{"id":5191,"date":"2022-08-18T20:06:13","date_gmt":"2022-08-18T20:06:13","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/?post_type=chapter&#038;p=5191"},"modified":"2022-08-18T20:07:31","modified_gmt":"2022-08-18T20:07:31","slug":"9b-in-class-activity","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/9b-in-class-activity\/","title":{"raw":"9B In-Class Activity","rendered":"9B In-Class Activity"},"content":{"raw":"Each month, the U.S. Bureau of Labor Statistics releases a report on the\u00a0employment situation in the United States.\u00a0Included in the report is an estimate of the\u00a0\u00a0nationwide unemployment rate\u2014the\u00a0\u00a0number of unemployed people as a\u00a0\u00a0percentage of the labor force (defined as\u00a0\u00a0the total number of employed individuals\u00a0\u00a0plus unemployed individuals who are\u00a0\u00a0actively looking for work).[footnote] U.S. Bureau of Labor Statistics. (n.d.). <em>Concepts and definition<\/em>s. https:\/\/www.bls.gov\/cps\/definitions.htm[\/footnote]\r\n\r\nFor example, at the start of the COVID-19 pandemic in the United States, the unemployment rate jumped from 3.5% in February\u00a0 2020 to 14.8% in April 2020.[footnote]U.S. Bureau of Labor Statistics. (n.d.). <em>Graphics for economic news releases.<\/em> https:\/\/www.bls.gov\/charts\/employment-situation\/civilian-unemployment-rate.htm[\/footnote]\r\n\r\n<img class=\"alignnone wp-image-5192\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/08\/18194653\/9B-In-Class.png\" alt=\"\" width=\"448\" height=\"298\" \/>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 1<\/h3>\r\nIf you would have taken a random sample of 1,000 individuals from the U.S. labor force in April 2020 and calculated the percentage of those individuals who were unemployed, do you think you would have gotten 14.8%? Explain.\r\n\r\n<\/div>\r\nIn this in-class activity, we will use the DCMP Sampling Distribution of the Sample Proportion tool to explore sampling variability of sample proportions. Our population will be the U.S. labor force, and we will assume that the true unemployment rate (the proportion of the U.S. labor force that is unemployed) is 0.15. In the tool:\r\n<ul>\r\n \t<li aria-level=\"1\">Set the Population Proportion ([latex] p[\/latex]) to 0.15.<\/li>\r\n \t<li>Set the Sample Size ([latex]n[\/latex]) to 50.<\/li>\r\n<\/ul>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 2<\/h3>\r\nUse the tool to select one random sample of size 50 from this population by clicking \u201c1\u201d and then \u201cDraw Sample(s).\u201d A bar graph of your randomly generated sample will be displayed directly under the graph of the population distribution.\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>What is the value of your sample proportion?<\/li>\r\n \t<li>What is the appropriate statistical notation for this value?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 3<\/h3>\r\nSelect four more random samples of size 50 and fill in the following table with your five randomly generated sample proportions.\r\n<div align=\"left\">\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Sample<\/td>\r\n<td>Sample\r\n\r\nProportion<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>5<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 4<\/h3>\r\nExplain why the values of sample proportions vary from sample to sample.\r\n\r\n<\/div>\r\nIn order to get a sense of the pattern of variation in sample proportions, we need to\u00a0 generate more than five samples. The distribution showing how sample proportions vary\u00a0 from sample to sample is called a sampling distribution of the sample proportion.\r\n\r\nIn statistics, we often talk about \u201cdistributions,\u201d and a \u201csampling distribution\u201d is just a\u00a0 distribution of sample statistics as they vary from sample to sample. An illustration of the\u00a0 distinction between the population distribution of the variable (whether or not an\u00a0 individual is unemployed), a single sample distribution of the variable (still, whether or\u00a0 not an individual is unemployed), and the sampling distribution of sample\r\n\r\nproportions (not the individual values of the variable but a statistic calculated from the values in a sample) are shown below.\r\n\r\n<img class=\"alignnone wp-image-5193\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/08\/18195248\/9B-In-Class-1-300x169.png\" alt=\"\" width=\"552\" height=\"311\" \/>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 5<\/h3>\r\nUse the DCMP Sampling Distribution of the Sample Proportion tool to generate 1,000 more sample proportions. A plot of the simulated sampling distribution of sample proportions will be displayed below the plot of the last randomly generated sample.\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>Describe the shape of the simulated sampling distribution of sample proportions.<\/li>\r\n \t<li>What is the mean of the simulated sampling distribution of sample\u00a0 proportions? Why does this value make sense?<\/li>\r\n \t<li>What is the standard deviation of the simulated sampling distribution of\u00a0 sample proportions? Write a sentence interpreting this value in the context of\u00a0 the problem.<\/li>\r\n<\/ol>\r\n<\/div>\r\nIf we were to simulate every possible sample, then we would be able to derive an exact\u00a0 sampling distribution, but that is not feasible. Luckily, we can use mathematical theory\u00a0 to derive expressions for the mean and standard deviation of the sampling distribution\u00a0 of a sample proportion:\r\n<div class=\"textbox\">\r\n\r\nSampling Distribution of a Sample Proportion\r\n\r\nWhen taking many random samples of size [latex]n[\/latex] from a population distribution with population proportion [latex]p:\r\n<ul>\r\n \t<li aria-level=\"1\">The mean of the distribution of sample proportions is [latex]p[\/latex].<\/li>\r\n \t<li aria-level=\"1\">The standard deviation of the distribution of sample proportions is [latex]\\sqrt{\\frac{p(1-p)}{n}}[\/latex].<\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 6<\/h3>\r\nIn our example, the sample size was [latex]n= 50[\/latex] and the population proportion was [latex]p = 0.15[\/latex]. Using the previous formulas, calculate the mean and standard deviation of the sampling distribution of sample proportions for our example. Round your answer to 2 decimal places.\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 7<\/h3>\r\nIn Question 5, you found the mean and standard deviation of one simulated distribution of sample proportions, which approximates the mean and standard deviation of the actual sampling distribution of sample proportions. The actual sampling distribution would result from considering every possible sample. Compare the values of the mean and standard deviation you calculated in Question 6 to the estimates of the mean and standard deviation from the simulation in Question 5.\r\n\r\n<\/div>\r\nThe previous exercise assumed that we knew the value of the population proportion and thus could calculate the mean and standard deviation of the sample proportion by formulas, or estimate the mean and standard deviation by simulation. In practice, however, 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:\r\n<div class=\"textbox\">\r\n\r\nEstimated mean of sample proportions = [latex]\\hat{p}[\/latex]\r\n\r\nEstimated standard deviation of sample proportions = [latex]\\sqrt{\\frac{\\hat{p}(1-\\hat{p})}{n}}[\/latex]\r\n\r\n<\/div>\r\nTo 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]:\r\n\r\n[latex]SE=\\sqrt{\\frac{\\hat{p}(1-\\hat{p})}{n}}[\/latex]\r\n\r\nSimulation provides one way to estimate the standard deviation of the sample proportion, and this formula gives another way.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 8<\/h3>\r\nSuppose you select a random sample of 50 individuals from the U.S. labor force and find that six of them are unemployed.\r\n\r\n&nbsp;\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>What is the value of the unemployment rate (proportion that are unemployed) for your random sample?<\/li>\r\n \t<li>Using only your sample data, estimate the true unemployment rate in the United States.<\/li>\r\n \t<li>Calculate the standard error of the sample proportion that are unemployed. Write a sentence interpreting this value in context.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 9<\/h3>\r\nGenerate another random sample of 50 individuals from the U.S. labor force where we assume [latex] p= 0.15[\/latex]. Then calculate the standard error using the sample proportion from that sample. How close is this value to the standard deviation of sample proportions you found in Question 6?\r\n\r\n<\/div>\r\n&nbsp;","rendered":"<p>Each month, the U.S. Bureau of Labor Statistics releases a report on the\u00a0employment situation in the United States.\u00a0Included in the report is an estimate of the\u00a0\u00a0nationwide unemployment rate\u2014the\u00a0\u00a0number of unemployed people as a\u00a0\u00a0percentage of the labor force (defined as\u00a0\u00a0the total number of employed individuals\u00a0\u00a0plus unemployed individuals who are\u00a0\u00a0actively looking for work).<a class=\"footnote\" title=\"U.S. Bureau of Labor Statistics. (n.d.). Concepts and definitions. https:\/\/www.bls.gov\/cps\/definitions.htm\" id=\"return-footnote-5191-1\" href=\"#footnote-5191-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a><\/p>\n<p>For example, at the start of the COVID-19 pandemic in the United States, the unemployment rate jumped from 3.5% in February\u00a0 2020 to 14.8% in April 2020.<a class=\"footnote\" title=\"U.S. Bureau of Labor Statistics. (n.d.). Graphics for economic news releases. https:\/\/www.bls.gov\/charts\/employment-situation\/civilian-unemployment-rate.htm\" id=\"return-footnote-5191-2\" href=\"#footnote-5191-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-5192\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/08\/18194653\/9B-In-Class.png\" alt=\"\" width=\"448\" height=\"298\" \/><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Question 1<\/h3>\n<p>If you would have taken a random sample of 1,000 individuals from the U.S. labor force in April 2020 and calculated the percentage of those individuals who were unemployed, do you think you would have gotten 14.8%? Explain.<\/p>\n<\/div>\n<p>In this in-class activity, we will use the DCMP Sampling Distribution of the Sample Proportion tool to explore sampling variability of sample proportions. Our population will be the U.S. labor force, and we will assume that the true unemployment rate (the proportion of the U.S. labor force that is unemployed) is 0.15. In the tool:<\/p>\n<ul>\n<li aria-level=\"1\">Set the Population Proportion ([latex]p[\/latex]) to 0.15.<\/li>\n<li>Set the Sample Size ([latex]n[\/latex]) to 50.<\/li>\n<\/ul>\n<div class=\"textbox key-takeaways\">\n<h3>Question 2<\/h3>\n<p>Use the tool to select one random sample of size 50 from this population by clicking \u201c1\u201d and then \u201cDraw Sample(s).\u201d A bar graph of your randomly generated sample will be displayed directly under the graph of the population distribution.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>What is the value of your sample proportion?<\/li>\n<li>What is the appropriate statistical notation for this value?<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Question 3<\/h3>\n<p>Select four more random samples of size 50 and fill in the following table with your five randomly generated sample proportions.<\/p>\n<div style=\"text-align: left;\">\n<table>\n<tbody>\n<tr>\n<td>Sample<\/td>\n<td>Sample<\/p>\n<p>Proportion<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Question 4<\/h3>\n<p>Explain why the values of sample proportions vary from sample to sample.<\/p>\n<\/div>\n<p>In order to get a sense of the pattern of variation in sample proportions, we need to\u00a0 generate more than five samples. The distribution showing how sample proportions vary\u00a0 from sample to sample is called a sampling distribution of the sample proportion.<\/p>\n<p>In statistics, we often talk about \u201cdistributions,\u201d and a \u201csampling distribution\u201d is just a\u00a0 distribution of sample statistics as they vary from sample to sample. An illustration of the\u00a0 distinction between the population distribution of the variable (whether or not an\u00a0 individual is unemployed), a single sample distribution of the variable (still, whether or\u00a0 not an individual is unemployed), and the sampling distribution of sample<\/p>\n<p>proportions (not the individual values of the variable but a statistic calculated from the values in a sample) are shown below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-5193\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/08\/18195248\/9B-In-Class-1-300x169.png\" alt=\"\" width=\"552\" height=\"311\" \/><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Question 5<\/h3>\n<p>Use the DCMP Sampling Distribution of the Sample Proportion tool to generate 1,000 more sample proportions. A plot of the simulated sampling distribution of sample proportions will be displayed below the plot of the last randomly generated sample.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>Describe the shape of the simulated sampling distribution of sample proportions.<\/li>\n<li>What is the mean of the simulated sampling distribution of sample\u00a0 proportions? Why does this value make sense?<\/li>\n<li>What is the standard deviation of the simulated sampling distribution of\u00a0 sample proportions? Write a sentence interpreting this value in the context of\u00a0 the problem.<\/li>\n<\/ol>\n<\/div>\n<p>If we were to simulate every possible sample, then we would be able to derive an exact\u00a0 sampling distribution, but that is not feasible. Luckily, we can use mathematical theory\u00a0 to derive expressions for the mean and standard deviation of the sampling distribution\u00a0 of a sample proportion:<\/p>\n<div class=\"textbox\">\n<p>Sampling Distribution of a Sample Proportion<\/p>\n<p>When taking many random samples of size [latex]n[\/latex] from a population distribution with population proportion [latex]p:  <\/p>\n<ul>\n<li aria-level=\"1\">The mean of the distribution of sample proportions is [latex]p[\/latex].<\/li>\n<li aria-level=\"1\">The standard deviation of the distribution of sample proportions is [latex]\\sqrt{\\frac{p(1-p)}{n}}[\/latex].<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Question 6<\/h3>\n<p>In our example, the sample size was [latex]n= 50[\/latex] and the population proportion was [latex]p = 0.15[\/latex]. Using the previous formulas, calculate the mean and standard deviation of the sampling distribution of sample proportions for our example. Round your answer to 2 decimal places.<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Question 7<\/h3>\n<p>In Question 5, you found the mean and standard deviation of one simulated distribution of sample proportions, which approximates the mean and standard deviation of the actual sampling distribution of sample proportions. The actual sampling distribution would result from considering every possible sample. Compare the values of the mean and standard deviation you calculated in Question 6 to the estimates of the mean and standard deviation from the simulation in Question 5.<\/p>\n<\/div>\n<p>The previous exercise assumed that we knew the value of the population proportion and thus could calculate the mean and standard deviation of the sample proportion by formulas, or estimate the mean and standard deviation by simulation. In practice, however, 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:<\/p>\n<div class=\"textbox\">\n<p>Estimated mean of sample proportions = [latex]\\hat{p}[\/latex]<\/p>\n<p>Estimated standard deviation of sample proportions = [latex]\\sqrt{\\frac{\\hat{p}(1-\\hat{p})}{n}}[\/latex]<\/p>\n<\/div>\n<p>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]:<\/p>\n<p>[latex]SE=\\sqrt{\\frac{\\hat{p}(1-\\hat{p})}{n}}[\/latex]<\/p>\n<p>Simulation provides one way to estimate the standard deviation of the sample proportion, and this formula gives another way.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Question 8<\/h3>\n<p>Suppose you select a random sample of 50 individuals from the U.S. labor force and find that six of them are unemployed.<\/p>\n<p>&nbsp;<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>What is the value of the unemployment rate (proportion that are unemployed) for your random sample?<\/li>\n<li>Using only your sample data, estimate the true unemployment rate in the United States.<\/li>\n<li>Calculate the standard error of the sample proportion that are unemployed. Write a sentence interpreting this value in context.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Question 9<\/h3>\n<p>Generate another random sample of 50 individuals from the U.S. labor force where we assume [latex]p= 0.15[\/latex]. Then calculate the standard error using the sample proportion from that sample. How close is this value to the standard deviation of sample proportions you found in Question 6?<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-5191-1\"> U.S. Bureau of Labor Statistics. (n.d.). <em>Concepts and definition<\/em>s. https:\/\/www.bls.gov\/cps\/definitions.htm <a href=\"#return-footnote-5191-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-5191-2\">U.S. Bureau of Labor Statistics. (n.d.). <em>Graphics for economic news releases.<\/em> https:\/\/www.bls.gov\/charts\/employment-situation\/civilian-unemployment-rate.htm <a href=\"#return-footnote-5191-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":574340,"menu_order":5,"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-5191","chapter","type-chapter","status-publish","hentry"],"part":5175,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5191","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\/574340"}],"version-history":[{"count":2,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5191\/revisions"}],"predecessor-version":[{"id":5195,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5191\/revisions\/5195"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/parts\/5175"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5191\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=5191"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=5191"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=5191"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=5191"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}