{"id":3931,"date":"2022-03-16T18:01:44","date_gmt":"2022-03-16T18:01:44","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/?post_type=chapter&#038;p=3931"},"modified":"2022-03-17T04:20:36","modified_gmt":"2022-03-17T04:20:36","slug":"instructor-guide-forming-connections-z-score-and-the-empirical-rule","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/instructor-guide-forming-connections-z-score-and-the-empirical-rule\/","title":{"raw":"Instructor Guide Z-Score and the Empirical Rule Forming Connections","rendered":"Instructor Guide Z-Score and the Empirical Rule Forming Connections"},"content":{"raw":"<h2>Overview<\/h2>\r\n<ul>\r\n \t<li>Students will\u00a0apply the Empirical Rule within a medical context to\u00a0determine \u201cunusual\u201d observations.<\/li>\r\n \t<li>Students are introduced to a quantifiable measure to\u00a0compare observations using standardized units and determine if an observation is\u00a0outside the range of usual values.<\/li>\r\n \t<li>This activity connects back to measures of center and spread, and connects forward to probability distributions and hypothesis testing.<\/li>\r\n \t<li><span style=\"background-color: #ffff99;\">V4, O3, S4, C5, O4, C3, V1, C4, S2, O2 \u2190 Link to EBTP descriptions\u00a0<\/span><\/li>\r\n<\/ul>\r\n<h3>Prerequisite assumptions<\/h3>\r\nStudents should be able to do each of the following after completing the <em>What to Know<\/em> assignment.\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Convert values into standardized scores.<\/li>\r\n \t<li>Use a value\u2019s standardized score to determine whether the value is above,\u00a0below, or equal to the mean.<\/li>\r\n \t<li>Explain the Empirical Rule.<\/li>\r\n<\/ul>\r\n<h3>Intended goals for this activity<\/h3>\r\nAfter completing this activity, students should understand that Standardized scores and the Empirical Rule can be used to determine if an\u00a0observation is usual or unusual.. They should be able to utilize standardized scores and the Empirical\u00a0 Rule to determine if an observation\u00a0is unusual and to\u00a0compare two observations by calculating and comparing the standardized score.\r\n<h2>Synchronous Delivery and Activity Flow<\/h2>\r\nThe sample activity delivery below assumes a face-to-face class meeting but can be adapted to a fully online or hybrid delivery by using break-out rooms for pairs and small groups.\r\n<h3>Frame the activity (3 minutes)<\/h3>\r\n<ul>\r\n \t<li>Have students read and answer Question 1 independently. Then,\u00a0have students discuss their answers in pairs before sharing with the\u00a0full group. V4<\/li>\r\n \t<li>Transition to the in-class activity by briefly discussing the objectives\u00a0for the activity.<\/li>\r\n<\/ul>\r\n<h3>Activity Flow (18 minutes)<\/h3>\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Use the last part of Question 1\u2014\u201cAre there alternatives to having\u00a0human volunteers participate in clinical trials?\u201d\u2014to lead into the two\u00a0paragraphs that follow the objectives.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">If it hasn\u2019t come up already, mention that one alternative to testing on\u00a0humans is testing on mice. Then have students read both paragraphs\u00a0individually.\r\n<ul>\r\n \t<li aria-level=\"1\">Make sure to give students enough time to process the information\u00a0they have read individually first. O3<\/li>\r\n \t<li aria-level=\"1\">In pairs, have students formulate any questions they might have\u00a0about this context.\u00a0Have the pairs share their questions and address them. Possible\u00a0questions may come up about the meaning of \"genome\" (an organism's complete set of genetic instructions) and \"toxicity\" (the degree to which a toxin or poison can harm humans or animals) and why organ weight is used to measure toxicity (organ weight changes are indicators of chemically-induced organ damage. Hence, changes in organ weight are often associated with treatment-related effects.). The student activity page also links to the website\u00a0https:\/\/www.yourgenome.org\/facts\/what-is-a-genome. S4<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Have students work on Questions 2 and 3 in pairs or small groups. You may wish to assign half the groups to Questions 2 and half to Question 3 before bringing everyone together to share their results. It may be helpful to display a graphical representation of the Empirical Rule. [reveal-answer q=\"782173\"]\u00a0This is the one they were introduced to in the\u00a0<em>What to Know<\/em> assignment\u00a0.[\/reveal-answer]\r\n[hidden-answer a=\"782173\"]\u00a0 <img class=\"alignnone size-medium wp-image-1033\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/11224126\/Picture59-300x295.jpg\" alt=\"A bar graph with the highest bars in the middle and lower bars to either side. In the center, the x-axis is labeled &quot;mu.&quot; Three bars to the left, it is labeled &quot;mu - sigma,&quot; three more bars to the left it is labeled &quot;mu - 2 sigma,&quot; and three more to the left, it's labeled &quot;mu - 3 sigma.&quot; Three to the right of the center, it is labeled &quot;mu + sigma.&quot; Three more to the right and it is labeled &quot;mu + 2 sigma&quot; and three more to the right, it is labeled &quot;mu + 3 sigma.&quot; The center six bars are all green and labeled as 68&amp; collectively. The three leftmost center bars are labeled 34.1%, and the three rightmost center bars are also labeled 34.1%. The next three bars out on either side of the center six are each labeled 13.6% and the center 12 are all labeled 95% collectively. Lastly, the next three out on either side of the center twelve are each labeled 2.1% and all 18 are collectively labeled 99.7% \u2248 100%\" width=\"300\" height=\"295\" \/>\u00a0 [\/hidden-answer]\r\n<ul>\r\n \t<li>If students have difficulty with these questions,\u00a0point back to the\u00a0listed means and standard deviations. Using the horizontal axis on\u00a0the chart, ask: \u201cWhat values are associated with 68% of the data?\u201d\u00a0[Answer: One standard deviation greater than the mean and\u00a0one standard deviation less than the mean ] C5<\/li>\r\n \t<li>Bring the whole-class together to summarize these questions before continuing. O4<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Students may have difficulty answering Question 4. It's okay to mention that\u00a0an observation that is two standard deviations from the mean is considered unusual. Point back to the chart and ask: \u201cWhat\u00a0percentage of the data is within two standard deviations of the\u00a0mean?\u201d [Answer: 95%] Follow up questions include:\r\n<ul>\r\n \t<li>\u201cIn this context, between which two values are\u00a095% of the data?\u201d [Answer: 0.825 g and 1.173 g for liver and 0.072 g\u00a0and 0.100 g for spleen]<\/li>\r\n \t<li>Have students identify a value that is not within 0.825 g\u00a0and 1.173 g for livers and not within 0.072 g and 0.100 g for spleens.<\/li>\r\n \t<li>Bring the whole-class together to debrief this question by having groups share a few unusual values. Make sure to discuss values that are more than two standard deviations above AND below the mean. C3<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>For Question 5, you can display the formula [latex]z=\\dfrac{\\bar{x}-\\mu}{\\sigma} and develop the inclination for students to talk about \"standard deviations from the mean.\"<\/li>\r\n<\/ul>\r\n<h3>Wrap-up\/transition (5 minutes)<\/h3>\r\n<ul>\r\n \t<li>Have students share their answers to Question 5. V1<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Ensure students are communicating appropriately using statistical\u00a0terms. In this question, the liver weight of [latex]1.07[\/latex]g is [latex]0.816[\/latex] standard\u00a0deviations above the mean and the spleen weight of [latex]0.104[\/latex]g is [latex]2.571[\/latex] standard deviations above the mean. C4<\/li>\r\n \t<li aria-level=\"1\">The spleen shows a higher level of toxicity because the weight of the\u00a0spleen is highly unusual at 2.571 standard deviations above the mean,\u00a0whereas the liver weight is within one standard deviation above the\u00a0mean.<\/li>\r\n \t<li aria-level=\"1\">Make sure students understand that z-scores do not have units associated with them.<\/li>\r\n \t<li aria-level=\"1\">Have students refer back to the Objectives for the activity and identify ones they recognized completing. S2, C4, O2<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Assign the homework or\u00a0<em>Practice<\/em>\u00a0and any <em>What to Know<\/em> pages for the <em>Forming Connections<\/em> activities you plan to complete in the next class meeting. C2<\/li>\r\n<\/ul>","rendered":"<h2>Overview<\/h2>\n<ul>\n<li>Students will\u00a0apply the Empirical Rule within a medical context to\u00a0determine \u201cunusual\u201d observations.<\/li>\n<li>Students are introduced to a quantifiable measure to\u00a0compare observations using standardized units and determine if an observation is\u00a0outside the range of usual values.<\/li>\n<li>This activity connects back to measures of center and spread, and connects forward to probability distributions and hypothesis testing.<\/li>\n<li><span style=\"background-color: #ffff99;\">V4, O3, S4, C5, O4, C3, V1, C4, S2, O2 \u2190 Link to EBTP descriptions\u00a0<\/span><\/li>\n<\/ul>\n<h3>Prerequisite assumptions<\/h3>\n<p>Students should be able to do each of the following after completing the <em>What to Know<\/em> assignment.<\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Convert values into standardized scores.<\/li>\n<li>Use a value\u2019s standardized score to determine whether the value is above,\u00a0below, or equal to the mean.<\/li>\n<li>Explain the Empirical Rule.<\/li>\n<\/ul>\n<h3>Intended goals for this activity<\/h3>\n<p>After completing this activity, students should understand that Standardized scores and the Empirical Rule can be used to determine if an\u00a0observation is usual or unusual.. They should be able to utilize standardized scores and the Empirical\u00a0 Rule to determine if an observation\u00a0is unusual and to\u00a0compare two observations by calculating and comparing the standardized score.<\/p>\n<h2>Synchronous Delivery and Activity Flow<\/h2>\n<p>The sample activity delivery below assumes a face-to-face class meeting but can be adapted to a fully online or hybrid delivery by using break-out rooms for pairs and small groups.<\/p>\n<h3>Frame the activity (3 minutes)<\/h3>\n<ul>\n<li>Have students read and answer Question 1 independently. Then,\u00a0have students discuss their answers in pairs before sharing with the\u00a0full group. V4<\/li>\n<li>Transition to the in-class activity by briefly discussing the objectives\u00a0for the activity.<\/li>\n<\/ul>\n<h3>Activity Flow (18 minutes)<\/h3>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Use the last part of Question 1\u2014\u201cAre there alternatives to having\u00a0human volunteers participate in clinical trials?\u201d\u2014to lead into the two\u00a0paragraphs that follow the objectives.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">If it hasn\u2019t come up already, mention that one alternative to testing on\u00a0humans is testing on mice. Then have students read both paragraphs\u00a0individually.\n<ul>\n<li aria-level=\"1\">Make sure to give students enough time to process the information\u00a0they have read individually first. O3<\/li>\n<li aria-level=\"1\">In pairs, have students formulate any questions they might have\u00a0about this context.\u00a0Have the pairs share their questions and address them. Possible\u00a0questions may come up about the meaning of &#8220;genome&#8221; (an organism&#8217;s complete set of genetic instructions) and &#8220;toxicity&#8221; (the degree to which a toxin or poison can harm humans or animals) and why organ weight is used to measure toxicity (organ weight changes are indicators of chemically-induced organ damage. Hence, changes in organ weight are often associated with treatment-related effects.). The student activity page also links to the website\u00a0https:\/\/www.yourgenome.org\/facts\/what-is-a-genome. S4<\/li>\n<\/ul>\n<\/li>\n<li>Have students work on Questions 2 and 3 in pairs or small groups. You may wish to assign half the groups to Questions 2 and half to Question 3 before bringing everyone together to share their results. It may be helpful to display a graphical representation of the Empirical Rule.\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q782173\">\u00a0This is the one they were introduced to in the\u00a0<em>What to Know<\/em> assignment\u00a0.<\/span><\/p>\n<div id=\"q782173\" class=\"hidden-answer\" style=\"display: none\">\u00a0 <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1033\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/11224126\/Picture59-300x295.jpg\" alt=\"A bar graph with the highest bars in the middle and lower bars to either side. In the center, the x-axis is labeled &quot;mu.&quot; Three bars to the left, it is labeled &quot;mu - sigma,&quot; three more bars to the left it is labeled &quot;mu - 2 sigma,&quot; and three more to the left, it's labeled &quot;mu - 3 sigma.&quot; Three to the right of the center, it is labeled &quot;mu + sigma.&quot; Three more to the right and it is labeled &quot;mu + 2 sigma&quot; and three more to the right, it is labeled &quot;mu + 3 sigma.&quot; The center six bars are all green and labeled as 68&amp; collectively. The three leftmost center bars are labeled 34.1%, and the three rightmost center bars are also labeled 34.1%. The next three bars out on either side of the center six are each labeled 13.6% and the center 12 are all labeled 95% collectively. Lastly, the next three out on either side of the center twelve are each labeled 2.1% and all 18 are collectively labeled 99.7% \u2248 100%\" width=\"300\" height=\"295\" \/>\u00a0 <\/div>\n<\/div>\n<ul>\n<li>If students have difficulty with these questions,\u00a0point back to the\u00a0listed means and standard deviations. Using the horizontal axis on\u00a0the chart, ask: \u201cWhat values are associated with 68% of the data?\u201d\u00a0[Answer: One standard deviation greater than the mean and\u00a0one standard deviation less than the mean ] C5<\/li>\n<li>Bring the whole-class together to summarize these questions before continuing. O4<\/li>\n<\/ul>\n<\/li>\n<li>Students may have difficulty answering Question 4. It&#8217;s okay to mention that\u00a0an observation that is two standard deviations from the mean is considered unusual. Point back to the chart and ask: \u201cWhat\u00a0percentage of the data is within two standard deviations of the\u00a0mean?\u201d [Answer: 95%] Follow up questions include:\n<ul>\n<li>\u201cIn this context, between which two values are\u00a095% of the data?\u201d [Answer: 0.825 g and 1.173 g for liver and 0.072 g\u00a0and 0.100 g for spleen]<\/li>\n<li>Have students identify a value that is not within 0.825 g\u00a0and 1.173 g for livers and not within 0.072 g and 0.100 g for spleens.<\/li>\n<li>Bring the whole-class together to debrief this question by having groups share a few unusual values. Make sure to discuss values that are more than two standard deviations above AND below the mean. C3<\/li>\n<\/ul>\n<\/li>\n<li>For Question 5, you can display the formula [latex]z=\\dfrac{\\bar{x}-\\mu}{\\sigma} and develop the inclination for students to talk about \"standard deviations from the mean.\"<\/li>\n<\/ul>\n<h3>Wrap-up\/transition (5 minutes)<\/h3>\n<ul>\n<li>Have students share their answers to Question 5. V1<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Ensure students are communicating appropriately using statistical\u00a0terms. In this question, the liver weight of [latex]1.07[\/latex]g is [latex]0.816[\/latex] standard\u00a0deviations above the mean and the spleen weight of [latex]0.104[\/latex]g is [latex]2.571[\/latex] standard deviations above the mean. C4<\/li>\n<li aria-level=\"1\">The spleen shows a higher level of toxicity because the weight of the\u00a0spleen is highly unusual at 2.571 standard deviations above the mean,\u00a0whereas the liver weight is within one standard deviation above the\u00a0mean.<\/li>\n<li aria-level=\"1\">Make sure students understand that z-scores do not have units associated with them.<\/li>\n<li aria-level=\"1\">Have students refer back to the Objectives for the activity and identify ones they recognized completing. S2, C4, O2<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Assign the homework or\u00a0<em>Practice<\/em>\u00a0and any <em>What to Know<\/em> pages for the <em>Forming Connections<\/em> activities you plan to complete in the next class meeting. C2<\/li>\n<\/ul>\n","protected":false},"author":25777,"menu_order":20,"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-3931","chapter","type-chapter","status-publish","hentry"],"part":3890,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/3931","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\/25777"}],"version-history":[{"count":7,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/3931\/revisions"}],"predecessor-version":[{"id":4066,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/3931\/revisions\/4066"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/parts\/3890"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/3931\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=3931"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=3931"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=3931"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=3931"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}