{"id":210412,"date":"2025-04-30T09:16:12","date_gmt":"2025-04-30T09:16:12","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=210412"},"modified":"2025-04-30T09:16:14","modified_gmt":"2025-04-30T09:16:14","slug":"in-constructing-a-confidence-interval-for-a-mean-with-unknown-variance-with-a-sample-of-25-items","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/04\/30\/in-constructing-a-confidence-interval-for-a-mean-with-unknown-variance-with-a-sample-of-25-items\/","title":{"rendered":"In constructing a confidence interval for a mean with unknown variance with a sample of 25 items"},"content":{"rendered":"\n

In constructing a confidence interval for a mean with unknown variance with a sample of 25 items, Bob used z instead of t. “Well, at least my interval will be wider than necessary, so it was a conservative error,” said he. Is Bob’s statement correct?<\/p>\n\n\n\n

A. Yes.
B. No.
C. It depends on \u00b5.<\/p>\n\n\n\n

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

The correct answer is B. No.<\/strong><\/p>\n\n\n\n

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

In constructing a confidence interval for a population mean when the population variance is unknown, the correct approach is to use the t-distribution<\/strong>, not the z-distribution<\/strong>. This is because when the population variance is unknown, we have to estimate it from the sample, and the sample mean follows a t-distribution rather than a normal distribution.<\/p>\n\n\n\n

The general formula for a confidence interval when using the t-distribution is:<\/p>\n\n\n\n

[
\\bar{x} \\pm t_{\\alpha\/2, n-1} \\times \\frac{s}{\\sqrt{n}}
]<\/p>\n\n\n\n

Where:<\/p>\n\n\n\n

    \n
  • (\\bar{x}) is the sample mean.<\/li>\n\n\n\n
  • (t_{\\alpha\/2, n-1}) is the critical value from the t-distribution with (n-1) degrees of freedom.<\/li>\n\n\n\n
  • (s) is the sample standard deviation.<\/li>\n\n\n\n
  • (n) is the sample size.<\/li>\n\n\n\n
  • (\\alpha) is the significance level (e.g., for a 95% confidence interval, (\\alpha = 0.05)).<\/li>\n<\/ul>\n\n\n\n

    The z-distribution<\/strong>, on the other hand, is used when the population variance is known or the sample size is large (typically (n > 30), where the central limit theorem applies and the sample mean is approximately normally distributed even if the population variance is unknown).<\/p>\n\n\n\n

    Bob’s mistake was using the z-distribution<\/strong> instead of the t-distribution<\/strong>. The z-distribution assumes that the population variance is known, which is not the case here. The critical value for the t-distribution is larger than the z-value for smaller sample sizes, so using the z-distribution instead of the t-distribution results in an incorrectly narrower confidence interval<\/strong>.<\/p>\n\n\n\n

    Bob incorrectly assumed that using the z-distribution would lead to a wider interval and thus a more conservative estimate. In fact, the z-distribution would result in a narrower<\/strong> confidence interval because the critical value associated with the z-distribution is smaller than that of the t-distribution (especially for smaller sample sizes like 25). Therefore, his statement is incorrect.<\/p>\n\n\n\n

    In summary:<\/p>\n\n\n\n

      \n
    • The correct distribution to use is the t-distribution<\/strong> when the population variance is unknown and the sample size is small.<\/li>\n\n\n\n
    • Using the z-distribution<\/strong> would lead to an incorrectly narrower confidence interval, not a wider one, as Bob assumed.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"

      In constructing a confidence interval for a mean with unknown variance with a sample of 25 items, Bob used z instead of t. “Well, at least my interval will be wider than necessary, so it was a conservative error,” said he. Is Bob’s statement correct? A. Yes.B. No.C. It depends on \u00b5. The correct answer […]<\/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-210412","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/210412","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=210412"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/210412\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=210412"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=210412"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=210412"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}