{"id":1300,"date":"2023-06-22T02:13:29","date_gmt":"2023-06-22T02:13:29","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/confidence-interval-for-difference-in-population-means-learn-it-1\/"},"modified":"2025-05-16T04:03:54","modified_gmt":"2025-05-16T04:03:54","slug":"confidence-interval-for-difference-in-population-means-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/confidence-interval-for-difference-in-population-means-learn-it-1\/","title":{"raw":"Confidence Interval for Difference in Population Means: Learn It 1","rendered":"Confidence Interval for Difference in Population Means: Learn It 1"},"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;Check the assumptions for a two-sample t confidence interval for population mean&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4865,&quot;3&quot;:{&quot;1&quot;:0},&quot;11&quot;:3,&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Check the assumptions for a two-sample [latex]t[\/latex] confidence interval for population mean.<\/span><\/li>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Calculate a confidence interval for the difference between two population means and explain what it means&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4865,&quot;3&quot;:{&quot;1&quot;:0},&quot;11&quot;:3,&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Calculate and explain a confidence interval for the difference between two population means.<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Independent vs. Paired Samples<\/h2>\r\n<p class=\"para\">When you are interested in estimating a difference in population means, you usually start with data from samples from each of the populations of interest. There are two different strategies for selecting the two samples. One strategy is to select a sample from one population and then independently select a sample from the second population. Using this strategy results in two samples where the individuals selected for the first sample do not influence the individuals selected for the second sample. This would be the case if you took a random sample from each population. Samples selected in this way are said to be [pb_glossary id=\"5622\"]independent samples[\/pb_glossary].<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]979[\/ohm2_question]<\/section>\r\n<p class=\"para\">The second strategy is to select samples where each observation in one sample is paired in a logical way with a particular observation in the second sample. In the question above, the observations would be paired by student\u2014there is a before keyboarding class typing speed and an after keyboarding class typing speed for each student. If samples are chosen in a way that results in the observations in one sample being paired with the observations in the other sample, the samples are said to be <b>paired samples <\/b>or <strong>matched pairs<\/strong>. Paired samples are also sometimes called [pb_glossary id=\"5623\"]dependent samples[\/pb_glossary].<\/p>\r\n<p class=\"para\">One common process that results in paired samples is when data are collected both before and after some intervention (like the keyboarding class). But there are other data collection methods that can result in paired samples. One example would be if participants in a study to evaluate the effect of exercise (light vs. moderate exercise) were paired by weight prior to the study, and then one person from each pair was assigned to each exercise group. This would result in exercise groups that were similar with respect to weight, and the two samples would be paired because there is a logical way to match an observation from the light exercise group with a particular observation from the moderate exercise group.<\/p>\r\n<p>It is important to make a distinction between independent samples and paired samples because the way the data from the samples are analyzed is different for these two cases.<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]980[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Check the assumptions for a two-sample t confidence interval for population mean&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4865,&quot;3&quot;:{&quot;1&quot;:0},&quot;11&quot;:3,&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Check the assumptions for a two-sample [latex]t[\/latex] confidence interval for population mean.<\/span><\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Calculate a confidence interval for the difference between two population means and explain what it means&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4865,&quot;3&quot;:{&quot;1&quot;:0},&quot;11&quot;:3,&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Calculate and explain a confidence interval for the difference between two population means.<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Independent vs. Paired Samples<\/h2>\n<p class=\"para\">When you are interested in estimating a difference in population means, you usually start with data from samples from each of the populations of interest. There are two different strategies for selecting the two samples. One strategy is to select a sample from one population and then independently select a sample from the second population. Using this strategy results in two samples where the individuals selected for the first sample do not influence the individuals selected for the second sample. This would be the case if you took a random sample from each population. Samples selected in this way are said to be <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_1300_5622\">independent samples<\/a>.<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm979\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=979&theme=lumen&iframe_resize_id=ohm979&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p class=\"para\">The second strategy is to select samples where each observation in one sample is paired in a logical way with a particular observation in the second sample. In the question above, the observations would be paired by student\u2014there is a before keyboarding class typing speed and an after keyboarding class typing speed for each student. If samples are chosen in a way that results in the observations in one sample being paired with the observations in the other sample, the samples are said to be <b>paired samples <\/b>or <strong>matched pairs<\/strong>. Paired samples are also sometimes called <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_1300_5623\">dependent samples<\/a>.<\/p>\n<p class=\"para\">One common process that results in paired samples is when data are collected both before and after some intervention (like the keyboarding class). But there are other data collection methods that can result in paired samples. One example would be if participants in a study to evaluate the effect of exercise (light vs. moderate exercise) were paired by weight prior to the study, and then one person from each pair was assigned to each exercise group. This would result in exercise groups that were similar with respect to weight, and the two samples would be paired because there is a logical way to match an observation from the light exercise group with a particular observation from the moderate exercise group.<\/p>\n<p>It is important to make a distinction between independent samples and paired samples because the way the data from the samples are analyzed is different for these two cases.<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm980\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=980&theme=lumen&iframe_resize_id=ohm980&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<div class=\"glossary\"><span class=\"screen-reader-text\" id=\"definition\">definition<\/span><template id=\"term_1300_5622\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_1300_5622\"><div tabindex=\"-1\"><p>Independent samples refer to a situation in statistics where the observations or measurements in one sample are not related or influenced by the observations or measurements in another sample.<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><template id=\"term_1300_5623\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_1300_5623\"><div tabindex=\"-1\"><p>Dependent samples, also known as paired samples, refer to a situation in statistics where the observations or measurements in one sample are related or connected to the observations or measurements in another sample.<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><\/div>","protected":false},"author":8,"menu_order":28,"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":"learn_it","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\/1300"}],"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\/1300\/revisions"}],"predecessor-version":[{"id":6796,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1300\/revisions\/6796"}],"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\/1300\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1300"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1300"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1300"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1300"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}