{"id":5329,"date":"2022-08-19T18:08:21","date_gmt":"2022-08-19T18:08:21","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/?post_type=chapter&#038;p=5329"},"modified":"2022-08-19T18:15:23","modified_gmt":"2022-08-19T18:15:23","slug":"12a-preview","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/12a-preview\/","title":{"raw":"12A Preview","rendered":"12A Preview"},"content":{"raw":"<strong>Preparing for the next class\u00a0<\/strong>\r\n\r\nIn the next in-class activity, you will need to use the DCMP Normal Distribution tool to\u00a0 calculate normal probabilities, simulate random samples from a population using the\u00a0 DCMP Sampling Distribution of the Sample Mean (Continuous Population) tool, find the\u00a0 mean and standard deviation of the sampling distribution of the sample mean, and use\u00a0 the mean and standard deviation of the sampling distribution of the sample mean to\u00a0 calculate and interpret a z-score for a sample mean.\r\n\r\nThe SAT is an assessment designed to evaluate a student\u2019s college-specific skills. SAT\u00a0 scores tend to follow an approximate normal distribution with mean of 1060 and a standard deviation of 195.[footnote]Institute of Education Sciences, National Center for Education Statistics. (2019). Digest of education\u00a0 statistics 2019 tables and figures: Table 226.40.\u00a0https:\/\/nces.ed.gov\/programs\/digest\/d19\/tables\/dt19_226.40.asp[\/footnote]\r\n\r\nGo to the DCMP Normal Distribution tool at\u00a0https:\/\/dcmathpathways.shinyapps.io\/NormalDist\/.\r\n<ul>\r\n \t<li>Select the Find Probability tab.<\/li>\r\n \t<li>Enter 1060 for the mean, [latex]\\mu[\/latex].<\/li>\r\n \t<li>Enter 195 for the standard deviation, [latex]\\sigma[\/latex].<\/li>\r\n<\/ul>\r\nThe normal distribution displayed now represents a model for the distribution of the SAT\u00a0 scores for all students who take the exam.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 1<\/h3>\r\n1) Use the tool to find the following probabilities.\r\n\r\na) What is the probability that a randomly selected student scores higher than a 1200 on the SAT?\r\n\r\nb) What proportion of students scores between 865 and 1255 on the SAT? Hint: Your answer should be consistent with the Empirical Rule.\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 2<\/h3>\r\n2) Instead of looking at the distribution of the SAT scores for individual students, you\u00a0 are interested in the distribution of the average SAT scores across classrooms.\u00a0 Suppose that the class size at a large high school is 10 students per class.\r\n\r\na) If you were to calculate the probability that the average SAT score for a\u00a0 randomly selected classroom is larger than 1200, would you predict this\u00a0 probability to be smaller, larger, or approximately the same as your answer\u00a0 to Question 1, Part A?\r\n\r\na) Smaller\r\n\r\nb) Larger\r\n\r\nc) Approximately the same\r\n\r\nb) If you were to calculate the proportion of classrooms with average SAT\u00a0 scores between 865 and 1255, would you predict this probability to be\u00a0 smaller, larger, or approximately the same as your answer to Question 1, Part B?\r\n\r\na) Smaller\r\n\r\nb) Larger\r\n\r\nc) Approximately the same\r\n\r\n<\/div>\r\nLet\u2019s use simulation to check your predictions from Question 2. Go to the DCMP\u00a0 Sampling Distribution of the Sample Mean (Continuous Population) tool at\u00a0 https:\/\/dcmathpathways.shinyapps.io\/SampDist_cont\/.\r\n<ul>\r\n \t<li>Under \u201cSelect Population Distribution,\u201d choose \u201cBell-Shaped.\u201d<\/li>\r\n \t<li>Set the population mean to 1060 and the population standard deviation to 195.\u00a0 (You will need to select the \u201cEnter values for \u00b5 and \u03c3\u201d option.)<\/li>\r\n<\/ul>\r\nThe population distribution displayed now represents a model for the distribution of the individual SAT scores for all students who take the exam.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 3<\/h3>\r\n3) Generate 1,000 samples of size 10 from this population. The simulated distribution shown at the bottom is the sampling distribution of the sample mean\u2014in this case, the distribution of the average SAT scores for your 1,000 randomly generated\u00a0 classrooms of 10 students. Select the \u201cFind Probability\u201d box and answer the following questions.\r\n\r\na) What proportion of your simulated classrooms have an average SAT score greater than 1200? Does this match your prediction from Question 2, Part A?\r\n\r\nHint: The data analysis tool will report the proportion of simulations resulting in a sample mean less than or equal to 1200. You will need to subtract that proportion\u00a0 from 1 to get your answer.\r\n\r\nb) What proportion of your simulated classrooms have an average SAT score between 865 and 1255? Does this match your prediction from Question 2, Part B?\r\n\r\nHint: You will need to find the proportion of simulations at or below 865 and then subtract that value from the proportion of simulations at or below 1255.\r\n\r\nc) Select the \u201cShow Normal Approximation\u201d box. This overlays a normal distribution with the same mean and standard deviation as the distribution of the simulated average SAT scores. What are the mean and standard deviation of your simulated average SAT scores? Does the distribution of the simulated average SAT scores seem to follow the normal curve?\r\n\r\n<\/div>\r\nWhen sampling from a normal population such as SAT scores, the distribution of the sample means will also have a normal distribution with the same mean; but, the variability in sample means will be less than the variability in individuals (similar to how\u00a0 variability in sample proportions will be less than the variability in individuals). There are\u00a0 mathematical formulas we can use to find the mean and standard deviation of the\u00a0 sampling distribution of the sample mean for samples of size [latex]n[\/latex]:\r\n\r\nMean of the sampling distribution of the sample mean [latex]=\\mu[\/latex]\r\n\r\nStandard deviation of the sampling distribution of the sample mean [latex]\\frac{\\sigma}{\\sqrt{n}}[\/latex]\r\n\r\nIn the formulas above, [latex]\\mu[\/latex] and [latex]\\sigma[\/latex] represent the mean and standard deviation of the original population, respectively. For the population distribution of the individual SAT scores, [latex]\\mu = 1060[\/latex] and [latex]\\sigma = 195[\/latex].\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 4<\/h3>\r\n4) Use the formulas above to calculate the mean and standard deviation of the sample\u00a0 mean SAT scores for samples of size [latex]n=10[\/latex]. Round to the nearest thousandth.\r\n\r\nHint: These values should be close to the simulated mean and standard deviation you found in Question 3, Part C.\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 5<\/h3>\r\n5) Go back to the DCMP Normal Distribution tool at\u00a0https:\/\/dcmathpathways.shinyapps.io\/NormalDist\/.\r\n<ul>\r\n \t<li>Select the Find Probability tab.<\/li>\r\n \t<li>Enter the mean you found in Question 4.<\/li>\r\n \t<li>Enter the standard deviation you found in Question 4.<\/li>\r\n<\/ul>\r\nThe normal distribution displayed now represents a model for the distribution of the mean SAT scores for all classrooms of 10 students who take the exam. Use the tool to find the following probabilities.\r\n\r\nPart A: What is the probability that the mean SAT score for a randomly selected\u00a0 classroom of 10 students is higher than 1200? Is this value similar to your\u00a0 answer to Question 3, Part A?\r\n\r\nPart B: What proportion of classrooms of 10 students have an average SAT score\u00a0 between 865 and 1255? Is this value similar to your answer to Question 3, Part B?\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 6<\/h3>\r\n6) Suppose one classroom\u2019s average SAT score is 950. Use the mean and standard\u00a0 deviation you found in Question 4 to calculate the z-score for a sample mean SAT\u00a0 score of 950. Write a sentence interpreting this value in context of the problem.<\/div>","rendered":"<p><strong>Preparing for the next class\u00a0<\/strong><\/p>\n<p>In the next in-class activity, you will need to use the DCMP Normal Distribution tool to\u00a0 calculate normal probabilities, simulate random samples from a population using the\u00a0 DCMP Sampling Distribution of the Sample Mean (Continuous Population) tool, find the\u00a0 mean and standard deviation of the sampling distribution of the sample mean, and use\u00a0 the mean and standard deviation of the sampling distribution of the sample mean to\u00a0 calculate and interpret a z-score for a sample mean.<\/p>\n<p>The SAT is an assessment designed to evaluate a student\u2019s college-specific skills. SAT\u00a0 scores tend to follow an approximate normal distribution with mean of 1060 and a standard deviation of 195.<a class=\"footnote\" title=\"Institute of Education Sciences, National Center for Education Statistics. (2019). Digest of education\u00a0 statistics 2019 tables and figures: Table 226.40.\u00a0https:\/\/nces.ed.gov\/programs\/digest\/d19\/tables\/dt19_226.40.asp\" id=\"return-footnote-5329-1\" href=\"#footnote-5329-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a><\/p>\n<p>Go to the DCMP Normal Distribution tool at\u00a0https:\/\/dcmathpathways.shinyapps.io\/NormalDist\/.<\/p>\n<ul>\n<li>Select the Find Probability tab.<\/li>\n<li>Enter 1060 for the mean, [latex]\\mu[\/latex].<\/li>\n<li>Enter 195 for the standard deviation, [latex]\\sigma[\/latex].<\/li>\n<\/ul>\n<p>The normal distribution displayed now represents a model for the distribution of the SAT\u00a0 scores for all students who take the exam.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Question 1<\/h3>\n<p>1) Use the tool to find the following probabilities.<\/p>\n<p>a) What is the probability that a randomly selected student scores higher than a 1200 on the SAT?<\/p>\n<p>b) What proportion of students scores between 865 and 1255 on the SAT? Hint: Your answer should be consistent with the Empirical Rule.<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Question 2<\/h3>\n<p>2) Instead of looking at the distribution of the SAT scores for individual students, you\u00a0 are interested in the distribution of the average SAT scores across classrooms.\u00a0 Suppose that the class size at a large high school is 10 students per class.<\/p>\n<p>a) If you were to calculate the probability that the average SAT score for a\u00a0 randomly selected classroom is larger than 1200, would you predict this\u00a0 probability to be smaller, larger, or approximately the same as your answer\u00a0 to Question 1, Part A?<\/p>\n<p>a) Smaller<\/p>\n<p>b) Larger<\/p>\n<p>c) Approximately the same<\/p>\n<p>b) If you were to calculate the proportion of classrooms with average SAT\u00a0 scores between 865 and 1255, would you predict this probability to be\u00a0 smaller, larger, or approximately the same as your answer to Question 1, Part B?<\/p>\n<p>a) Smaller<\/p>\n<p>b) Larger<\/p>\n<p>c) Approximately the same<\/p>\n<\/div>\n<p>Let\u2019s use simulation to check your predictions from Question 2. Go to the DCMP\u00a0 Sampling Distribution of the Sample Mean (Continuous Population) tool at\u00a0 https:\/\/dcmathpathways.shinyapps.io\/SampDist_cont\/.<\/p>\n<ul>\n<li>Under \u201cSelect Population Distribution,\u201d choose \u201cBell-Shaped.\u201d<\/li>\n<li>Set the population mean to 1060 and the population standard deviation to 195.\u00a0 (You will need to select the \u201cEnter values for \u00b5 and \u03c3\u201d option.)<\/li>\n<\/ul>\n<p>The population distribution displayed now represents a model for the distribution of the individual SAT scores for all students who take the exam.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Question 3<\/h3>\n<p>3) Generate 1,000 samples of size 10 from this population. The simulated distribution shown at the bottom is the sampling distribution of the sample mean\u2014in this case, the distribution of the average SAT scores for your 1,000 randomly generated\u00a0 classrooms of 10 students. Select the \u201cFind Probability\u201d box and answer the following questions.<\/p>\n<p>a) What proportion of your simulated classrooms have an average SAT score greater than 1200? Does this match your prediction from Question 2, Part A?<\/p>\n<p>Hint: The data analysis tool will report the proportion of simulations resulting in a sample mean less than or equal to 1200. You will need to subtract that proportion\u00a0 from 1 to get your answer.<\/p>\n<p>b) What proportion of your simulated classrooms have an average SAT score between 865 and 1255? Does this match your prediction from Question 2, Part B?<\/p>\n<p>Hint: You will need to find the proportion of simulations at or below 865 and then subtract that value from the proportion of simulations at or below 1255.<\/p>\n<p>c) Select the \u201cShow Normal Approximation\u201d box. This overlays a normal distribution with the same mean and standard deviation as the distribution of the simulated average SAT scores. What are the mean and standard deviation of your simulated average SAT scores? Does the distribution of the simulated average SAT scores seem to follow the normal curve?<\/p>\n<\/div>\n<p>When sampling from a normal population such as SAT scores, the distribution of the sample means will also have a normal distribution with the same mean; but, the variability in sample means will be less than the variability in individuals (similar to how\u00a0 variability in sample proportions will be less than the variability in individuals). There are\u00a0 mathematical formulas we can use to find the mean and standard deviation of the\u00a0 sampling distribution of the sample mean for samples of size [latex]n[\/latex]:<\/p>\n<p>Mean of the sampling distribution of the sample mean [latex]=\\mu[\/latex]<\/p>\n<p>Standard deviation of the sampling distribution of the sample mean [latex]\\frac{\\sigma}{\\sqrt{n}}[\/latex]<\/p>\n<p>In the formulas above, [latex]\\mu[\/latex] and [latex]\\sigma[\/latex] represent the mean and standard deviation of the original population, respectively. For the population distribution of the individual SAT scores, [latex]\\mu = 1060[\/latex] and [latex]\\sigma = 195[\/latex].<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Question 4<\/h3>\n<p>4) Use the formulas above to calculate the mean and standard deviation of the sample\u00a0 mean SAT scores for samples of size [latex]n=10[\/latex]. Round to the nearest thousandth.<\/p>\n<p>Hint: These values should be close to the simulated mean and standard deviation you found in Question 3, Part C.<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Question 5<\/h3>\n<p>5) Go back to the DCMP Normal Distribution tool at\u00a0https:\/\/dcmathpathways.shinyapps.io\/NormalDist\/.<\/p>\n<ul>\n<li>Select the Find Probability tab.<\/li>\n<li>Enter the mean you found in Question 4.<\/li>\n<li>Enter the standard deviation you found in Question 4.<\/li>\n<\/ul>\n<p>The normal distribution displayed now represents a model for the distribution of the mean SAT scores for all classrooms of 10 students who take the exam. Use the tool to find the following probabilities.<\/p>\n<p>Part A: What is the probability that the mean SAT score for a randomly selected\u00a0 classroom of 10 students is higher than 1200? Is this value similar to your\u00a0 answer to Question 3, Part A?<\/p>\n<p>Part B: What proportion of classrooms of 10 students have an average SAT score\u00a0 between 865 and 1255? Is this value similar to your answer to Question 3, Part B?<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Question 6<\/h3>\n<p>6) Suppose one classroom\u2019s average SAT score is 950. Use the mean and standard\u00a0 deviation you found in Question 4 to calculate the z-score for a sample mean SAT\u00a0 score of 950. Write a sentence interpreting this value in context of the problem.<\/p><\/div>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-5329-1\">Institute of Education Sciences, National Center for Education Statistics. (2019). Digest of education\u00a0 statistics 2019 tables and figures: Table 226.40.\u00a0https:\/\/nces.ed.gov\/programs\/digest\/d19\/tables\/dt19_226.40.asp <a href=\"#return-footnote-5329-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":23592,"menu_order":3,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-5329","chapter","type-chapter","status-publish","hentry"],"part":5315,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5329","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/users\/23592"}],"version-history":[{"count":3,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5329\/revisions"}],"predecessor-version":[{"id":5332,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5329\/revisions\/5332"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/parts\/5315"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5329\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=5329"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=5329"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=5329"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=5329"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}