{"id":5414,"date":"2022-08-22T19:42:02","date_gmt":"2022-08-22T19:42:02","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/?post_type=chapter&#038;p=5414"},"modified":"2022-08-22T20:12:50","modified_gmt":"2022-08-22T20:12:50","slug":"12b-inclass","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/12b-inclass\/","title":{"raw":"12B InClass","rendered":"12B InClass"},"content":{"raw":"Many undergraduate students are employed\u00a0at the same time they are enrolled in school.\u00a0In 2018, the National Center for Education\u00a0Statistics reported that 43% of full-time\u00a0students worked.[footnote]U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics.\u00a0 (2020). College student employment. Retrieved from https:\/\/nces.ed.gov\/programs\/coe\/pdf\/coe_ssa.pdf[\/footnote] Being employed while in\u00a0school can help a student pay for tuition,\u00a0housing, and other expenses, but it can also\u00a0be associated (either positively or negatively)\u00a0with a student\u2019s academic performance.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 3<\/h3>\r\n1) Do you think working while in school has\u00a0an overall positive or negative association\u00a0with academic performance for full-time college students? Explain.<\/div>\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"300\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/26121004\/Picture281-300x200.jpg\" alt=\"A man wearing an apron smiling in a cafe.\" width=\"300\" height=\"200\" \/> Credit: iStock\/Six_Characters[\/caption]\r\n\r\nA random sample of 15 employed full-time students at a large university was selected\u00a0 for a survey on employment. The following is the number of hours (in increasing order) worked per week for each of those 15 students:\r\n\r\n2 5 7 10 16 19 19 22 23 26 27 27 31 40 50\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 2<\/h3>\r\n2) Go to the DCMP Explore Quantitative Data tool at\u00a0https:\/\/dcmathpathways.shinyapps.io\/EDA_quantitative\/.\r\n<ul>\r\n \t<li>Select \u201cYour Own\u201d under \u201cEnter Data.\u201d<\/li>\r\n \t<li>Enter a descriptive name for the variable (e.g., Hours Worked per Week). \u2022 Enter the previous observations into the tool.<\/li>\r\n<\/ul>\r\na) Examine the histogram, boxplot, dotplot, and summary statistics of these\u00a0 data. Write a few sentences describing the features of the data distribution.\u00a0 Address the shape, center, spread, and outliers.\r\n\r\nb) What is the sample mean hours worked per week? What is the appropriate\u00a0 symbol for this value?\r\n\r\nc) What is the sample standard deviation? What is the appropriate symbol for\u00a0 this value?\r\n\r\nd) Write a sentence interpreting the standard deviation in the context of the\u00a0 problem.\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 3<\/h3>\r\n3) Suppose the hours worked per week for all employed full-time students at this\u00a0 university varies according to an approximate normal distribution with mean \u00b5 and\u00a0 standard deviation \u03c3.\r\n\r\nConsider the standardized sample mean, or z-statistic:\r\n\r\n[latex]z=\\frac{\\bar{x}-[mean\\;of\\;\\bar{x}}{[std.\\;deviation\\;of\\;\\bar{x}}=\\frac{\\bar{x}-\\mu}{\\sigma\/\\sqrt{n}}[\/latex]\r\n\r\na) Which of the following quantities in the z-statistic are parameters and which\u00a0 are statistics: [latex]\\bar{x},\\mu,\\sigma[\/latex]?\r\n\r\nb) If we were to take repeated random samples of size [latex]n[\/latex] = 15 from this\u00a0 population, which of the quantities in the z-statistic could change from\u00a0 sample to sample? There may be more than one correct answer.\r\n\r\na)\u00a0[latex]\\bar{x}[\/latex]\r\n\r\nb)\u00a0[latex]\\mu[\/latex]\r\n\r\nc)\u00a0[latex]\\sigma[\/latex]\r\n\r\nd)\u00a0[latex]n[\/latex]\r\n\r\nc) What distribution does the z-statistic follow across many random samples?\r\n\r\n<\/div>\r\nSuppose we know that the mean hours worked per week for all employed full-time\u00a0 students at this university is\u00a0[latex]\\mu=19[\/latex] hours, but we do not know the value of the\u00a0 population standard deviation, [latex]\\sigma[\/latex]. If we want to calculate the value of a standardized sample mean, however, we need to know [latex]\\sigma[\/latex]. So, instead of [latex]\\sigma[\/latex], let\u2019s substitute in our best estimate for [latex]\\sigma[\/latex]: the sample standard deviation, [latex]s[\/latex].\r\n\r\nRecall that an estimate of the standard deviation of a statistic is called the standard\u00a0 error of that statistic. Since we estimate the standard deviation of the sample mean\u00a0[latex]\\sigma\/\\sqrt{n}[\/latex] by substituting in [latex]s[\/latex] for [latex]\\sigma[\/latex], the standard error of the sample mean is\r\n\r\n[latex]SE(\\bar{x})=\\frac{s}{\\sqrt{n}}[\/latex]\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 4<\/h3>\r\n4) Now, consider the quantity\r\n\r\n[latex]t=\\frac{\\bar{x}-[mean\\;of\\;\\bar{x}]}{[std.\\;error\\;of\\\\;\\bar{x}}=\\frac{\\bar{x}-\\mu}{s\/\\sqrt{n}}[\/latex]\r\n\r\nThis quantity is called a t-statistic.\r\n\r\n(a) Which of the following quantities in the t-statistic are parameters and which\u00a0 are statistics: [latex]\\bar{x},\\mu,s[\/latex]?\r\n\r\n(b) If we were to take repeated random samples of size [latex]n[\/latex] = 15 from this\u00a0 population, which of the quantities in the t-statistic could change from sample\u00a0 to sample? There may be more than one correct answer.\r\n\r\na)\u00a0[latex]n\\bar{x}[\/latex]\r\n\r\nb)\u00a0[latex]\\mu[\/latex]\r\n\r\nc)\u00a0[latex]s[\/latex]\r\n\r\nd)\u00a0[latex]n[\/latex]\r\n\r\n(c) Go to the DCMP t Distribution tool at\u00a0https:\/\/dcmathpathways.shinyapps.io\/tdist\/. Under the Explore tab, select\u00a0 \u201cShow Standard Normal Curve.\u201d Compare the t Distributions for 3, 8, and 14\u00a0 degrees of freedom. Explain what happens to the t Distribution curve as the\u00a0 degrees of freedom increase.\r\n\r\n<\/div>\r\n<div align=\"left\">\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>t Distribution\u00a0<\/strong>\r\n\r\nWhen taking many, many random samples of size [latex]n[\/latex] from a population distribution with mean [latex]\\mu[\/latex] and standard deviation [latex]\\sigma[\/latex], the t-statistic\r\n\r\n[latex]n[\/latex]\r\n\r\nwill follow a <strong>t Distribution with [latex]n-1[\/latex] degrees of freedom<\/strong> if the population distribution\u00a0 is normal or if the population distribution is not too skewed and the sample size is large (e.g., [latex]n[\/latex] \u2265 30).<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nSince the t-statistic exhibits more sampling variability than the z-statistic, its distribution\u00a0 has slightly more variability than a standard normal distribution. However, as the sample\u00a0 size increases, there is less sampling variability associated with the standard error of\u00a0 the sample mean, so its distribution gets closer to a standard normal distribution.\r\n\r\nA picture of the t Distribution for various degrees of freedom, along with the standard\u00a0 normal distribution for reference, is shown below.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 5<\/h3>\r\n5) Assume the hours worked per week for all employed full-time students at this\u00a0 university varies according to an approximate normal distribution with mean [latex]\\mu[\/latex] = 19 hours.\r\n\r\na) Calculate the standard error of the sample mean for the observed sample data shown before Question 2. Write a sentence interpreting this value in the\u00a0 context of the problem.\r\n\r\n<span style=\"font-size: 1rem; text-align: initial;\">b) Calculate the value of the t-statistic for the observed sample data shown before Question 2. Write a sentence interpreting this value in the context of the problem.\u00a0<\/span>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 6<\/h3>\r\n6) Consider the t-statistic for Question 5, Part B.\r\n\r\na) What distribution did the t-statistic value come from? Explain why the\u00a0 conditions for using the t Distribution are met, and then sketch the\u00a0 distribution.\r\n\r\nb) The probability of observing 0.77 or higher on a standard normal distribution\u00a0 is 0.2206. Would you predict the probability of observing 0.77 or higher on\u00a0 the distribution you specified in Part A to be less than, equal to, or larger\u00a0 than 0.2206? Explain.\r\n\r\nc) Using the Find Probability tab in the DCMP t Distribution tool, calculate the\u00a0 probability of observing a t-statistic of 0.77 or higher for the distribution you\u00a0 specified in Part A. Does this match your prediction from Part B?\r\n\r\n<\/div>","rendered":"<p>Many undergraduate students are employed\u00a0at the same time they are enrolled in school.\u00a0In 2018, the National Center for Education\u00a0Statistics reported that 43% of full-time\u00a0students worked.<a class=\"footnote\" title=\"U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics.\u00a0 (2020). College student employment. Retrieved from https:\/\/nces.ed.gov\/programs\/coe\/pdf\/coe_ssa.pdf\" id=\"return-footnote-5414-1\" href=\"#footnote-5414-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a> Being employed while in\u00a0school can help a student pay for tuition,\u00a0housing, and other expenses, but it can also\u00a0be associated (either positively or negatively)\u00a0with a student\u2019s academic performance.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Question 3<\/h3>\n<p>1) Do you think working while in school has\u00a0an overall positive or negative association\u00a0with academic performance for full-time college students? Explain.<\/p><\/div>\n<div style=\"width: 310px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/26121004\/Picture281-300x200.jpg\" alt=\"A man wearing an apron smiling in a cafe.\" width=\"300\" height=\"200\" \/><\/p>\n<p class=\"wp-caption-text\">Credit: iStock\/Six_Characters<\/p>\n<\/div>\n<p>A random sample of 15 employed full-time students at a large university was selected\u00a0 for a survey on employment. The following is the number of hours (in increasing order) worked per week for each of those 15 students:<\/p>\n<p>2 5 7 10 16 19 19 22 23 26 27 27 31 40 50<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Question 2<\/h3>\n<p>2) Go to the DCMP Explore Quantitative Data tool at\u00a0https:\/\/dcmathpathways.shinyapps.io\/EDA_quantitative\/.<\/p>\n<ul>\n<li>Select \u201cYour Own\u201d under \u201cEnter Data.\u201d<\/li>\n<li>Enter a descriptive name for the variable (e.g., Hours Worked per Week). \u2022 Enter the previous observations into the tool.<\/li>\n<\/ul>\n<p>a) Examine the histogram, boxplot, dotplot, and summary statistics of these\u00a0 data. Write a few sentences describing the features of the data distribution.\u00a0 Address the shape, center, spread, and outliers.<\/p>\n<p>b) What is the sample mean hours worked per week? What is the appropriate\u00a0 symbol for this value?<\/p>\n<p>c) What is the sample standard deviation? What is the appropriate symbol for\u00a0 this value?<\/p>\n<p>d) Write a sentence interpreting the standard deviation in the context of the\u00a0 problem.<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Question 3<\/h3>\n<p>3) Suppose the hours worked per week for all employed full-time students at this\u00a0 university varies according to an approximate normal distribution with mean \u00b5 and\u00a0 standard deviation \u03c3.<\/p>\n<p>Consider the standardized sample mean, or z-statistic:<\/p>\n<p>[latex]z=\\frac{\\bar{x}-[mean\\;of\\;\\bar{x}}{[std.\\;deviation\\;of\\;\\bar{x}}=\\frac{\\bar{x}-\\mu}{\\sigma\/\\sqrt{n}}[\/latex]<\/p>\n<p>a) Which of the following quantities in the z-statistic are parameters and which\u00a0 are statistics: [latex]\\bar{x},\\mu,\\sigma[\/latex]?<\/p>\n<p>b) If we were to take repeated random samples of size [latex]n[\/latex] = 15 from this\u00a0 population, which of the quantities in the z-statistic could change from\u00a0 sample to sample? There may be more than one correct answer.<\/p>\n<p>a)\u00a0[latex]\\bar{x}[\/latex]<\/p>\n<p>b)\u00a0[latex]\\mu[\/latex]<\/p>\n<p>c)\u00a0[latex]\\sigma[\/latex]<\/p>\n<p>d)\u00a0[latex]n[\/latex]<\/p>\n<p>c) What distribution does the z-statistic follow across many random samples?<\/p>\n<\/div>\n<p>Suppose we know that the mean hours worked per week for all employed full-time\u00a0 students at this university is\u00a0[latex]\\mu=19[\/latex] hours, but we do not know the value of the\u00a0 population standard deviation, [latex]\\sigma[\/latex]. If we want to calculate the value of a standardized sample mean, however, we need to know [latex]\\sigma[\/latex]. So, instead of [latex]\\sigma[\/latex], let\u2019s substitute in our best estimate for [latex]\\sigma[\/latex]: the sample standard deviation, [latex]s[\/latex].<\/p>\n<p>Recall that an estimate of the standard deviation of a statistic is called the standard\u00a0 error of that statistic. Since we estimate the standard deviation of the sample mean\u00a0[latex]\\sigma\/\\sqrt{n}[\/latex] by substituting in [latex]s[\/latex] for [latex]\\sigma[\/latex], the standard error of the sample mean is<\/p>\n<p>[latex]SE(\\bar{x})=\\frac{s}{\\sqrt{n}}[\/latex]<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Question 4<\/h3>\n<p>4) Now, consider the quantity<\/p>\n<p>[latex]t=\\frac{\\bar{x}-[mean\\;of\\;\\bar{x}]}{[std.\\;error\\;of\\\\;\\bar{x}}=\\frac{\\bar{x}-\\mu}{s\/\\sqrt{n}}[\/latex]<\/p>\n<p>This quantity is called a t-statistic.<\/p>\n<p>(a) Which of the following quantities in the t-statistic are parameters and which\u00a0 are statistics: [latex]\\bar{x},\\mu,s[\/latex]?<\/p>\n<p>(b) If we were to take repeated random samples of size [latex]n[\/latex] = 15 from this\u00a0 population, which of the quantities in the t-statistic could change from sample\u00a0 to sample? There may be more than one correct answer.<\/p>\n<p>a)\u00a0[latex]n\\bar{x}[\/latex]<\/p>\n<p>b)\u00a0[latex]\\mu[\/latex]<\/p>\n<p>c)\u00a0[latex]s[\/latex]<\/p>\n<p>d)\u00a0[latex]n[\/latex]<\/p>\n<p>(c) Go to the DCMP t Distribution tool at\u00a0https:\/\/dcmathpathways.shinyapps.io\/tdist\/. Under the Explore tab, select\u00a0 \u201cShow Standard Normal Curve.\u201d Compare the t Distributions for 3, 8, and 14\u00a0 degrees of freedom. Explain what happens to the t Distribution curve as the\u00a0 degrees of freedom increase.<\/p>\n<\/div>\n<div style=\"text-align: left;\">\n<table>\n<tbody>\n<tr>\n<td><strong>t Distribution\u00a0<\/strong><\/p>\n<p>When taking many, many random samples of size [latex]n[\/latex] from a population distribution with mean [latex]\\mu[\/latex] and standard deviation [latex]\\sigma[\/latex], the t-statistic<\/p>\n<p>[latex]n[\/latex]<\/p>\n<p>will follow a <strong>t Distribution with [latex]n-1[\/latex] degrees of freedom<\/strong> if the population distribution\u00a0 is normal or if the population distribution is not too skewed and the sample size is large (e.g., [latex]n[\/latex] \u2265 30).<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Since the t-statistic exhibits more sampling variability than the z-statistic, its distribution\u00a0 has slightly more variability than a standard normal distribution. However, as the sample\u00a0 size increases, there is less sampling variability associated with the standard error of\u00a0 the sample mean, so its distribution gets closer to a standard normal distribution.<\/p>\n<p>A picture of the t Distribution for various degrees of freedom, along with the standard\u00a0 normal distribution for reference, is shown below.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Question 5<\/h3>\n<p>5) Assume the hours worked per week for all employed full-time students at this\u00a0 university varies according to an approximate normal distribution with mean [latex]\\mu[\/latex] = 19 hours.<\/p>\n<p>a) Calculate the standard error of the sample mean for the observed sample data shown before Question 2. Write a sentence interpreting this value in the\u00a0 context of the problem.<\/p>\n<p><span style=\"font-size: 1rem; text-align: initial;\">b) Calculate the value of the t-statistic for the observed sample data shown before Question 2. Write a sentence interpreting this value in the context of the problem.\u00a0<\/span><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Question 6<\/h3>\n<p>6) Consider the t-statistic for Question 5, Part B.<\/p>\n<p>a) What distribution did the t-statistic value come from? Explain why the\u00a0 conditions for using the t Distribution are met, and then sketch the\u00a0 distribution.<\/p>\n<p>b) The probability of observing 0.77 or higher on a standard normal distribution\u00a0 is 0.2206. Would you predict the probability of observing 0.77 or higher on\u00a0 the distribution you specified in Part A to be less than, equal to, or larger\u00a0 than 0.2206? Explain.<\/p>\n<p>c) Using the Find Probability tab in the DCMP t Distribution tool, calculate the\u00a0 probability of observing a t-statistic of 0.77 or higher for the distribution you\u00a0 specified in Part A. Does this match your prediction from Part B?<\/p>\n<\/div>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-5414-1\">U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics.\u00a0 (2020). College student employment. Retrieved from https:\/\/nces.ed.gov\/programs\/coe\/pdf\/coe_ssa.pdf <a href=\"#return-footnote-5414-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":23592,"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-5414","chapter","type-chapter","status-publish","hentry"],"part":5315,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5414","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\/5414\/revisions"}],"predecessor-version":[{"id":5417,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5414\/revisions\/5417"}],"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\/5414\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=5414"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=5414"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=5414"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=5414"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}