p><strong>sampling distribution of a statistic<\/strong>\u00a0<\/p>\r\n<p style=\"padding-left: 40px;\">a probability distribution that describes the long-term behavior of the statistic.<\/p>\r\n<p><strong>sampling distribution of a sample proportion<\/strong>\u00a0<\/p>\r\n<p style=\"padding-left: 40px;\">a probability distribution that describes how sample proportions vary from sample to sample<\/p>\r\n<p><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">standard deviation<\/strong><\/p>\r\n<p style=\"padding-left: 40px;\">a measure that describes the variability of a population<\/p>\r\n<p><strong>standard error<\/strong><\/p>\r\n<p style=\"padding-left: 40px;\">an estimate of the variability across the samples of a population<\/p>\r\n<p><strong>statistics<\/strong><\/p>\r\n<p style=\"padding-left: 40px;\">numbers that are calculated from a sample<\/p>","rendered":"<p style=\"text-align: right;\"><span style=\"font-size: 14pt;\"><a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Statistics+Exemplar\/Cheat+Sheets\/Module+8_+Cheat+Sheet.pdf\" target=\"_blank\" rel=\"noopener\">Download a PDF of this page here.<\/a><\/span><\/p>\n<h4 style=\"text-align: right;\"><a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Statistics+Exemplar\/Cheat+Sheets\/Spanish\/Mo%CC%81dulo+9_+Hoja+de+trucos+-+Espan%CC%83ol.pdf\" target=\"_blank\" rel=\"noopener\">Download the Spanish version here<\/a><\/h4>\n<h2>Essential Concepts<\/h2>\n<ul>\n<li style=\"font-weight: 400;\">One of the goals of statistical inference is to draw a conclusion about a population on the basis of a random sample from the population. Random samples vary, so we need to understand how much they vary and how they relate to the population. Our ultimate goal is to create a probability model that describes the long-run behavior of sample measurements. We use this model to make inferences about the population.<\/li>\n<li style=\"font-weight: 400;\">When we want to describe the characteristics of a sample, we call the values statistics. However, when we want to describe the characteristics of a population, we call those values parameters.<\/li>\n<li style=\"font-weight: 400;\">We can use mathematical theory to derive expressions for the mean and standard deviation of the sampling distribution of the sample proportion. When taking many random samples of size [latex]n[\/latex] from a population distribution with population proportion [latex]p[\/latex]:<\/li>\n<\/ul>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li style=\"font-weight: 400;\">The mean of the distribution of sample proportions is [latex]p[\/latex].<\/li>\n<li style=\"font-weight: 400;\">The standard deviation of the distribution of sample proportions is [latex]\\sqrt{\\frac{p(1-p)}{n}}[\/latex].<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<ul>\n<li style=\"font-weight: 400;\">In order to get a sense of the pattern of variation in sample proportions, we need to generate more than five samples. The distribution showing how sample proportions vary from sample to sample is called a sampling distribution of the sample proportion.<\/li>\n<li>The\u00a0Central Limit Theorem\u00a0states that, as the sample size gets larger, the distribution of the sample proportion will become closer to a normal distribution.<\/li>\n<li style=\"font-weight: 400;\">When taking many random samples of size [latex]n[\/latex] from a population distribution with population proportion [latex]p[\/latex]:<\/li>\n<\/ul>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li style=\"font-weight: 400;\">The mean of the distribution of sample proportions is [latex]p[\/latex].<\/li>\n<li style=\"font-weight: 400;\">The standard deviation of the distribution of sample proportions is [latex]\\sqrt{\\frac{p(1-p)}{n}}[\/latex].<\/li>\n<li style=\"font-weight: 400;\">If [latex]np\\geq 10[\/latex] and [latex]n(1-p) \\geq 10[\/latex], then the Central Limit Theorem states that the distribution of the sample proportions follows an approximate normal distribution with mean [latex]p[\/latex] and standard deviation [latex]\\sqrt{\\frac{p(1-p)}{n}}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li style=\"font-weight: 400;\">In practice, we do not know the population proportion, nor do we have the luxury of taking thousands of random samples. Instead, we observe a single random sample. In this case, we need to estimate the mean and standard deviation of the sample proportion:<\/li>\n<\/ul>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li style=\"font-weight: 400;\">The estimated mean of the distribution of sample proportions is [latex]\\hat{p}[\/latex].<\/li>\n<li style=\"font-weight: 400;\">To distinguish it from the true standard deviation of sample proportions, we call the estimated standard deviation of sample proportions the standard error of [latex]\\hat{p}[\/latex]:<\/li>\n<\/ul>\n<\/li>\n<li>If the desired standard deviation is known, then we can calculate the sample size needed by working backwards from the standard deviation formula.<\/li>\n<\/ul>\n<h2>Key Equations<\/h2>\n<p><strong>sample size<\/strong><\/p>\n<p style=\"padding-left: 40px;\">[latex]n = \\frac{p(1-p)}{(\\sigma_{\\hat{p}})^2}[\/latex]<\/p>\n<p><strong>standard error:<\/strong><\/p>\n<p style=\"padding-left: 40px;\">[latex]SE = \\sqrt{\\frac{\\hat{p}(1-\\hat{p})}{n}}[\/latex]<\/p>\n<p><strong>standard deviation<\/strong><\/p>\n<p style=\"padding-left: 40px;\">[latex]\\sigma_\\hat{p}=\\sqrt{\\frac{p(1-p)}{n}}[\/latex]<\/p>\n<h2>Glossary<\/h2>\n<p><strong>Central Limit Theorem<\/strong><\/p>\n<p style=\"padding-left: 40px;\">If [latex]np\\geq 10[\/latex] and [latex]n(1-p) \\geq 10[\/latex], then the Central Limit Theorem states that the distribution of the sample proportions follows an approximate normal distribution with mean p and standard deviation [latex]\\sqrt{\\frac{p(1-p)}{n}}[\/latex]<\/p>\n<p><strong>parameters<\/strong><\/p>\n<p style=\"padding-left: 40px;\">numbers that describe a population<\/p>\n<p><strong>population<\/strong><\/p>\n<p style=\"padding-left: 40px;\">the population is the entire collection of individuals or objects that you want to learn about<\/p>\n<p><strong>sample<\/strong><\/p>\n<p style=\"padding-left: 40px;\">a sample is a part of the population that is selected for study<\/p>\n<p><strong>sampling distribution of a statistic<\/strong>\u00a0<\/p>\n<p style=\"padding-left: 40px;\">a probability distribution that describes the long-term behavior of the statistic.<\/p>\n<p><strong>sampling distribution of a sample proportion<\/strong>\u00a0<\/p>\n<p style=\"padding-left: 40px;\">a probability distribution that describes how sample proportions vary from sample to sample<\/p>\n<p><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">standard deviation<\/strong><\/p>\n<p style=\"padding-left: 40px;\">a measure that describes the variability of a population<\/p>\n<p><strong>standard error<\/strong><\/p>\n<p style=\"padding-left: 40px;\">an estimate of the variability across the samples of a population<\/p>\n<p><strong>statistics<\/strong><\/p>\n<p style=\"padding-left: 40px;\">numbers that are calculated from a sample<\/p>\n","protected":false},"author":8,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":1126,"module-header":"cheat_sheet","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\/1128"}],"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\/8"}],"version-history":[{"count":14,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1128\/revisions"}],"predecessor-version":[{"id":6238,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1128\/revisions\/6238"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/1126"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1128\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1128"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1128"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1128"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1128"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}