{"id":209237,"date":"2025-04-28T18:15:28","date_gmt":"2025-04-28T18:15:28","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=209237"},"modified":"2025-04-28T18:15:32","modified_gmt":"2025-04-28T18:15:32","slug":"find-the-critical-value-za-2-that-corresponds-to-a-90-confidence-level","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/04\/28\/find-the-critical-value-za-2-that-corresponds-to-a-90-confidence-level\/","title":{"rendered":"Find The Critical Value Za\/2 That Corresponds To A 90% Confidence Level"},"content":{"rendered":"\n

Find The Critical Value Za\/2 That Corresponds To A 90% Confidence Level.<\/p>\n\n\n\n

The correct answer and explanation is :<\/strong><\/mark><\/p>\n\n\n\n

Answer:<\/h3>\n\n\n\n

The critical value ( z_{\\alpha\/2} ) that corresponds to a 90% confidence level is 1.645<\/strong>.<\/p>\n\n\n\n

\n\n\n\n

Detailed Explanation:<\/h3>\n\n\n\n

In statistics, when constructing confidence intervals for a population mean (especially when the population standard deviation is known or the sample size is large), we often use the standard normal distribution<\/strong> (z-distribution). The critical value ( z_{\\alpha\/2} )<\/strong> is a key part of building these intervals. It marks the point(s) on the z-distribution that capture the middle percentage of the curve corresponding to the desired confidence level.<\/p>\n\n\n\n

Step-by-Step Process:<\/h4>\n\n\n\n
    \n
  1. Understand Confidence Level:<\/strong>
    A 90% confidence level<\/strong> means that we want the middle 90% of the normal distribution curve to fall between two z-scores, one negative and one positive (because the standard normal curve is symmetric).<\/li>\n\n\n\n
  2. Find ( \\alpha ):<\/strong>
    In statistics, ( \\alpha ) (alpha) represents the total probability of making a Type I error<\/strong> (rejecting a true null hypothesis). It is the area in the two tails of the normal distribution outside<\/em> the confidence interval.
    [
    \\alpha = 1 – \\text{Confidence Level} = 1 – 0.90 = 0.10
    ]<\/li>\n\n\n\n
  3. Find ( \\alpha\/2 ):<\/strong>
    Since the normal curve is symmetric, the probability is split equally between the two tails.
    [
    \\alpha\/2 = 0.10\/2 = 0.05
    ]
    So, each tail will have 5% of the area.<\/li>\n\n\n\n
  4. Find ( z_{\\alpha\/2} ):<\/strong>
    We now need to find the z-value such that 5%<\/strong> of the area is to the right of it. Equivalently, 95%<\/strong> of the area is to the left of it (because 100% – 5% = 95%).<\/li>\n\n\n\n
  5. Using the Z-table:<\/strong>
    Using a standard normal distribution table (or a calculator), we find that the z-score corresponding to an area of 0.9500<\/strong> to the left is approximately 1.645<\/strong>.<\/li>\n<\/ol>\n\n\n\n

    Important Concept:<\/h4>\n\n\n\n
      \n
    • A 90% confidence interval captures the middle 90% of data, meaning that 5% of the data lies in each tail.<\/li>\n\n\n\n
    • ( z_{\\alpha\/2} ) is positive 1.645<\/strong> for the right side and negative 1.645<\/strong> for the left side.<\/li>\n\n\n\n
    • In confidence interval formulas, we typically use the positive critical value.<\/li>\n<\/ul>\n\n\n\n

      Final Summary:<\/h4>\n\n\n\n

      Thus, for a 90% confidence level, the critical value ( z_{\\alpha\/2} ) is 1.645<\/strong>.
      This value is used to calculate margins of error and create confidence intervals for population parameters when the data distribution is normal or approximately normal.<\/p>\n\n\n\n

      \n","protected":false},"excerpt":{"rendered":"

      Find The Critical Value Za\/2 That Corresponds To A 90% Confidence Level. The correct answer and explanation is : Answer: The critical value ( z_{\\alpha\/2} ) that corresponds to a 90% confidence level is 1.645. Detailed Explanation: In statistics, when constructing confidence intervals for a population mean (especially when the population standard deviation is known […]<\/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-209237","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/209237","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=209237"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/209237\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=209237"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=209237"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=209237"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}