{"id":1014,"date":"2023-06-22T01:45:03","date_gmt":"2023-06-22T01:45:03","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/module-8-cheat-sheet\/"},"modified":"2025-02-10T23:50:30","modified_gmt":"2025-02-10T23:50:30","slug":"module-8-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/module-8-cheat-sheet\/","title":{"raw":"Module 7: Cheat Sheet","rendered":"Module 7: 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+7_+Cheat+Sheet.pdf\" target=\"_blank\" rel=\"noopener\">Download a PDF of this page here.<\/a><\/span><\/p>\r\n<h2>Essential Concepts<\/h2>\r\n<ul>\r\n\t<li style=\"font-weight: 400;\">A probability model includes all possible outcomes of a chance experiment and the probabilities associated with those outcomes. A probability model is also known as a probability distribution. For a probability distribution to be valid:<br \/>\r\n<ul>\r\n\t<li style=\"font-weight: 400;\">The outcomes are random events.<\/li>\r\n\t<li style=\"font-weight: 400;\">All outcomes are assigned a probability. The probabilities are numbers between 0 and 1.<\/li>\r\n\t<li style=\"font-weight: 400;\">The sum of all of the probabilities is 1.<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>In a uniform probability model, each outcome has equal probability (for example, rolling a die where each face has a 1\/6 probability of occuring)<\/li>\r\n\t<li style=\"font-weight: 400;\">For a discrete random variable:\r\n\r\n<ul>\r\n\t<li style=\"font-weight: 400;\">The values associated with the random variable of interest are numerical and discrete.<\/li>\r\n\t<li style=\"font-weight: 400;\">All possible values of the random variable are listed in a table or graph with each value having an associated probability greater than or equal to 0 and less than or equal to 1.<\/li>\r\n\t<li style=\"font-weight: 400;\">The sum of all probabilities in the table or graph equals 1.<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li style=\"font-weight: 400;\">A continuous probability distribution is a probability distribution for a continuous random variable (an infinite and uncountable random variable). For all continuous random variables, the probability distribution can be approximated by a smooth curve called a probability density curve. The probabilities of a continuous probability distribution are represented as the area under a density curve.<\/li>\r\n\t<li>A normal distribution is a mathematical model with a smooth bell-shaped curve to describe the bell-shaped data distributions.<\/li>\r\n\t<li>A normal distribution has the following characteristics:\r\n\r\n<ul>\r\n\t<li>[latex]X[\/latex] is a continuous random variable.<\/li>\r\n\t<li>The mean is the center of the distribution which is symmetrical, the left side is a mirror image of the right side centered at the mean.<\/li>\r\n\t<li>Bell shaped: There is one peak (unimodal) at the mean.<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>The standard deviation of a normal distribution, [latex] \\sigma [\/latex], will change depending on how spread out or flat the curve appears.<\/li>\r\n\t<li>A normal distribution with a mean ([latex] \\mu [\/latex]) = 0 and a standard deviation ([latex] \\sigma [\/latex]) = 1 is called the standard normal distribution (or [latex]z[\/latex] distribution).<\/li>\r\n\t<li>To compare [latex]x[\/latex]-values from different distributions, we can standardize the values into a standard normal distribution by converting the [latex]x[\/latex]-values into their respective [latex]z[\/latex]-scores.<\/li>\r\n\t<li>When working with probability distributions, we often need to find the probability of values falling in certain regions of the distribution. There are three main types of probability calculations:\u00a0\r\n\r\n<ul>\r\n\t<li>Upper tail [latex]P(X&gt;a)[\/latex] finds the probability of the top percent, or events that exceed given outcome.<\/li>\r\n\t<li>Lower tail [latex]P(X&lt;a)[\/latex] finds the probability of the bottom percent, or events below the given outcome. Lower tail is also used to find the percentile.<\/li>\r\n\t<li>Interval find the probability of an outcome occurring between two events. The probability of an outcome between two values can be found as [latex]P(a&lt;X&lt;b)=P(X&lt;a)-P(X&lt;b)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<h2>Key Equations<\/h2>\r\n<p><strong>[latex]z[\/latex]-score<\/strong><\/p>\r\n<p style=\"text-align: left; padding-left: 30px;\">[latex]z=\\dfrac{x-\\mu}{\\sigma}[\/latex], where [latex]x[\/latex] represents the value of the observation, [latex]\\mu[\/latex] represents the population mean, [latex]\\sigma[\/latex] represents the population standard deviation, and [latex]z[\/latex] represents the standardized value, or z-score.<\/p>\r\n<h2>Glossary<\/h2>\r\n<p><strong>continuous<\/strong>\u00a0<strong>probability distribution<\/strong><\/p>\r\n<p style=\"padding-left: 40px;\">a probability distribution for a continuous random variable (an infinite and uncountable random variable)<\/p>\r\n<p><strong>discrete<\/strong>\u00a0<strong>probability distribution<\/strong><\/p>\r\n<p style=\"padding-left: 40px;\">a type of probability distribution that shows all possible values of a discrete random variable (countable or finite outcomes) along with the probabilities associated with those outcomes<\/p>\r\n<p><strong>interval probability <\/strong><\/p>\r\n<p style=\"padding-left: 40px;\">the probability of observing a value between two specified values<\/p>\r\n<p><strong>lower tail probability <\/strong><\/p>\r\n<p style=\"padding-left: 40px;\">the probability of observing a value less than a specified value; the area under a probability curve to the left of a specified point<\/p>\r\n<p><strong>normal distribution<\/strong><\/p>\r\n<p style=\"padding-left: 40px;\">a mathematical model with a smooth bell-shaped curve to describe the bell-shaped data distributions<\/p>\r\n<p><strong>probability density curve<\/strong><\/p>\r\n<p style=\"padding-left: 40px;\">the probability distribution that can be approximated by a smooth curve<\/p>\r\n<p><strong>probability model (probability distribution)<\/strong><\/p>\r\n<p style=\"padding-left: 40px;\">all possible outcomes of a chance experiment and the probabilities associated with those outcomes<\/p>\r\n<p><strong>standard normal distribution<\/strong><\/p>\r\n<p style=\"padding-left: 40px;\">a normal distribution with a mean ([latex] \\mu [\/latex]) = 0 and a standard deviation ([latex] \\sigma [\/latex]) = 1<\/p>\r\n<p><strong>uniform probability <\/strong><\/p>\r\n<p style=\"padding-left: 40px;\">model a probability model where all possible outcomes have equal probability of occurring (e.g., rolling a fair die where each number has probability 1\/6)<\/p>\r\n<p><strong>upper tail probability <\/strong><\/p>\r\n<p style=\"padding-left: 40px;\">the probability of observing a value greater than a specified value; the area under a probability curve to the right of a specified point<\/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+7_+Cheat+Sheet.pdf\" target=\"_blank\" rel=\"noopener\">Download a PDF of this page here.<\/a><\/span><\/p>\n<h2>Essential Concepts<\/h2>\n<ul>\n<li style=\"font-weight: 400;\">A probability model includes all possible outcomes of a chance experiment and the probabilities associated with those outcomes. A probability model is also known as a probability distribution. For a probability distribution to be valid:\n<ul>\n<li style=\"font-weight: 400;\">The outcomes are random events.<\/li>\n<li style=\"font-weight: 400;\">All outcomes are assigned a probability. The probabilities are numbers between 0 and 1.<\/li>\n<li style=\"font-weight: 400;\">The sum of all of the probabilities is 1.<\/li>\n<\/ul>\n<\/li>\n<li>In a uniform probability model, each outcome has equal probability (for example, rolling a die where each face has a 1\/6 probability of occuring)<\/li>\n<li style=\"font-weight: 400;\">For a discrete random variable:\n<ul>\n<li style=\"font-weight: 400;\">The values associated with the random variable of interest are numerical and discrete.<\/li>\n<li style=\"font-weight: 400;\">All possible values of the random variable are listed in a table or graph with each value having an associated probability greater than or equal to 0 and less than or equal to 1.<\/li>\n<li style=\"font-weight: 400;\">The sum of all probabilities in the table or graph equals 1.<\/li>\n<\/ul>\n<\/li>\n<li style=\"font-weight: 400;\">A continuous probability distribution is a probability distribution for a continuous random variable (an infinite and uncountable random variable). For all continuous random variables, the probability distribution can be approximated by a smooth curve called a probability density curve. The probabilities of a continuous probability distribution are represented as the area under a density curve.<\/li>\n<li>A normal distribution is a mathematical model with a smooth bell-shaped curve to describe the bell-shaped data distributions.<\/li>\n<li>A normal distribution has the following characteristics:\n<ul>\n<li>[latex]X[\/latex] is a continuous random variable.<\/li>\n<li>The mean is the center of the distribution which is symmetrical, the left side is a mirror image of the right side centered at the mean.<\/li>\n<li>Bell shaped: There is one peak (unimodal) at the mean.<\/li>\n<\/ul>\n<\/li>\n<li>The standard deviation of a normal distribution, [latex]\\sigma[\/latex], will change depending on how spread out or flat the curve appears.<\/li>\n<li>A normal distribution with a mean ([latex]\\mu[\/latex]) = 0 and a standard deviation ([latex]\\sigma[\/latex]) = 1 is called the standard normal distribution (or [latex]z[\/latex] distribution).<\/li>\n<li>To compare [latex]x[\/latex]-values from different distributions, we can standardize the values into a standard normal distribution by converting the [latex]x[\/latex]-values into their respective [latex]z[\/latex]-scores.<\/li>\n<li>When working with probability distributions, we often need to find the probability of values falling in certain regions of the distribution. There are three main types of probability calculations:\u00a0\n<ul>\n<li>Upper tail [latex]P(X>a)[\/latex] finds the probability of the top percent, or events that exceed given outcome.<\/li>\n<li>Lower tail [latex]P(X<a)[\/latex] finds the probability of the bottom percent, or events below the given outcome. Lower tail is also used to find the percentile.<\/li>\n<li>Interval find the probability of an outcome occurring between two events. The probability of an outcome between two values can be found as [latex]P(a<X<b)=P(X<a)-P(X<b)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h2>Key Equations<\/h2>\n<p><strong>[latex]z[\/latex]-score<\/strong><\/p>\n<p style=\"text-align: left; padding-left: 30px;\">[latex]z=\\dfrac{x-\\mu}{\\sigma}[\/latex], where [latex]x[\/latex] represents the value of the observation, [latex]\\mu[\/latex] represents the population mean, [latex]\\sigma[\/latex] represents the population standard deviation, and [latex]z[\/latex] represents the standardized value, or z-score.<\/p>\n<h2>Glossary<\/h2>\n<p><strong>continuous<\/strong>\u00a0<strong>probability distribution<\/strong><\/p>\n<p style=\"padding-left: 40px;\">a probability distribution for a continuous random variable (an infinite and uncountable random variable)<\/p>\n<p><strong>discrete<\/strong>\u00a0<strong>probability distribution<\/strong><\/p>\n<p style=\"padding-left: 40px;\">a type of probability distribution that shows all possible values of a discrete random variable (countable or finite outcomes) along with the probabilities associated with those outcomes<\/p>\n<p><strong>interval probability <\/strong><\/p>\n<p style=\"padding-left: 40px;\">the probability of observing a value between two specified values<\/p>\n<p><strong>lower tail probability <\/strong><\/p>\n<p style=\"padding-left: 40px;\">the probability of observing a value less than a specified value; the area under a probability curve to the left of a specified point<\/p>\n<p><strong>normal distribution<\/strong><\/p>\n<p style=\"padding-left: 40px;\">a mathematical model with a smooth bell-shaped curve to describe the bell-shaped data distributions<\/p>\n<p><strong>probability density curve<\/strong><\/p>\n<p style=\"padding-left: 40px;\">the probability distribution that can be approximated by a smooth curve<\/p>\n<p><strong>probability model (probability distribution)<\/strong><\/p>\n<p style=\"padding-left: 40px;\">all possible outcomes of a chance experiment and the probabilities associated with those outcomes<\/p>\n<p><strong>standard normal distribution<\/strong><\/p>\n<p style=\"padding-left: 40px;\">a normal distribution with a mean ([latex]\\mu[\/latex]) = 0 and a standard deviation ([latex]\\sigma[\/latex]) = 1<\/p>\n<p><strong>uniform probability <\/strong><\/p>\n<p style=\"padding-left: 40px;\">model a probability model where all possible outcomes have equal probability of occurring (e.g., rolling a fair die where each number has probability 1\/6)<\/p>\n<p><strong>upper tail probability <\/strong><\/p>\n<p style=\"padding-left: 40px;\">the probability of observing a value greater than a specified value; the area under a probability curve to the right of a specified point<\/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":3053,"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\/1014"}],"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":22,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1014\/revisions"}],"predecessor-version":[{"id":6235,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1014\/revisions\/6235"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/3053"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1014\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1014"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1014"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1014"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1014"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}