Find the critical value za\/2 that corresponds to the given confidence level.
80%
Za\/2<\/p>\n\n\n\n
The correct answer and explanation is :<\/strong><\/mark><\/p>\n\n\n\n Let’s solve this carefully:<\/p>\n\n\n\n The confidence level given is 80%<\/strong>.<\/p>\n\n\n\n The critical value ( z_{\\alpha\/2} ) corresponds to the value on the standard normal distribution such that the area between (-z_{\\alpha\/2}) and (z_{\\alpha\/2}) is the confidence level. In other words, you have 80% of the distribution’s area between (-z_{\\alpha\/2}) and (z_{\\alpha\/2}), leaving 20% of the area in the two tails combined (because 100% – 80% = 20%).<\/p>\n\n\n\n Since the two tails share the 20% equally, each tail has an area of:<\/p>\n\n\n\n [ So, the area to the left of ( z_{\\alpha\/2} ) must be:<\/p>\n\n\n\n [ Thus, we need to find the z-score<\/strong> that corresponds to a cumulative area of 0.90<\/strong>.<\/p>\n\n\n\n Using a standard normal distribution table<\/strong> or a calculator, the z-score for an area of 0.90 is approximately:<\/p>\n\n\n\n [ \u2705 Final Answer:<\/strong><\/p>\n\n\n\n [ The concept of a critical value<\/strong> ( z_{\\alpha\/2} ) is fundamental when working with confidence intervals, particularly in statistics and inferential analysis. The critical value represents the number of standard deviations a data point is from the mean in a normal distribution. For a given confidence level, it marks the point beyond which a certain percentage of the distribution lies in the tails.<\/p>\n\n\n\n In this question, the confidence level is 80%, meaning we want the middle 80% of the normal distribution to be captured between (-z_{\\alpha\/2}) and (z_{\\alpha\/2}). To find ( z_{\\alpha\/2} ), we must first determine the total area left in the two tails of the distribution, which is 100% – 80% = 20%. Since the normal distribution is symmetric, this 20% is divided equally between the left and right tails, meaning each tail has 10% of the area.<\/p>\n\n\n\n To find the correct critical value, we look for the z-score that corresponds to an area of 90% (0.90) to the left of it. In a z-table (standard normal table), 0.90 corresponds approximately to a z-score of 1.28. This means 90% of the distribution lies to the left of 1.28, leaving 10% to the right, which matches the desired division of the tails.<\/p>\n\n\n\n Understanding critical values like ( z_{\\alpha\/2} ) is crucial because they directly influence the width of confidence intervals. A lower confidence level (like 80%) results in a smaller critical value and a narrower confidence interval, while a higher confidence level (like 95% or 99%) leads to a larger critical value and a wider interval. This trade-off is important in research, where precision and certainty must be balanced.<\/p>\n\n\n\n Thus, for an 80% confidence level, the corresponding critical value ( z_{\\alpha\/2} ) is 1.28<\/strong>.<\/p>\n","protected":false},"excerpt":{"rendered":" Find the critical value za\/2 that corresponds to the given confidence level.80%Za\/2 The correct answer and explanation is : Let’s solve this carefully: The confidence level given is 80%. The critical value ( z_{\\alpha\/2} ) corresponds to the value on the standard normal distribution such that the area between (-z_{\\alpha\/2}) and (z_{\\alpha\/2}) is the confidence […]<\/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-209239","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/209239","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=209239"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/209239\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=209239"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=209239"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=209239"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\\frac{20\\%}{2} = 10\\% = 0.10
]<\/p>\n\n\n\n
1 – 0.10 = 0.90
]<\/p>\n\n\n\n
z_{\\alpha\/2} = 1.28
]<\/p>\n\n\n\n
z_{\\alpha\/2} = 1.28
]<\/p>\n\n\n\n
\n\n\n\nDetailed Explanation (300+ Words):<\/h3>\n\n\n\n