{"id":1460,"date":"2023-06-22T02:28:52","date_gmt":"2023-06-22T02:28:52","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/confidence-interval-and-prediction-interval-learn-it-2\/"},"modified":"2025-05-17T02:36:50","modified_gmt":"2025-05-17T02:36:50","slug":"confidence-interval-and-prediction-interval-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/confidence-interval-and-prediction-interval-learn-it-2\/","title":{"raw":"Confidence Interval and Prediction Interval - Learn It 2","rendered":"Confidence Interval and Prediction Interval &#8211; Learn It 2"},"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;Find and interpret the confidence interval for the mean response&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;}\">Find and interpret the confidence interval for the mean response<\/span><\/li>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Find and interpret the prediction interval for the mean response&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;}\">Find and interpret the prediction interval for an individual response<\/span><\/li>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Identify whether a confidence interval or a prediction interval is more appropriate in context of the problem&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;}\">Identify whether a confidence interval or a prediction interval is more appropriate in context of the problem<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Intervals for the Mean Response<\/h2>\r\n<p>Though the regression equation is used to calculate the expected body mass given the flipper length, we know that multiple penguins can have the same flipper length and different body masses. (This occurs quite frequently in our data set!) Therefore, if we are trying to predict the weight of an individual penguin, it makes sense to calculate an interval that takes the variability in the actual penguin weights into account.<\/p>\r\n<p>In addition, thinking about what we have learned about sample variability in previous activities, we know that if we randomly select another sample with [latex]324[\/latex] penguins, the equation of the line of best line will be different\u2014so the predicted body mass for a given flipper length (the point estimate) will change.<\/p>\r\n<p>Before calculating the interval for predicted values, however, we need to first consider the type of prediction we\u2019re most interested in obtaining.<\/p>\r\n<p>There are two ways we can use the linear regression equation:<\/p>\r\n<ol>\r\n\t<li>To estimate the <strong>mean value of the response<\/strong> when the explanatory variable is equal to a particular value, [latex]x_0[\/latex]<\/li>\r\n\t<li>To predict the <strong>value of the response for an individual observation<\/strong> when the explanatory variable is equal to [latex]x_0[\/latex]<\/li>\r\n<\/ol>\r\n<p>The type of interval calculated will depend on how we want to use the linear regression equation.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3><strong>confidence interval for the mean response<\/strong><\/h3>\r\n<p>When the objective is to estimate the mean value of the response variable for a particular value of the explanatory variable, [latex]x_0[\/latex], we will calculate a <strong>confidence interval for the mean response<\/strong>, where [latex]x_0[\/latex]\u00a0is the confidence level associated with the interval<strong>. <\/strong>This interval gives us a range of plausible values of the mean response for the subset of the population with a value of the explanatory variable equal to [latex]x_0[\/latex].<\/p>\r\n<\/section>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3><strong>prediction<\/strong><strong> interval for an individual response<\/strong><\/h3>\r\n<p>When the objective is to predict the value of the response variable for an individual observation with the explanatory variable equal to [latex]x_0[\/latex], we will calculate a [latex]C[\/latex]<strong>% prediction<\/strong><strong> interval for an individual response, <\/strong>where [latex]C[\/latex] is the confidence level associated with the interval. This interval gives us a range of plausible values of the response for an individual observation that has a value of the explanatory variable equal to [latex]x_0[\/latex].<\/p>\r\n<\/section>\r\n<section class=\"textbox interact\"><strong>Step 1: <\/strong>Access spreadsheet <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Statistics+Exemplar\/Data+Set+Files\/Penguins_FINAL.xlsx\">Penguins<\/a>.<strong><br \/>\r\nStep 2:<\/strong> <span style=\"background-color: initial; font-size: 0.9em; word-spacing: normal;\">Under \u201cEnter Data,\u201d select \u201cEnter Own.\u201d<br \/>\r\n<\/span><strong>Step 3: <\/strong>Copy and paste the appropriate explanatory variable ([latex]x[\/latex]) and response variable ([latex]y[\/latex]).<strong><br \/>\r\nStep 4: <\/strong>Check the \"Confidence\/Prediction Interval\" and input the value of the explanatory variable under<em style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial; word-spacing: normal;\"> \u201c<\/em><span style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial; word-spacing: normal;\">x-value.<\/span><em style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial; word-spacing: normal;\">\u201d<\/em><span style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial; word-spacing: normal;\"> Select the appropriate level of confidence, [latex]C[\/latex], by moving the slider.<\/span><\/section>\r\n<p><iframe src=\"https:\/\/lumen-learning.shinyapps.io\/linear_regression\/\" width=\"100%\" height=\"1050\" frameborder=\"no\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><br \/>\r\n[<a href=\"https:\/\/lumen-learning.shinyapps.io\/linear_regression\/\" 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]3063[\/ohm2_question]<\/section>\r\n<section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]3064[\/ohm2_question]<\/section>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Find and interpret the confidence interval for the mean response&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;}\">Find and interpret the confidence interval for the mean response<\/span><\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Find and interpret the prediction interval for the mean response&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;}\">Find and interpret the prediction interval for an individual response<\/span><\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Identify whether a confidence interval or a prediction interval is more appropriate in context of the problem&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;}\">Identify whether a confidence interval or a prediction interval is more appropriate in context of the problem<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Intervals for the Mean Response<\/h2>\n<p>Though the regression equation is used to calculate the expected body mass given the flipper length, we know that multiple penguins can have the same flipper length and different body masses. (This occurs quite frequently in our data set!) Therefore, if we are trying to predict the weight of an individual penguin, it makes sense to calculate an interval that takes the variability in the actual penguin weights into account.<\/p>\n<p>In addition, thinking about what we have learned about sample variability in previous activities, we know that if we randomly select another sample with [latex]324[\/latex] penguins, the equation of the line of best line will be different\u2014so the predicted body mass for a given flipper length (the point estimate) will change.<\/p>\n<p>Before calculating the interval for predicted values, however, we need to first consider the type of prediction we\u2019re most interested in obtaining.<\/p>\n<p>There are two ways we can use the linear regression equation:<\/p>\n<ol>\n<li>To estimate the <strong>mean value of the response<\/strong> when the explanatory variable is equal to a particular value, [latex]x_0[\/latex]<\/li>\n<li>To predict the <strong>value of the response for an individual observation<\/strong> when the explanatory variable is equal to [latex]x_0[\/latex]<\/li>\n<\/ol>\n<p>The type of interval calculated will depend on how we want to use the linear regression equation.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3><strong>confidence interval for the mean response<\/strong><\/h3>\n<p>When the objective is to estimate the mean value of the response variable for a particular value of the explanatory variable, [latex]x_0[\/latex], we will calculate a <strong>confidence interval for the mean response<\/strong>, where [latex]x_0[\/latex]\u00a0is the confidence level associated with the interval<strong>. <\/strong>This interval gives us a range of plausible values of the mean response for the subset of the population with a value of the explanatory variable equal to [latex]x_0[\/latex].<\/p>\n<\/section>\n<section class=\"textbox keyTakeaway\">\n<h3><strong>prediction<\/strong><strong> interval for an individual response<\/strong><\/h3>\n<p>When the objective is to predict the value of the response variable for an individual observation with the explanatory variable equal to [latex]x_0[\/latex], we will calculate a [latex]C[\/latex]<strong>% prediction<\/strong><strong> interval for an individual response, <\/strong>where [latex]C[\/latex] is the confidence level associated with the interval. This interval gives us a range of plausible values of the response for an individual observation that has a value of the explanatory variable equal to [latex]x_0[\/latex].<\/p>\n<\/section>\n<section class=\"textbox interact\"><strong>Step 1: <\/strong>Access spreadsheet <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Statistics+Exemplar\/Data+Set+Files\/Penguins_FINAL.xlsx\">Penguins<\/a>.<strong><br \/>\nStep 2:<\/strong> <span style=\"background-color: initial; font-size: 0.9em; word-spacing: normal;\">Under \u201cEnter Data,\u201d select \u201cEnter Own.\u201d<br \/>\n<\/span><strong>Step 3: <\/strong>Copy and paste the appropriate explanatory variable ([latex]x[\/latex]) and response variable ([latex]y[\/latex]).<strong><br \/>\nStep 4: <\/strong>Check the &#8220;Confidence\/Prediction Interval&#8221; and input the value of the explanatory variable under<em style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial; word-spacing: normal;\"> \u201c<\/em><span style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial; word-spacing: normal;\">x-value.<\/span><em style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial; word-spacing: normal;\">\u201d<\/em><span style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial; word-spacing: normal;\"> Select the appropriate level of confidence, [latex]C[\/latex], by moving the slider.<\/span><\/section>\n<p><iframe loading=\"lazy\" src=\"https:\/\/lumen-learning.shinyapps.io\/linear_regression\/\" width=\"100%\" height=\"1050\" frameborder=\"no\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><br \/>\n[<a href=\"https:\/\/lumen-learning.shinyapps.io\/linear_regression\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm3063\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=3063&theme=lumen&iframe_resize_id=ohm3063&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm3064\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=3064&theme=lumen&iframe_resize_id=ohm3064&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/section>\n","protected":false},"author":8,"menu_order":20,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":1438,"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\/1460"}],"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\/1460\/revisions"}],"predecessor-version":[{"id":6898,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1460\/revisions\/6898"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/1438"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1460\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1460"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1460"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1460"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1460"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}