{"id":151686,"date":"2024-10-10T11:25:47","date_gmt":"2024-10-10T11:25:47","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=151686"},"modified":"2024-10-10T11:25:50","modified_gmt":"2024-10-10T11:25:50","slug":"based-on-an-old-survey-46-of-blue-collar-workers-do-not-contribute-money-to-the-401k-plan-at-their-job-for-this-year-you-would-like-to-obtain-a-new-sample-of-blue-collar-workers-to-estimate-the-prop","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2024\/10\/10\/based-on-an-old-survey-46-of-blue-collar-workers-do-not-contribute-money-to-the-401k-plan-at-their-job-for-this-year-you-would-like-to-obtain-a-new-sample-of-blue-collar-workers-to-estimate-the-prop\/","title":{"rendered":"Based on an old survey 46% of blue collar workers do not contribute money to the 401k plan at their job. for this year, you would like to obtain a new sample of blue collar workers to estimate the proportion who do not contribute to their 401k accounts."},"content":{"rendered":"\n<p>Based on an old survey 46% of blue collar workers do not contribute money to the 401k plan at their job. for this year, you would like to obtain a new sample of blue collar workers to estimate the proportion who do not contribute to their 401k accounts. You would like to be 98% confident that your estimate is within 4.6% of the true population proportion. How large of a sample size is required?<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\">The correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To determine the required sample size for estimating the proportion of blue-collar workers who do not contribute to their 401(k) accounts with a specific level of confidence and margin of error, we can use the formula for sample size in proportion estimates:<\/p>\n\n\n\n<p>[<br>n = \\left( \\frac{Z^2 \\cdot p \\cdot (1 &#8211; p)}{E^2} \\right)<br>]<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( n ) is the required sample size.<\/li>\n\n\n\n<li>( Z ) is the Z-score corresponding to the desired confidence level (98% confidence level).<\/li>\n\n\n\n<li>( p ) is the estimated population proportion.<\/li>\n\n\n\n<li>( E ) is the margin of error.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Identify the values<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Confidence level = 98%. The Z-score for a 98% confidence interval can be found using statistical tables or a calculator. It is approximately 2.33.<\/li>\n\n\n\n<li>Estimated population proportion, ( p ) = 0.46 (from the old survey).<\/li>\n\n\n\n<li>Margin of error, ( E ) = 0.046 (or 4.6%).<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Substitute into the formula<\/h3>\n\n\n\n<p>Now we can plug these values into the formula:<\/p>\n\n\n\n<p>[<br>n = \\left( \\frac{2.33^2 \\cdot 0.46 \\cdot (1 &#8211; 0.46)}{0.046^2} \\right)<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Calculate the required sample size<\/h3>\n\n\n\n<p>First, calculate the components:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( 2.33^2 = 5.4289 )<\/li>\n\n\n\n<li>( 0.46 \\cdot (1 &#8211; 0.46) = 0.46 \\cdot 0.54 = 0.2484 )<\/li>\n\n\n\n<li>( 0.046^2 = 0.002116 )<\/li>\n<\/ul>\n\n\n\n<p>Now, calculate ( n ):<\/p>\n\n\n\n<p>[<br>n = \\left( \\frac{5.4289 \\cdot 0.2484}{0.002116} \\right)<br>]<\/p>\n\n\n\n<p>[<br>n = \\left( \\frac{1.34887756}{0.002116} \\right) \\approx 637.38<br>]<\/p>\n\n\n\n<p>Since the sample size must be a whole number, round up to the nearest whole number:<\/p>\n\n\n\n<p>[<br>n \\approx 638<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p>To estimate the proportion of blue-collar workers who do not contribute to their 401(k) accounts within a 4.6% margin of error and with 98% confidence, we need a sample size of approximately 638 workers. The formula for calculating sample size in proportion estimation involves the Z-score, which accounts for the level of confidence, the estimated proportion based on previous data, and the desired margin of error. A higher confidence level (98%) requires a larger sample size than a lower confidence level, as it ensures that the estimate will likely be within the specified margin of error. In this case, using a Z-score of 2.33 and the estimated population proportion of 46%, the calculation shows that surveying 638 workers will give a precise estimate of the true proportion within the desired margin of 4.6%.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Based on an old survey 46% of blue collar workers do not contribute money to the 401k plan at their job. for this year, you would like to obtain a new sample of blue collar workers to estimate the proportion who do not contribute to their 401k accounts. You would like to be 98% confident [&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-151686","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/151686","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=151686"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/151686\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=151686"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=151686"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=151686"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}