{"id":1271,"date":"2023-06-22T02:13:06","date_gmt":"2023-06-22T02:13:06","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/sampling-distribution-of-a-sample-mean-background-youll-need-1\/"},"modified":"2025-05-16T03:54:16","modified_gmt":"2025-05-16T03:54:16","slug":"sampling-distribution-of-a-sample-mean-background-youll-need-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/sampling-distribution-of-a-sample-mean-background-youll-need-1\/","title":{"raw":"Module 11: Background You'll Need 1","rendered":"Module 11: Background You&#8217;ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Calculate and interpret [latex]z[\/latex]-scores.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2><strong>[latex]z[\/latex]-scores<\/strong><\/h2>\r\n<section class=\"textbox recall\">\r\n<p class=\"Para\">A<b> [latex]z[\/latex]-score<\/b>\u00a0or <b>standardized score <\/b>measures a value\u2019s distance from the mean in units of standard deviation. A positive [latex]z[\/latex]-score indicates the value is above the mean, whereas a negative [latex]z[\/latex]-score indicates the value is below the mean.<\/p>\r\n<p style=\"text-align: center;\">[latex]z=\\dfrac{\\text{value}-\\text{mean}}{\\text{standard deviation}}[\/latex]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1491[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1493[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1495[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1496[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1497[\/ohm2_question]<\/section>\r\n<section class=\"textbox recall\">A<strong> [latex]z[\/latex]-score<\/strong> of a value is calculated by subtracting the mean and then dividing by the standard deviation. Similarly, we can calculate the [latex]z[\/latex]-score of a sample mean by:\r\n\r\n<p style=\"text-align: center;\">[latex]z=\\dfrac{\\bar{x}-[\\text{mean of } \\bar{x}'s]}{\\text{std. deviation of } \\bar{x}'s} [\/latex] [latex]= \\dfrac{\\bar{x}-\\mu}{\\frac{\\sigma}{\\sqrt{n}}}[\/latex]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1794[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Calculate and interpret [latex]z[\/latex]-scores.<\/li>\n<\/ul>\n<\/section>\n<h2><strong>[latex]z[\/latex]-scores<\/strong><\/h2>\n<section class=\"textbox recall\">\n<p class=\"Para\">A<b> [latex]z[\/latex]-score<\/b>\u00a0or <b>standardized score <\/b>measures a value\u2019s distance from the mean in units of standard deviation. A positive [latex]z[\/latex]-score indicates the value is above the mean, whereas a negative [latex]z[\/latex]-score indicates the value is below the mean.<\/p>\n<p style=\"text-align: center;\">[latex]z=\\dfrac{\\text{value}-\\text{mean}}{\\text{standard deviation}}[\/latex]<\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1491\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1491&theme=lumen&iframe_resize_id=ohm1491&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1493\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1493&theme=lumen&iframe_resize_id=ohm1493&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1495\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1495&theme=lumen&iframe_resize_id=ohm1495&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1496\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1496&theme=lumen&iframe_resize_id=ohm1496&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1497\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1497&theme=lumen&iframe_resize_id=ohm1497&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox recall\">A<strong> [latex]z[\/latex]-score<\/strong> of a value is calculated by subtracting the mean and then dividing by the standard deviation. Similarly, we can calculate the [latex]z[\/latex]-score of a sample mean by:<\/p>\n<p style=\"text-align: center;\">[latex]z=\\dfrac{\\bar{x}-[\\text{mean of } \\bar{x}'s]}{\\text{std. deviation of } \\bar{x}'s}[\/latex] [latex]= \\dfrac{\\bar{x}-\\mu}{\\frac{\\sigma}{\\sqrt{n}}}[\/latex]<\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1794\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1794&theme=lumen&iframe_resize_id=ohm1794&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":8,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":1268,"module-header":"background_you_need","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1271"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/users\/8"}],"version-history":[{"count":6,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1271\/revisions"}],"predecessor-version":[{"id":6781,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1271\/revisions\/6781"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/1268"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1271\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1271"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1271"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1271"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1271"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}