{"id":1184,"date":"2023-06-22T01:59:48","date_gmt":"2023-06-22T01:59:48","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/confidence-interval-for-proportions-continued-dig-deeper\/"},"modified":"2025-05-16T03:14:50","modified_gmt":"2025-05-16T03:14:50","slug":"confidence-interval-for-proportions-continued-dig-deeper","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/confidence-interval-for-proportions-continued-dig-deeper\/","title":{"raw":"Confidence Interval for Proportions (continued): Fresh Take","rendered":"Confidence Interval for Proportions (continued): 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;Calculate a confidence interval and explain what it means&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;}\">Calculate a confidence interval and explain what it means.<\/span><\/li>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Recognize common misinterpretations of confidence intervals&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;}\">Recognize common misinterpretations of confidence intervals.<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<p>The purpose of a confidence interval is to estimate a population parameter on the basis of a sample statistic. Sample statistics vary, so there are always errors in our estimates, but we never know how much. We therefore use the standard error, which is the average error in our sample estimates, to create a margin of error. The margin of error is related to our confidence that the interval contains the population parameter.<\/p>\r\n<p>[footnote]https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/why-it-matters-confidence-intervals\/[\/footnote]Here we add these ideas to the Big Picture to show how probability connects to inference.<\/p>\r\n<p><img class=\"aligncenter wp-image-6723 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/11\/16201839\/m7_link_prob_statistical_inference_topic_7_1_m7_intro_inference_3_image1.png\" alt=\"\" width=\"742\" height=\"690\" \/><\/p>\r\n<p><iframe src=\"https:\/\/lumen-learning.shinyapps.io\/inference_prop\/\" width=\"100%\" height=\"850\"><\/iframe><\/p>\r\n<p>[<a href=\"https:\/\/lumen-learning.shinyapps.io\/inference_prop\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\r\n<section class=\"textbox example\">Suppose 250 randomly selected people are surveyed to determine if they own a tablet. Of the 250 surveyed, 98 reported owning a tablet.\r\n\r\n<ul>\r\n\t<li>Using a 95% confidence level, compute and interpret a confidence interval estimate for the true proportion of people who own tablets.<\/li>\r\n<\/ul>\r\n<p>[reveal-answer q=\"715955\"]Show answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"715955\"]Using statistical tool above, we found the [latex]95\\%[\/latex] confidence interval is [latex](0.3315, 0.4525)[\/latex].<\/p>\r\n<p>Interpretation: We are [latex]95\\%[\/latex]confident that the true proportion of people who own tablets is between [latex]33.15\\%[\/latex] and [latex]45.25\\%[\/latex].[\/hidden-answer]<\/p>\r\n<ul>\r\n\t<li>Using a 99% confidence level, compute and interpret a confidence interval estimate for the true proportion of people who own tablets.<\/li>\r\n<\/ul>\r\n<p>[reveal-answer q=\"583857\"]Show answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"583857\"]<\/p>\r\n<p>Using statistical tool above, we found the [latex]99\\%[\/latex] confidence interval is [latex](0.3125, 0.4715)[\/latex].<\/p>\r\n<p>Interpretation: We are [latex]99\\%[\/latex]confident that the true proportion of people who own tablets is between [latex]31.25\\%[\/latex] and [latex]47.15\\%[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h3><strong>Is yawning contagious?<\/strong><\/h3>\r\n<p>In one experiment, [latex]34[\/latex] participants saw a person near them yawn. The experimenters recorded whether or not the participants yawned. [latex]10[\/latex] participants did yawn (\"Yes\") and [latex]24[\/latex] participants didn't yawn (\"No\").<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1678[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1680[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]920[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Calculate a confidence interval and explain what it means&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;}\">Calculate a confidence interval and explain what it means.<\/span><\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Recognize common misinterpretations of confidence intervals&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;}\">Recognize common misinterpretations of confidence intervals.<\/span><\/li>\n<\/ul>\n<\/section>\n<p>The purpose of a confidence interval is to estimate a population parameter on the basis of a sample statistic. Sample statistics vary, so there are always errors in our estimates, but we never know how much. We therefore use the standard error, which is the average error in our sample estimates, to create a margin of error. The margin of error is related to our confidence that the interval contains the population parameter.<\/p>\n<p><a class=\"footnote\" title=\"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/why-it-matters-confidence-intervals\/\" id=\"return-footnote-1184-1\" href=\"#footnote-1184-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a>Here we add these ideas to the Big Picture to show how probability connects to inference.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-6723 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/11\/16201839\/m7_link_prob_statistical_inference_topic_7_1_m7_intro_inference_3_image1.png\" alt=\"\" width=\"742\" height=\"690\" \/><\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/lumen-learning.shinyapps.io\/inference_prop\/\" width=\"100%\" height=\"850\"><\/iframe><\/p>\n<p>[<a href=\"https:\/\/lumen-learning.shinyapps.io\/inference_prop\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\n<section class=\"textbox example\">Suppose 250 randomly selected people are surveyed to determine if they own a tablet. Of the 250 surveyed, 98 reported owning a tablet.<\/p>\n<ul>\n<li>Using a 95% confidence level, compute and interpret a confidence interval estimate for the true proportion of people who own tablets.<\/li>\n<\/ul>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q715955\">Show answer<\/button><\/p>\n<div id=\"q715955\" class=\"hidden-answer\" style=\"display: none\">Using statistical tool above, we found the [latex]95\\%[\/latex] confidence interval is [latex](0.3315, 0.4525)[\/latex].<\/p>\n<p>Interpretation: We are [latex]95\\%[\/latex]confident that the true proportion of people who own tablets is between [latex]33.15\\%[\/latex] and [latex]45.25\\%[\/latex].<\/p><\/div>\n<\/div>\n<ul>\n<li>Using a 99% confidence level, compute and interpret a confidence interval estimate for the true proportion of people who own tablets.<\/li>\n<\/ul>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q583857\">Show answer<\/button><\/p>\n<div id=\"q583857\" class=\"hidden-answer\" style=\"display: none\">\n<p>Using statistical tool above, we found the [latex]99\\%[\/latex] confidence interval is [latex](0.3125, 0.4715)[\/latex].<\/p>\n<p>Interpretation: We are [latex]99\\%[\/latex]confident that the true proportion of people who own tablets is between [latex]31.25\\%[\/latex] and [latex]47.15\\%[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h3><strong>Is yawning contagious?<\/strong><\/h3>\n<p>In one experiment, [latex]34[\/latex] participants saw a person near them yawn. The experimenters recorded whether or not the participants yawned. [latex]10[\/latex] participants did yawn (&#8220;Yes&#8221;) and [latex]24[\/latex] participants didn&#8217;t yawn (&#8220;No&#8221;).<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1678\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1678&theme=lumen&iframe_resize_id=ohm1678&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1680\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1680&theme=lumen&iframe_resize_id=ohm1680&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm920\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=920&theme=lumen&iframe_resize_id=ohm920&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-1184-1\">https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/why-it-matters-confidence-intervals\/ <a href=\"#return-footnote-1184-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":8,"menu_order":21,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":1163,"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\/1184"}],"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":8,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1184\/revisions"}],"predecessor-version":[{"id":6733,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1184\/revisions\/6733"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/1163"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1184\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1184"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1184"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1184"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1184"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}