{"id":186133,"date":"2025-01-24T06:57:57","date_gmt":"2025-01-24T06:57:57","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=186133"},"modified":"2025-01-24T06:57:59","modified_gmt":"2025-01-24T06:57:59","slug":"consider-a-binomial-experiment-with-n-20-and-p-70-2","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/24\/consider-a-binomial-experiment-with-n-20-and-p-70-2\/","title":{"rendered":"Consider a binomial experiment with n = 20 and p = .70"},"content":{"rendered":"\n<p>Consider a binomial experiment with n = 20 and p = .70.<\/p>\n\n\n\n<p>a. Compute f (12).<\/p>\n\n\n\n<p>b. Compute f (16).<\/p>\n\n\n\n<p>c. Compute P(x \u2265 16).<\/p>\n\n\n\n<p>d. Compute P(x \u2264 15).<\/p>\n\n\n\n<p>e. Compute E(x).<\/p>\n\n\n\n<p>f. Compute Var(x) and \u03c3.<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>In this problem, we&#8217;re dealing with a binomial experiment where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( n = 20 ) (number of trials),<\/li>\n\n\n\n<li>( p = 0.70 ) (probability of success).<\/li>\n<\/ul>\n\n\n\n<p>The binomial probability mass function (PMF) is given by:<\/p>\n\n\n\n<p>[<br>f(x) = P(X = x) = \\binom{n}{x} p^x (1 &#8211; p)^{n &#8211; x}<br>]<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( \\binom{n}{x} ) is the binomial coefficient, which can be calculated as ( \\frac{n!}{x!(n &#8211; x)!} ).<\/li>\n\n\n\n<li>( p^x ) is the probability of getting exactly ( x ) successes.<\/li>\n\n\n\n<li>( (1 &#8211; p)^{n &#8211; x} ) is the probability of getting exactly ( n &#8211; x ) failures.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">a. Compute ( f(12) )<\/h3>\n\n\n\n<p>To calculate ( f(12) ), use the binomial PMF formula:<\/p>\n\n\n\n<p>[<br>f(12) = \\binom{20}{12} (0.70)^{12} (0.30)^{8}<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">b. Compute ( f(16) )<\/h3>\n\n\n\n<p>Similarly, for ( f(16) ):<\/p>\n\n\n\n<p>[<br>f(16) = \\binom{20}{16} (0.70)^{16} (0.30)^{4}<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">c. Compute ( P(x \\geq 16) )<\/h3>\n\n\n\n<p>To calculate the probability that ( x \\geq 16 ), sum the probabilities from ( x = 16 ) to ( x = 20 ):<\/p>\n\n\n\n<p>[<br>P(x \\geq 16) = f(16) + f(17) + f(18) + f(19) + f(20)<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">d. Compute ( P(x \\leq 15) )<\/h3>\n\n\n\n<p>To calculate the probability that ( x \\leq 15 ), sum the probabilities from ( x = 0 ) to ( x = 15 ):<\/p>\n\n\n\n<p>[<br>P(x \\leq 15) = f(0) + f(1) + \\cdots + f(15)<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">e. Compute ( E(x) )<\/h3>\n\n\n\n<p>The expected value ( E(x) ) for a binomial distribution is given by:<\/p>\n\n\n\n<p>[<br>E(x) = n \\cdot p = 20 \\cdot 0.70 = 14<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">f. Compute ( \\text{Var}(x) ) and ( \\sigma )<\/h3>\n\n\n\n<p>The variance ( \\text{Var}(x) ) for a binomial distribution is given by:<\/p>\n\n\n\n<p>[<br>\\text{Var}(x) = n \\cdot p \\cdot (1 &#8211; p) = 20 \\cdot 0.70 \\cdot 0.30 = 4.2<br>]<\/p>\n\n\n\n<p>The standard deviation ( \\sigma ) is the square root of the variance:<\/p>\n\n\n\n<p>[<br>\\sigma = \\sqrt{4.2} \\approx 2.05<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Summary<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( f(12) ) and ( f(16) ) are computed using the binomial PMF formula.<\/li>\n\n\n\n<li>( P(x \\geq 16) ) and ( P(x \\leq 15) ) require summing appropriate probabilities.<\/li>\n\n\n\n<li>( E(x) = 14 ) and ( \\text{Var}(x) = 4.2 ), with ( \\sigma \\approx 2.05 ).<\/li>\n<\/ul>\n\n\n\n<p>These calculations give insight into the expected outcome and variability of the binomial distribution.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Consider a binomial experiment with n = 20 and p = .70. a. Compute f (12). b. Compute f (16). c. Compute P(x \u2265 16). d. Compute P(x \u2264 15). e. Compute E(x). f. Compute Var(x) and \u03c3. The Correct Answer and Explanation is : In this problem, we&#8217;re dealing with a binomial experiment where: [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-186133","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/186133","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=186133"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/186133\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=186133"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=186133"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=186133"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}