{"id":5418,"date":"2022-08-22T20:27:50","date_gmt":"2022-08-22T20:27:50","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/?post_type=chapter&#038;p=5418"},"modified":"2022-08-22T20:41:39","modified_gmt":"2022-08-22T20:41:39","slug":"12b-preview","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/12b-preview\/","title":{"raw":"12B Preview","rendered":"12B Preview"},"content":{"raw":"Preparing for the next class\r\n\r\nIn the next in-class activity, you will need to calculate and interpret a standardized\u00a0 sample mean, calculate and interpret the standard deviation of a sample mean, and\u00a0 calculate probabilities involving standardized statistics.\r\n\r\nGo to the DCMP Sampling Distribution of the Sample Mean (Continuous Population)\u00a0 tool at https:\/\/dcmathpathways.shinyapps.io\/SampDist_cont\/. You will use this data\u00a0 analysis tool to simulate random samples of colleges and examine the mean annual\u00a0 cost of attendance for each sample. Enter the following inputs:\r\n<ul>\r\n \t<li>Select Population Distribution: Real Population Data<\/li>\r\n \t<li>Select Example: College Cost[footnote]Agresti, A., Franklin, C. A., &amp; Klingenberg, B. (2021). Statistics: The art and science of learning from\u00a0 data. Pearson.[\/footnote]<\/li>\r\n<\/ul>\r\nThe population distribution shown is the distribution of the average annual costs of\u00a0 attending college (in U.S. dollars) for a population of 1,909 public and private four-year\u00a0 U.S. colleges.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 1<\/h3>\r\n1) What shape is the population distribution of the average annual costs of attending\u00a0 college?\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 2<\/h3>\r\n2) Generate 1,000 random samples of size [latex]n[\/latex] = 20.\r\n\r\na) The histogram labeled \u201cData Distribution (Histogram from last generated\u00a0 sample)\u201d is a histogram of the last randomly generated sample. What is the sample mean for this sample?\r\n\r\nb) The same sample mean generated from the last randomly generated sample should be shown on the sampling distribution of the sample mean.\r\n\r\nUse the mean and standard deviation of the simulated sampling distribution of the sample mean (from the 1,000 simulated samples) to calculate a z score for the last observed sample mean. Write a sentence interpreting this\u00a0 value.\r\n\r\nc) Write a sentence interpreting the value of the standard deviation of the sampling distribution of the sample mean that you used to calculate the z score in Part B.\r\n\r\n<\/div>\r\nWhen we calculate a z-score for a statistic, as in Question 1, Part B, we call this a\u00a0 standardized statistic. Question 1 used simulation to estimate the mean and standard\u00a0 deviation of the sample mean. Mathematically, though, we know the exact formulas for\u00a0 these values. The standardized sample mean is\r\n\r\n<span style=\"font-size: 1rem; text-align: initial;\">[latex]z=\\frac{\\bar{x}-[mean\\;of\\;\\bar{x}]}{[std.\\;deviation\\;of\\;\\bar{x}]}=\\frac{\\bar{x}-\\mu}{\\sigma\/\\sqrt{n}}[\/latex]<\/span>\r\n\r\n<span style=\"font-size: 1rem; text-align: initial;\">where [latex]\\bar{x}[\/latex] is the sample mean, [latex]\\mu[\/latex] is the population mean, [latex]\\sigma[\/latex] is the population standard\u00a0 deviation, and [latex]n[\/latex] is the sample size. The statistic is \u201cstandardized\u201d since it is centered to have a mean of 0 and scaled to have a standard deviation of 1.\u00a0\u00a0<\/span>\r\n\r\nIf the population distribution is normal or the sample size is sufficiently large, this\u00a0 standardized statistic will follow a standard normal distribution\u2014a normal distribution with a mean of 0 and a standard deviation of 1.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 3<\/h3>\r\n3) What are the population mean and population standard deviation of the variable average annual cost of attending college? Use appropriate statistical notation.\r\n\r\nHint: These values are displayed in the main title of the Population Distribution in the tool.\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 4<\/h3>\r\n4) Using the values from Question 3, calculate the mean and standard deviation of the sampling distribution of the sample mean for samples of size [latex]n[\/latex] = 20.<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 5<\/h3>\r\n5) Using your answers from Question 4, calculate the value of the standardized statistic for your observed sample mean from Question 2, Part A. How does this value\u00a0 compare to your answer from Question 2, Part B?<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 6<\/h3>\r\n6) Now, go to the DCMP Normal Distribution tool at\u00a0https:\/\/dcmathpathways.shinyapps.io\/NormalDist\/. Select the Find Probability tab.\r\n\r\na) Enter your mean and standard deviation from Question 4 for the mean and\u00a0 standard deviation of the normal distribution. Calculate the probability of\u00a0 observing the value of your observed sample mean from Question 2, Part A\u00a0 or something smaller.\r\n\r\nb) Enter the mean and standard deviation of a standard normal distribution.\u00a0 What is the probability of observing the value of your standardized statistic\u00a0 from Question 5 or something smaller? How does this probability compare to\u00a0 the probability from Part A?\r\n\r\n<\/div>","rendered":"<p>Preparing for the next class<\/p>\n<p>In the next in-class activity, you will need to calculate and interpret a standardized\u00a0 sample mean, calculate and interpret the standard deviation of a sample mean, and\u00a0 calculate probabilities involving standardized statistics.<\/p>\n<p>Go to the DCMP Sampling Distribution of the Sample Mean (Continuous Population)\u00a0 tool at https:\/\/dcmathpathways.shinyapps.io\/SampDist_cont\/. You will use this data\u00a0 analysis tool to simulate random samples of colleges and examine the mean annual\u00a0 cost of attendance for each sample. Enter the following inputs:<\/p>\n<ul>\n<li>Select Population Distribution: Real Population Data<\/li>\n<li>Select Example: College Cost<a class=\"footnote\" title=\"Agresti, A., Franklin, C. A., &amp; Klingenberg, B. (2021). Statistics: The art and science of learning from\u00a0 data. Pearson.\" id=\"return-footnote-5418-1\" href=\"#footnote-5418-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a><\/li>\n<\/ul>\n<p>The population distribution shown is the distribution of the average annual costs of\u00a0 attending college (in U.S. dollars) for a population of 1,909 public and private four-year\u00a0 U.S. colleges.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Question 1<\/h3>\n<p>1) What shape is the population distribution of the average annual costs of attending\u00a0 college?<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Question 2<\/h3>\n<p>2) Generate 1,000 random samples of size [latex]n[\/latex] = 20.<\/p>\n<p>a) The histogram labeled \u201cData Distribution (Histogram from last generated\u00a0 sample)\u201d is a histogram of the last randomly generated sample. What is the sample mean for this sample?<\/p>\n<p>b) The same sample mean generated from the last randomly generated sample should be shown on the sampling distribution of the sample mean.<\/p>\n<p>Use the mean and standard deviation of the simulated sampling distribution of the sample mean (from the 1,000 simulated samples) to calculate a z score for the last observed sample mean. Write a sentence interpreting this\u00a0 value.<\/p>\n<p>c) Write a sentence interpreting the value of the standard deviation of the sampling distribution of the sample mean that you used to calculate the z score in Part B.<\/p>\n<\/div>\n<p>When we calculate a z-score for a statistic, as in Question 1, Part B, we call this a\u00a0 standardized statistic. Question 1 used simulation to estimate the mean and standard\u00a0 deviation of the sample mean. Mathematically, though, we know the exact formulas for\u00a0 these values. The standardized sample mean is<\/p>\n<p><span style=\"font-size: 1rem; text-align: initial;\">[latex]z=\\frac{\\bar{x}-[mean\\;of\\;\\bar{x}]}{[std.\\;deviation\\;of\\;\\bar{x}]}=\\frac{\\bar{x}-\\mu}{\\sigma\/\\sqrt{n}}[\/latex]<\/span><\/p>\n<p><span style=\"font-size: 1rem; text-align: initial;\">where [latex]\\bar{x}[\/latex] is the sample mean, [latex]\\mu[\/latex] is the population mean, [latex]\\sigma[\/latex] is the population standard\u00a0 deviation, and [latex]n[\/latex] is the sample size. The statistic is \u201cstandardized\u201d since it is centered to have a mean of 0 and scaled to have a standard deviation of 1.\u00a0\u00a0<\/span><\/p>\n<p>If the population distribution is normal or the sample size is sufficiently large, this\u00a0 standardized statistic will follow a standard normal distribution\u2014a normal distribution with a mean of 0 and a standard deviation of 1.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Question 3<\/h3>\n<p>3) What are the population mean and population standard deviation of the variable average annual cost of attending college? Use appropriate statistical notation.<\/p>\n<p>Hint: These values are displayed in the main title of the Population Distribution in the tool.<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Question 4<\/h3>\n<p>4) Using the values from Question 3, calculate the mean and standard deviation of the sampling distribution of the sample mean for samples of size [latex]n[\/latex] = 20.<\/p><\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Question 5<\/h3>\n<p>5) Using your answers from Question 4, calculate the value of the standardized statistic for your observed sample mean from Question 2, Part A. How does this value\u00a0 compare to your answer from Question 2, Part B?<\/p><\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Question 6<\/h3>\n<p>6) Now, go to the DCMP Normal Distribution tool at\u00a0https:\/\/dcmathpathways.shinyapps.io\/NormalDist\/. Select the Find Probability tab.<\/p>\n<p>a) Enter your mean and standard deviation from Question 4 for the mean and\u00a0 standard deviation of the normal distribution. Calculate the probability of\u00a0 observing the value of your observed sample mean from Question 2, Part A\u00a0 or something smaller.<\/p>\n<p>b) Enter the mean and standard deviation of a standard normal distribution.\u00a0 What is the probability of observing the value of your standardized statistic\u00a0 from Question 5 or something smaller? How does this probability compare to\u00a0 the probability from Part A?<\/p>\n<\/div>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-5418-1\">Agresti, A., Franklin, C. A., &amp; Klingenberg, B. (2021). Statistics: The art and science of learning from\u00a0 data. Pearson. <a href=\"#return-footnote-5418-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":23592,"menu_order":6,"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-5418","chapter","type-chapter","status-publish","hentry"],"part":5315,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5418","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\/5418\/revisions"}],"predecessor-version":[{"id":5421,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5418\/revisions\/5421"}],"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\/5418\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=5418"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=5418"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=5418"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=5418"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}