{"id":3370,"date":"2015-04-12T00:11:44","date_gmt":"2015-04-12T05:11:44","guid":{"rendered":"http:\/\/www.ssc.wisc.edu\/~jfrees\/?page_id=3370"},"modified":"2015-08-17T14:41:11","modified_gmt":"2015-08-17T19:41:11","slug":"confidence-intervals","status":"publish","type":"page","link":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/regression\/basic-linear-regression\/2-5-statistical-inference\/confidence-intervals\/","title":{"rendered":"Confidence Intervals"},"content":{"rendered":"<p> Investigators often cite the formal hypothesis testing mechanism to respond to the question &#8220;Does the explanatory variable have a real influence on the response?&#8221; A natural follow-up question is &#8220;To what extent does <em>x<\/em> affect <em>y<\/em>?&#8221; To a certain degree, one could respond using the size of the <em>t<\/em>-ratio or the <em>p<\/em>-value. However, in many instances a <em>confidence interval<\/em> for the slope is more useful. <\/p>\n<p> To introduce confidence intervals for the slope, recall that \\(b_1\\) is our point estimator of the true, unknown slope \\(\\beta_1\\). Section 2.4 argued that this estimator has standard error \\(se(b_1)\\) and that \\(\\left( b_1-\\beta_1\\right) \/se(b_1)\\) follows a <em>t<\/em>-distribution with <em>n<\/em>-2 degrees of freedom. Probability statements can be inverted to yield confidence intervals. Using this logic, we have the following confidence interval for the slope \\(\\beta_1\\). <\/p>\n<p> <div class=\"scbb-content-box scbb-content-box-gray\"><em>Definition<\/em>. A \\(100(1-\\alpha)\\)% confidence interval for the slope \\(\\beta_1\\) is \\begin{equation}\\label{E2:ConfIntb1} b_1\\pm t_{n-2,1-\\alpha \/2} ~se(b_1). \\end{equation} <\/div><br \/>\n As with hypothesis testing, \\(t_{n-2,1-\\alpha \/2}\\) is the (1-\\( \\alpha \\)\/2)th percentile from the <em>t<\/em>-distribution with <em>df=n<\/em>-2 degrees of freedom. Because of the two-sided nature of confidence intervals, the percentile is 1 &#8211; (1 &#8211; confidence level) \/ 2. In this text, for notational simplicity we generally use a 95% confidence interval, so the percentile is 1-(1-.0.95)\/2 = 0.975. The confidence interval provides a range of reliability that measures the usefulness of the estimate. <\/p>\n<p> In Section 2.1, we established that the least squares slope estimate for the lottery sales example is \\(b_1\\)=0.647. The interpretation is that if a zip code&#8217;s population differs by 1,000, then we expect mean lottery sales to differ by $647. How reliable is this estimate? It turns out that \\( se(b_1)=0.0488\\) and thus an approximate 95% confidence interval for the slope is \\begin{equation*} 0.647\\pm (2.011)(.0488), \\end{equation*} or (0.549, 0.745). Similarly, if population differs by 1,000, a 95% confidence interval for the expected change in sales is (549, 745). Here, we use the \\(t\\)-value \\(t_{48,0.975}=2.011\\) because there are 48 (= <em>n<\/em>-2) degrees of freedom and, for a 95% confidence interval, we need the 97.5th percentile. <\/p>\n<p><div class=\"alignleft\"><a href=\"https:\/\/users.ssc.wisc.edu\/~ewfrees\/regression\/basic-linear-regression\/2-5-statistical-inference\/is-the-explanatory-variable-important-the-t-test\/\" title=\"Is the Explanatory Variable Important?: The <em>t<\/em>-Test\">&#9668 Previous page<\/a><\/div><div class=\"alignright\"><a href=\"https:\/\/users.ssc.wisc.edu\/~ewfrees\/regression\/basic-linear-regression\/2-5-statistical-inference\/prediction-intervals\/\" title=\"Prediction Intervals\">Next page &#9658<\/a><\/div><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Investigators often cite the formal hypothesis testing mechanism to respond to the question &#8220;Does the explanatory variable have a real influence on the response?&#8221; A natural follow-up question is &#8220;To what extent does x affect &hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":3343,"menu_order":2,"comment_status":"closed","ping_status":"closed","template":"","meta":{"jetpack_post_was_ever_published":false},"jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/P8cLPd-Sm","acf":[],"_links":{"self":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/3370"}],"collection":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/comments?post=3370"}],"version-history":[{"count":6,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/3370\/revisions"}],"predecessor-version":[{"id":4894,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/3370\/revisions\/4894"}],"up":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/3343"}],"wp:attachment":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/media?parent=3370"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}