{"id":3918,"date":"2023-06-02T16:36:06","date_gmt":"2023-06-02T16:36:06","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=3918"},"modified":"2024-10-18T20:57:12","modified_gmt":"2024-10-18T20:57:12","slug":"modeling-cheat-sheet","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/modeling-cheat-sheet\/","title":{"raw":"Introduction to Modeling: Cheat Sheet","rendered":"Introduction to Modeling: Cheat Sheet"},"content":{"raw":"<h2>Essential Concepts<\/h2>\r\n<ul>\r\n\t<li>Linear growth involves adding a constant amount to the previous value in a sequence.<\/li>\r\n\t<li>In linear growth, the next value in a sequence [latex](P_n)[\/latex] is found by adding a constant difference to the previous value [latex](P_{n-1})[\/latex]. This is known as the recursive formula for linear growth: [latex]P_n =P_{n\u22121}+d[\/latex]<\/li>\r\n\t<li>The explicit formula for linear growth is [latex]P_n=P_0+dn[\/latex], where [latex]P_0[\/latex] is the initial value, [latex]d[\/latex] is the constant difference, and [latex]n[\/latex] is the number of time periods.<\/li>\r\n\t<li>Use the explicit formula to directly calculate future values in a linear growth scenario.<\/li>\r\n\t<li>Exponential growth occurs when a quantity increases by a fixed percentage of the whole in each time period.<\/li>\r\n\t<li>The recursive formula for exponential growth is [latex]P_n = (1+r)P_{n-1}[\/latex], and the explicit formula is [latex]P_n = (1+r)^{n}P_0[\/latex] or equivalently, [latex]P_n= P_0(1+r)^{n}[\/latex], where [latex]P_0[\/latex] is the initial amount, [latex]r[\/latex] is the growth rate, and [latex]n[\/latex] is the number of time periods.<\/li>\r\n\t<li>Using the explicit formula to quickly calculate future values in an exponential growth scenario.<\/li>\r\n\t<li>The term [latex](1+r)[\/latex] is the growth multiplier, where [latex]r[\/latex] is the growth rate.<\/li>\r\n\t<li>Mathematical models are tools that represent real-world situations using mathematical language and symbols to simplify, explain, and predict phenomena. These models can be physical, conceptual, or mathematical, each serving a unique purpose in interpreting and understanding real-world scenarios.<\/li>\r\n\t<li>The process of modeling involves identifying variables, collecting data, and formulating a model that captures the relationship between variables.<\/li>\r\n\t<li>Models are not perfect representations and should be used with an understanding of their limitations and assumptions.<\/li>\r\n\t<li>The independent variable<em>,\u00a0<\/em>representing the input quantity is graphed on the horizontal axis.<\/li>\r\n\t<li>The\u00a0dependent variable, representing the\u00a0output quantity\u00a0is graphed on the vertical axis.<\/li>\r\n\t<li>Scatterplots are graphical representations of data points in two dimensions, typically used to observe relationships between variables.<\/li>\r\n\t<li>Creating a scatterplot involves plotting data points on a graph, where each point represents a pair of values from the dataset.<\/li>\r\n\t<li>Spreadsheet software like Microsoft Excel can be used to create scatterplots and analyze data efficiently.<\/li>\r\n\t<li>Scatterplots help in visually identifying trends, patterns, or correlations in the data, which are crucial for developing mathematical models.<\/li>\r\n\t<li>Trendlines in scatterplots represent the best fit for the given data, often used to identify the underlying trend in the dataset.<\/li>\r\n\t<li>Adjusting data inputs, like changing years to the number of years since a starting point, can significantly affect the model's accuracy and relevance.<\/li>\r\n\t<li>Linear growth occurs when the output quantity changes by a constant amount per unit of input, resulting in a straight-line pattern in a graph.<\/li>\r\n\t<li>A linear model can be created to describe real-world situations of linear growth or decay, using the slope-intercept form of a linear equation, [latex]y=mx+b[\/latex].\u00a0<\/li>\r\n\t<li>The slope ([latex]m[\/latex]) in the linear equation represents the rate of change, and the y-intercept ([latex]b[\/latex]) represents the starting value.<\/li>\r\n\t<li>Spreadsheet software can be used to build models for making predictions in scenarios of linear growth, enhancing efficiency and accuracy.<\/li>\r\n\t<li>Linear regression involves finding the best-fitting line for a set of data, often using technology to ensure accuracy.<\/li>\r\n\t<li>The process of least squares regression is used to determine this line by minimizing the distances between the data points and the line itself.<\/li>\r\n\t<li>The correlation coefficient, denoted as [latex]r[\/latex], is a numerical measure between [latex]-1[\/latex] and [latex]1[\/latex] that indicates the strength and direction of the relationship between variables. A positive correlation coefficient, [latex]r \\gt 0 [\/latex], would indicate a positive slope while a negative correlation coefficient [latex]r \\lt 0[\/latex], would indicate a negative slope.\u00a0<\/li>\r\n\t<li>The coefficient of determination, known as [latex]R^2[\/latex], quantifies how well the regression line represents the data.<\/li>\r\n\t<li>[latex]R^2[\/latex] is equivalent to the square of the correlation coefficient\u00a0[latex]r[\/latex] and will always be a positive number between [latex]0\\%[\/latex] and\u00a0[latex]100\\%[\/latex].<\/li>\r\n\t<li>[latex]R^2[\/latex] should be interpreted and written as a percentage.<\/li>\r\n\t<li>In scatterplots, a strong linear relationship is shown by data points lying close to a line, while a weaker relationship is indicated by more spread out data points.<\/li>\r\n\t<li>Extrapolating involves making predictions outside the known data set, a practice that should be approached with caution due to the potential for unreliable predictions.<\/li>\r\n\t<li>Interpolating refers to making predictions within the known data, offering a safer and more reliable approach to modeling and forecasting.<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<p><strong>coefficient of determination<\/strong>, <strong>[latex]R^2[\/latex] or<\/strong> <strong>[latex]r^2[\/latex]<\/strong><\/p>\r\n<p>a measure of the proportion of the variation of a response variable in linearly related bivariate data that can be explained by its relationship with the explanatory variable that ranges from [latex]0[\/latex] and [latex]1[\/latex]<\/p>\r\n<p><span style=\"color: #000000;\"><strong>common difference<\/strong><\/span><\/p>\r\n<p>the amount that the population changes each time [latex]n[\/latex] increases by [latex]1[\/latex]<\/p>\r\n<p><strong>correlation<\/strong> <strong>c<\/strong><strong>oefficient, [latex]r[\/latex]<\/strong><\/p>\r\n<p>a value between [latex]-1[\/latex] and [latex]1[\/latex] that is returned by the least squares method and measures the correlation between the input and output variables of a model<\/p>\r\n<p><strong>linear regression<\/strong><\/p>\r\n<p>the process of finding the equation of the line that best \"fits\" the data<\/p>\r\n<p><span style=\"color: #000000;\"><strong>recursive relationship<\/strong><\/span><\/p>\r\n<p>a formula that relates the next value in a sequence to the previous values<\/p>\r\n<h2>Key Equations<\/h2>\r\n<p><strong>explicit form of exponential growth<\/strong><\/p>\r\n<p>[latex]P_n= P_0(1+r)^{n}[\/latex]<\/p>\r\n<p><span style=\"color: #000000;\"><strong>explicit form of linear growth<\/strong><\/span><\/p>\r\n<p>[latex]P_n = P_0 + d n[\/latex]<\/p>\r\n<p><strong>recursive form of exponential growth<\/strong><\/p>\r\n<p>[latex]P_n = (1+r)P_{n-1}[\/latex]<\/p>\r\n<p><span style=\"color: #000000;\"><strong>recursive form of linear growth<\/strong><\/span><\/p>\r\n<p>[latex]P_n = P_{n-1} + d[\/latex]<\/p>","rendered":"<h2>Essential Concepts<\/h2>\n<ul>\n<li>Linear growth involves adding a constant amount to the previous value in a sequence.<\/li>\n<li>In linear growth, the next value in a sequence [latex](P_n)[\/latex] is found by adding a constant difference to the previous value [latex](P_{n-1})[\/latex]. This is known as the recursive formula for linear growth: [latex]P_n =P_{n\u22121}+d[\/latex]<\/li>\n<li>The explicit formula for linear growth is [latex]P_n=P_0+dn[\/latex], where [latex]P_0[\/latex] is the initial value, [latex]d[\/latex] is the constant difference, and [latex]n[\/latex] is the number of time periods.<\/li>\n<li>Use the explicit formula to directly calculate future values in a linear growth scenario.<\/li>\n<li>Exponential growth occurs when a quantity increases by a fixed percentage of the whole in each time period.<\/li>\n<li>The recursive formula for exponential growth is [latex]P_n = (1+r)P_{n-1}[\/latex], and the explicit formula is [latex]P_n = (1+r)^{n}P_0[\/latex] or equivalently, [latex]P_n= P_0(1+r)^{n}[\/latex], where [latex]P_0[\/latex] is the initial amount, [latex]r[\/latex] is the growth rate, and [latex]n[\/latex] is the number of time periods.<\/li>\n<li>Using the explicit formula to quickly calculate future values in an exponential growth scenario.<\/li>\n<li>The term [latex](1+r)[\/latex] is the growth multiplier, where [latex]r[\/latex] is the growth rate.<\/li>\n<li>Mathematical models are tools that represent real-world situations using mathematical language and symbols to simplify, explain, and predict phenomena. These models can be physical, conceptual, or mathematical, each serving a unique purpose in interpreting and understanding real-world scenarios.<\/li>\n<li>The process of modeling involves identifying variables, collecting data, and formulating a model that captures the relationship between variables.<\/li>\n<li>Models are not perfect representations and should be used with an understanding of their limitations and assumptions.<\/li>\n<li>The independent variable<em>,\u00a0<\/em>representing the input quantity is graphed on the horizontal axis.<\/li>\n<li>The\u00a0dependent variable, representing the\u00a0output quantity\u00a0is graphed on the vertical axis.<\/li>\n<li>Scatterplots are graphical representations of data points in two dimensions, typically used to observe relationships between variables.<\/li>\n<li>Creating a scatterplot involves plotting data points on a graph, where each point represents a pair of values from the dataset.<\/li>\n<li>Spreadsheet software like Microsoft Excel can be used to create scatterplots and analyze data efficiently.<\/li>\n<li>Scatterplots help in visually identifying trends, patterns, or correlations in the data, which are crucial for developing mathematical models.<\/li>\n<li>Trendlines in scatterplots represent the best fit for the given data, often used to identify the underlying trend in the dataset.<\/li>\n<li>Adjusting data inputs, like changing years to the number of years since a starting point, can significantly affect the model&#8217;s accuracy and relevance.<\/li>\n<li>Linear growth occurs when the output quantity changes by a constant amount per unit of input, resulting in a straight-line pattern in a graph.<\/li>\n<li>A linear model can be created to describe real-world situations of linear growth or decay, using the slope-intercept form of a linear equation, [latex]y=mx+b[\/latex].\u00a0<\/li>\n<li>The slope ([latex]m[\/latex]) in the linear equation represents the rate of change, and the y-intercept ([latex]b[\/latex]) represents the starting value.<\/li>\n<li>Spreadsheet software can be used to build models for making predictions in scenarios of linear growth, enhancing efficiency and accuracy.<\/li>\n<li>Linear regression involves finding the best-fitting line for a set of data, often using technology to ensure accuracy.<\/li>\n<li>The process of least squares regression is used to determine this line by minimizing the distances between the data points and the line itself.<\/li>\n<li>The correlation coefficient, denoted as [latex]r[\/latex], is a numerical measure between [latex]-1[\/latex] and [latex]1[\/latex] that indicates the strength and direction of the relationship between variables. A positive correlation coefficient, [latex]r \\gt 0[\/latex], would indicate a positive slope while a negative correlation coefficient [latex]r \\lt 0[\/latex], would indicate a negative slope.\u00a0<\/li>\n<li>The coefficient of determination, known as [latex]R^2[\/latex], quantifies how well the regression line represents the data.<\/li>\n<li>[latex]R^2[\/latex] is equivalent to the square of the correlation coefficient\u00a0[latex]r[\/latex] and will always be a positive number between [latex]0\\%[\/latex] and\u00a0[latex]100\\%[\/latex].<\/li>\n<li>[latex]R^2[\/latex] should be interpreted and written as a percentage.<\/li>\n<li>In scatterplots, a strong linear relationship is shown by data points lying close to a line, while a weaker relationship is indicated by more spread out data points.<\/li>\n<li>Extrapolating involves making predictions outside the known data set, a practice that should be approached with caution due to the potential for unreliable predictions.<\/li>\n<li>Interpolating refers to making predictions within the known data, offering a safer and more reliable approach to modeling and forecasting.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<p><strong>coefficient of determination<\/strong>, <strong>[latex]R^2[\/latex] or<\/strong> <strong>[latex]r^2[\/latex]<\/strong><\/p>\n<p>a measure of the proportion of the variation of a response variable in linearly related bivariate data that can be explained by its relationship with the explanatory variable that ranges from [latex]0[\/latex] and [latex]1[\/latex]<\/p>\n<p><span style=\"color: #000000;\"><strong>common difference<\/strong><\/span><\/p>\n<p>the amount that the population changes each time [latex]n[\/latex] increases by [latex]1[\/latex]<\/p>\n<p><strong>correlation<\/strong> <strong>c<\/strong><strong>oefficient, [latex]r[\/latex]<\/strong><\/p>\n<p>a value between [latex]-1[\/latex] and [latex]1[\/latex] that is returned by the least squares method and measures the correlation between the input and output variables of a model<\/p>\n<p><strong>linear regression<\/strong><\/p>\n<p>the process of finding the equation of the line that best &#8220;fits&#8221; the data<\/p>\n<p><span style=\"color: #000000;\"><strong>recursive relationship<\/strong><\/span><\/p>\n<p>a formula that relates the next value in a sequence to the previous values<\/p>\n<h2>Key Equations<\/h2>\n<p><strong>explicit form of exponential growth<\/strong><\/p>\n<p>[latex]P_n= P_0(1+r)^{n}[\/latex]<\/p>\n<p><span style=\"color: #000000;\"><strong>explicit form of linear growth<\/strong><\/span><\/p>\n<p>[latex]P_n = P_0 + d n[\/latex]<\/p>\n<p><strong>recursive form of exponential growth<\/strong><\/p>\n<p>[latex]P_n = (1+r)P_{n-1}[\/latex]<\/p>\n<p><span style=\"color: #000000;\"><strong>recursive form of linear growth<\/strong><\/span><\/p>\n<p>[latex]P_n = P_{n-1} + d[\/latex]<\/p>\n","protected":false},"author":23,"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":87,"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\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/3918"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/23"}],"version-history":[{"count":12,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/3918\/revisions"}],"predecessor-version":[{"id":11837,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/3918\/revisions\/11837"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/87"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/3918\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=3918"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=3918"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=3918"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=3918"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}