{"id":5333,"date":"2022-08-19T18:49:55","date_gmt":"2022-08-19T18:49:55","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/?post_type=chapter&#038;p=5333"},"modified":"2022-08-19T18:54:35","modified_gmt":"2022-08-19T18:54:35","slug":"12b-coreq","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/12b-coreq\/","title":{"raw":"12B Coreq","rendered":"12B Coreq"},"content":{"raw":"In the next preview assignment and in the next class, you will need to generate random\u00a0 samples from a population, find values of the sample mean and sample standard deviation, and calculate z-scores and another similar standardized score for sample\u00a0 means.\r\n\r\n<strong>Calculating and Interpreting z-scores of Sample Means<\/strong>\r\n\r\nData collected by the Centers for Disease Control and Prevention show that the\u00a0 average birthweight for babies in the United States is 7.17 pounds and the standard\u00a0 deviation of birthweights is 1.30 pounds.[footnote]Centers for Disease Control and Prevention. (n.d.). Natality for 2016\u20132019 (expanded).\u00a0 https:\/\/wonder.cdc.gov\/controller\/datarequest\/D149;jsessionid=7AB7525C7DC02FF1F19D38C125AC[\/footnote] Assume that birthweights in the United States follow an approximate normal distribution.\r\n\r\nGo to the DCMP Sampling Distribution of the Sample Mean (Continuous Population) tool at https:\/\/dcmathpathways.shinyapps.io\/SampDist_cont\/. You will use this tool to\u00a0 simulate random samples of births and examine the mean birthweight for each sample.\r\n\r\nEnter the following inputs:\r\n<ul>\r\n \t<li>Select Population Distribution: Bell-Shaped<\/li>\r\n \t<li>Enter 7.17 and 1.30 for the population mean and standard deviation, respectively. (You will need to select the \u201cEnter values for [latex]\\mu[\/latex] and [latex]\\sigma[\/latex]\u201d option.)<\/li>\r\n<\/ul>\r\nThe population distribution shown is our model for the distribution of all the birthweights\u00a0 in the United States.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 1<\/h3>\r\n1) Enter [latex]n=5[\/latex] for the sample size and generate a single sample of five babies.\r\n\r\na) What is the value of the sample mean birthweight for your random sample?\u00a0 What is the correct notation for this value?\r\n\r\nb) What is the value of the sample standard deviation of birthweights for your\u00a0 random sample? What is the correct notation for this value?\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 2<\/h3>\r\n2) Select four more random samples of five babies and write down the sample means\u00a0 in the table below.\r\n<div align=\"left\">\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Sample 2<\/td>\r\n<td>Sample 3<\/td>\r\n<td>Sample 4<\/td>\r\n<td>Sample 5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/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 3<\/h3>\r\n3) Now, compare the sample results.\r\n\r\nPart A: Are all of the sample means the same?\r\n\r\nPart B: If you selected another random sample of five babies, would it be possible to get a different sample mean?\r\n\r\n<\/div>\r\nBecause the birthweights of different babies will vary from sample to sample, the\u00a0 sample mean birthweights will also vary from sample to sample. The tendency of\u00a0 samples to have different statistics (means, proportions) than the population as a whole\u00a0 due to randomness is called sampling variability, and the distribution of these\u00a0 statistics is called a sampling distribution.\r\n\r\nIn the case of sample means, if we sample from a normal population as the one seen\u00a0 here, the sampling distribution of the sample means will also have a normal distribution.\u00a0 If the mean and standard deviation of the population are \u00b5 and \u03c3, respectively, the mean\u00a0 and standard deviation of the sample means for random samples of size [latex]n[\/latex] are:\r\n\r\nMean of the sample means [latex]=\\mu[\/latex]\r\n\r\nStandard deviation of the sample means = [latex]\\frac{\\sigma}{\\sqrt{n}}[\/latex]\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 4<\/h3>\r\n4) Find the values of the mean and standard deviation of the sampling distribution of the sample mean birthweights for random samples of five babies.\r\n\r\n<\/div>\r\nRecall that a z-score of a value is calculated by subtracting the mean and then dividing by the standard deviation. Similarly, we can calculate the z-score of a sample mean by:\r\n\r\n*missing latex* (not missing, but not working)\r\n\r\n[latex]z=\\frac{\\bar{x}-[mean\\;of\\;\\bar{x}'s]}{[std.\\;deviation\\;of\\;\\bar{x}'s]}=\\frac{\\bar{x}-\\mu}{\\sigma\\sqrt{n}}\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 5<\/h3>\r\n5) Calculate the z-score of the sample mean generated in Question 1.\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 6<\/h3>\r\n6) How many standard deviations from the population mean birthweight of all babies born in the United States is the sample mean generated in Question 1?\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 7<\/h3>\r\n7) In Questions 1\u20136, you considered the following symbols: [latex]\\mu, \\bar{x}, \\sigma,[\/latex] and [latex]s[\/latex].\r\n\r\nPart A: Which of the symbols represents a mean? Which represents a standard\u00a0 deviation?\r\n\r\nPart B: Which of the symbols are parameters? Which are statistics?\r\n\r\nPart C: If you were to take another random sample of five babies, which of these\u00a0 symbols\u2019 values could change? Which would remain the same?\r\n\r\n<\/div>\r\n&nbsp;","rendered":"<p>In the next preview assignment and in the next class, you will need to generate random\u00a0 samples from a population, find values of the sample mean and sample standard deviation, and calculate z-scores and another similar standardized score for sample\u00a0 means.<\/p>\n<p><strong>Calculating and Interpreting z-scores of Sample Means<\/strong><\/p>\n<p>Data collected by the Centers for Disease Control and Prevention show that the\u00a0 average birthweight for babies in the United States is 7.17 pounds and the standard\u00a0 deviation of birthweights is 1.30 pounds.<a class=\"footnote\" title=\"Centers for Disease Control and Prevention. (n.d.). Natality for 2016\u20132019 (expanded).\u00a0 https:\/\/wonder.cdc.gov\/controller\/datarequest\/D149;jsessionid=7AB7525C7DC02FF1F19D38C125AC\" id=\"return-footnote-5333-1\" href=\"#footnote-5333-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a> Assume that birthweights in the United States follow an approximate normal distribution.<\/p>\n<p>Go to the DCMP Sampling Distribution of the Sample Mean (Continuous Population) tool at https:\/\/dcmathpathways.shinyapps.io\/SampDist_cont\/. You will use this tool to\u00a0 simulate random samples of births and examine the mean birthweight for each sample.<\/p>\n<p>Enter the following inputs:<\/p>\n<ul>\n<li>Select Population Distribution: Bell-Shaped<\/li>\n<li>Enter 7.17 and 1.30 for the population mean and standard deviation, respectively. (You will need to select the \u201cEnter values for [latex]\\mu[\/latex] and [latex]\\sigma[\/latex]\u201d option.)<\/li>\n<\/ul>\n<p>The population distribution shown is our model for the distribution of all the birthweights\u00a0 in the United States.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Question 1<\/h3>\n<p>1) Enter [latex]n=5[\/latex] for the sample size and generate a single sample of five babies.<\/p>\n<p>a) What is the value of the sample mean birthweight for your random sample?\u00a0 What is the correct notation for this value?<\/p>\n<p>b) What is the value of the sample standard deviation of birthweights for your\u00a0 random sample? What is the correct notation for this value?<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Question 2<\/h3>\n<p>2) Select four more random samples of five babies and write down the sample means\u00a0 in the table below.<\/p>\n<div style=\"text-align: left;\">\n<table>\n<tbody>\n<tr>\n<td>Sample 2<\/td>\n<td>Sample 3<\/td>\n<td>Sample 4<\/td>\n<td>Sample 5<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Question 3<\/h3>\n<p>3) Now, compare the sample results.<\/p>\n<p>Part A: Are all of the sample means the same?<\/p>\n<p>Part B: If you selected another random sample of five babies, would it be possible to get a different sample mean?<\/p>\n<\/div>\n<p>Because the birthweights of different babies will vary from sample to sample, the\u00a0 sample mean birthweights will also vary from sample to sample. The tendency of\u00a0 samples to have different statistics (means, proportions) than the population as a whole\u00a0 due to randomness is called sampling variability, and the distribution of these\u00a0 statistics is called a sampling distribution.<\/p>\n<p>In the case of sample means, if we sample from a normal population as the one seen\u00a0 here, the sampling distribution of the sample means will also have a normal distribution.\u00a0 If the mean and standard deviation of the population are \u00b5 and \u03c3, respectively, the mean\u00a0 and standard deviation of the sample means for random samples of size [latex]n[\/latex] are:<\/p>\n<p>Mean of the sample means [latex]=\\mu[\/latex]<\/p>\n<p>Standard deviation of the sample means = [latex]\\frac{\\sigma}{\\sqrt{n}}[\/latex]<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Question 4<\/h3>\n<p>4) Find the values of the mean and standard deviation of the sampling distribution of the sample mean birthweights for random samples of five babies.<\/p>\n<\/div>\n<p>Recall that a z-score of a value is calculated by subtracting the mean and then dividing by the standard deviation. Similarly, we can calculate the z-score of a sample mean by:<\/p>\n<p>*missing latex* (not missing, but not working)<\/p>\n<p>[latex]z=\\frac{\\bar{x}-[mean\\;of\\;\\bar{x}'s]}{[std.\\;deviation\\;of\\;\\bar{x}'s]}=\\frac{\\bar{x}-\\mu}{\\sigma\\sqrt{n}}  <\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Question 5<\/h3>\n<p>  5) Calculate the z-score of the sample mean generated in Question 1.    <\/p><\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Question 6<\/h3>\n<p>  6) How many standard deviations from the population mean birthweight of all babies born in the United States is the sample mean generated in Question 1?    <\/p><\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Question 7<\/h3>\n<p>  7) In Questions 1\u20136, you considered the following symbols: [latex]\\mu, \\bar{x}, \\sigma,[\/latex] and [latex]s[\/latex].<\/p>\n<p>Part A: Which of the symbols represents a mean? Which represents a standard\u00a0 deviation?<\/p>\n<p>Part B: Which of the symbols are parameters? Which are statistics?<\/p>\n<p>Part C: If you were to take another random sample of five babies, which of these\u00a0 symbols\u2019 values could change? Which would remain the same?<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-5333-1\">Centers for Disease Control and Prevention. (n.d.). Natality for 2016\u20132019 (expanded).\u00a0 https:\/\/wonder.cdc.gov\/controller\/datarequest\/D149;jsessionid=7AB7525C7DC02FF1F19D38C125AC <a href=\"#return-footnote-5333-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":23592,"menu_order":4,"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-5333","chapter","type-chapter","status-publish","hentry"],"part":5315,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5333","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":4,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5333\/revisions"}],"predecessor-version":[{"id":5337,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5333\/revisions\/5337"}],"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\/5333\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=5333"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=5333"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=5333"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=5333"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}