{"id":1265,"date":"2023-06-22T02:09:48","date_gmt":"2023-06-22T02:09:48","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/connecting-tests-and-intervals-dig-deeper\/"},"modified":"2025-05-16T03:50:43","modified_gmt":"2025-05-16T03:50:43","slug":"connecting-tests-and-intervals-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/connecting-tests-and-intervals-fresh-take\/","title":{"raw":"Connecting Tests and Intervals: Fresh Take","rendered":"Connecting Tests and Intervals: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Explain how a confidence interval is related to a two-sided hypothesis test&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Explain how a confidence interval is related to a two-sided hypothesis test.<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2><strong>Two-Sided Hypothesis Test<\/strong><\/h2>\r\n<ol>\r\n\t<li>\r\n<p><strong>Null and Alternative Hypotheses:<\/strong><\/p>\r\n<ul>\r\n\t<li>Null Hypothesis ([latex]H_0[\/latex]): Often states that there is no effect, no difference, or no relationship.<\/li>\r\n\t<li>Alternative Hypothesis ([latex]H_A[\/latex]): States that there is a significant effect, difference, or relationship.<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>\r\n<p><strong>Test Statistic and Critical Region: <\/strong><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">A test statistic is calculated, and a critical region is defined on both tails of the distribution. If the test statistic falls into either tail, the null hypothesis is rejected.<\/span><\/p>\r\n<\/li>\r\n<\/ol>\r\n<h2>Two-Sided Confidence Interval<\/h2>\r\n<ol>\r\n\t<li>\r\n<p><strong>Estimation: <\/strong><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">A confidence interval provides a range of values within which we are reasonably confident the true population parameter lies.<\/span><\/p>\r\n<\/li>\r\n\t<li>\r\n<p><strong>Margin of Error: <\/strong><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">The width of the confidence interval is determined by the margin of error, which is influenced by the level of confidence (e.g., 95%).<\/span><\/p>\r\n<\/li>\r\n<\/ol>\r\n<h2>Connection<\/h2>\r\n<ul>\r\n\t<li>\r\n<p><strong>Rejecting the Null Hypothesis: <\/strong><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">If the null hypothesis value is not within the confidence interval, it suggests evidence against the null hypothesis in a two-sided test.<\/span><\/p>\r\n<\/li>\r\n\t<li>\r\n<p><strong>Fail to Reject the Null Hypothesis: <\/strong><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">If the null hypothesis value falls within the confidence interval, it may lead to a non-rejection of the null hypothesis in a two-sided test.<\/span><\/p>\r\n<\/li>\r\n<\/ul>\r\n<section class=\"textbox example\">\r\n<p>Suppose a two-sided hypothesis test is conducted for the difference in proportions ([latex]p_1 -p_2[\/latex]), and the null hypothesis is [latex]H_0: p_1 -p_2 = 0[\/latex].<\/p>\r\n<p>If the 95% confidence interval does not include zero, it would lead to rejecting the null hypothesis at the 5% significance level.<\/p>\r\n<\/section>\r\n<p>In this way, the range of values provided by the confidence interval informs the decision-making process in a two-sided hypothesis test.<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1775[\/ohm2_question]<\/section>\r\n<p>[reveal-answer q=\"782856\"]Steps to obtain the confidence interval.[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"782856\"]<\/p>\r\n<ul>\r\n\t<li>Under <strong>Enter Data<\/strong>, select <strong>Number of Successes<\/strong>.<\/li>\r\n\t<li>Input the information for Group 1 and Group 2 accordingly.<\/li>\r\n\t<li>Adjust the <strong>Confidence Level<\/strong>\u00a0according to your scenario.[\/hidden-answer]<\/li>\r\n<\/ul>\r\n<p><iframe src=\"https:\/\/lumen-learning.shinyapps.io\/2sample_prop\/ \" width=\"100%\" height=\"1300\" frameborder=\"no\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\"><\/span><\/iframe><br \/>\r\n[<a href=\"https:\/\/lumen-learning.shinyapps.io\/2sample_prop\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1776[\/ohm2_question]<\/section>\r\n<section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2817[\/ohm2_question]<\/section>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1777[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1778[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Explain how a confidence interval is related to a two-sided hypothesis test&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Explain how a confidence interval is related to a two-sided hypothesis test.<\/span><\/li>\n<\/ul>\n<\/section>\n<h2><strong>Two-Sided Hypothesis Test<\/strong><\/h2>\n<ol>\n<li>\n<p><strong>Null and Alternative Hypotheses:<\/strong><\/p>\n<ul>\n<li>Null Hypothesis ([latex]H_0[\/latex]): Often states that there is no effect, no difference, or no relationship.<\/li>\n<li>Alternative Hypothesis ([latex]H_A[\/latex]): States that there is a significant effect, difference, or relationship.<\/li>\n<\/ul>\n<\/li>\n<li>\n<p><strong>Test Statistic and Critical Region: <\/strong><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">A test statistic is calculated, and a critical region is defined on both tails of the distribution. If the test statistic falls into either tail, the null hypothesis is rejected.<\/span><\/p>\n<\/li>\n<\/ol>\n<h2>Two-Sided Confidence Interval<\/h2>\n<ol>\n<li>\n<p><strong>Estimation: <\/strong><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">A confidence interval provides a range of values within which we are reasonably confident the true population parameter lies.<\/span><\/p>\n<\/li>\n<li>\n<p><strong>Margin of Error: <\/strong><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">The width of the confidence interval is determined by the margin of error, which is influenced by the level of confidence (e.g., 95%).<\/span><\/p>\n<\/li>\n<\/ol>\n<h2>Connection<\/h2>\n<ul>\n<li>\n<p><strong>Rejecting the Null Hypothesis: <\/strong><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">If the null hypothesis value is not within the confidence interval, it suggests evidence against the null hypothesis in a two-sided test.<\/span><\/p>\n<\/li>\n<li>\n<p><strong>Fail to Reject the Null Hypothesis: <\/strong><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">If the null hypothesis value falls within the confidence interval, it may lead to a non-rejection of the null hypothesis in a two-sided test.<\/span><\/p>\n<\/li>\n<\/ul>\n<section class=\"textbox example\">\n<p>Suppose a two-sided hypothesis test is conducted for the difference in proportions ([latex]p_1 -p_2[\/latex]), and the null hypothesis is [latex]H_0: p_1 -p_2 = 0[\/latex].<\/p>\n<p>If the 95% confidence interval does not include zero, it would lead to rejecting the null hypothesis at the 5% significance level.<\/p>\n<\/section>\n<p>In this way, the range of values provided by the confidence interval informs the decision-making process in a two-sided hypothesis test.<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1775\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1775&theme=lumen&iframe_resize_id=ohm1775&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q782856\">Steps to obtain the confidence interval.<\/button><\/p>\n<div id=\"q782856\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li>Under <strong>Enter Data<\/strong>, select <strong>Number of Successes<\/strong>.<\/li>\n<li>Input the information for Group 1 and Group 2 accordingly.<\/li>\n<li>Adjust the <strong>Confidence Level<\/strong>\u00a0according to your scenario.<\/div>\n<\/div>\n<\/li>\n<\/ul>\n<p><iframe loading=\"lazy\" src=\"https:\/\/lumen-learning.shinyapps.io\/2sample_prop\/\" width=\"100%\" height=\"1300\" frameborder=\"no\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\"><\/span><\/iframe><br \/>\n[<a href=\"https:\/\/lumen-learning.shinyapps.io\/2sample_prop\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1776\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1776&theme=lumen&iframe_resize_id=ohm1776&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2817\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2817&theme=lumen&iframe_resize_id=ohm2817&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1777\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1777&theme=lumen&iframe_resize_id=ohm1777&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1778\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1778&theme=lumen&iframe_resize_id=ohm1778&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":8,"menu_order":46,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":1205,"module-header":"fresh_take","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\/1265"}],"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":9,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1265\/revisions"}],"predecessor-version":[{"id":6780,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1265\/revisions\/6780"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/1205"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1265\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1265"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1265"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1265"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1265"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}