{"id":265240,"date":"2025-07-21T19:49:51","date_gmt":"2025-07-21T19:49:51","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=265240"},"modified":"2025-07-21T19:49:55","modified_gmt":"2025-07-21T19:49:55","slug":"what-is-the-critical-value-of-z-for-a-99-confidence-interval-for-a-known-standard-deviation","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/21\/what-is-the-critical-value-of-z-for-a-99-confidence-interval-for-a-known-standard-deviation\/","title":{"rendered":"What is the critical value of z for a 99% confidence interval for a known standard deviation"},"content":{"rendered":"\n

What is the critical value of z for a 99% confidence interval for a known standard deviation? B) A 99% confidence interval for the mean of a population was found to be (123.33, 140.67). If the sample size was 40, find the margin of error for this estimation. C) A 95% confidence interval for a population mean is (12.9, 18.6). Which of the following is the sample mean used to calculate this interval?<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Critical Value of z for a 99% Confidence Interval<\/h3>\n\n\n\n

To find the critical value of z for a 99% confidence interval, we need to consider that a 99% confidence interval means that 99% of the data lies within the interval, and the remaining 1% is split equally on both sides of the distribution. Therefore, there is 0.5% of the data in each tail of the standard normal distribution.<\/p>\n\n\n\n

For a 99% confidence interval, we look up the z-value corresponding to a cumulative probability of 0.995 (because 0.5% is in the right tail, so 99.5% is to the left). From the standard normal distribution table or using a calculator, the z-value is approximately 2.576<\/strong>. This means the critical value of z for a 99% confidence interval is 2.576<\/strong>.<\/p>\n\n\n\n

B) Margin of Error for the Confidence Interval<\/h3>\n\n\n\n

The margin of error (ME) is the half-width of the confidence interval. It can be calculated using the formula:Margin of Error=Upper Bound\u2212Lower Bound2\\text{Margin of Error} = \\frac{\\text{Upper Bound} – \\text{Lower Bound}}{2}Margin of Error=2Upper Bound\u2212Lower Bound\u200b<\/p>\n\n\n\n

For the given confidence interval of (123.33, 140.67), the margin of error is:ME=140.67\u2212123.332=17.342=8.67\\text{ME} = \\frac{140.67 – 123.33}{2} = \\frac{17.34}{2} = 8.67ME=2140.67\u2212123.33\u200b=217.34\u200b=8.67<\/p>\n\n\n\n

Therefore, the margin of error for this estimation is 8.67<\/strong>.<\/p>\n\n\n\n

C) Sample Mean from the Confidence Interval<\/h3>\n\n\n\n

The sample mean (x\u02c9\\bar{x}x\u02c9) is the midpoint of the confidence interval. To find the sample mean, we average the upper and lower bounds of the confidence interval:x\u02c9=Upper Bound+Lower Bound2\\bar{x} = \\frac{\\text{Upper Bound} + \\text{Lower Bound}}{2}x\u02c9=2Upper Bound+Lower Bound\u200b<\/p>\n\n\n\n

For the given confidence interval of (12.9, 18.6):x\u02c9=18.6+12.92=31.52=15.75\\bar{x} = \\frac{18.6 + 12.9}{2} = \\frac{31.5}{2} = 15.75x\u02c9=218.6+12.9\u200b=231.5\u200b=15.75<\/p>\n\n\n\n

Thus, the sample mean used to calculate the 95% confidence interval is 15.75<\/strong>.<\/p>\n\n\n\n

Summary of Answers:<\/h3>\n\n\n\n
    \n
  • A)<\/strong> The critical value of z for a 99% confidence interval is 2.576<\/strong>.<\/li>\n\n\n\n
  • B)<\/strong> The margin of error for the given confidence interval is 8.67<\/strong>.<\/li>\n\n\n\n
  • C)<\/strong> The sample mean for the 95% confidence interval is 15.75<\/strong>.<\/li>\n<\/ul>\n\n\n\n
    \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

    What is the critical value of z for a 99% confidence interval for a known standard deviation? B) A 99% confidence interval for the mean of a population was found to be (123.33, 140.67). If the sample size was 40, find the margin of error for this estimation. C) A 95% confidence interval for a […]<\/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-265240","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/265240","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=265240"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/265240\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=265240"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=265240"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=265240"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}