A random sample of 16 ATM transactions at the Last National Bank of Flat Rock revealed a mean transaction time of 2.8 minutes with a standard deviation of 1.2 minutes. The width (in minutes) of the 95 percent confidence interval for the true mean transaction time is:<\/p>\n\n\n\n
A. \u00b1 0.639
B. \u00b1 0.588
C. \u00b1 0.300
D. \u00b1 2.131<\/p>\n\n\n\n
The correct answer and explanation is :<\/strong><\/mark><\/p>\n\n\n\n To determine the width<\/strong> of a 95% confidence interval for the true mean transaction time<\/strong>, we use the t-distribution<\/strong> because the sample size is small (n < 30) and the population standard deviation is unknown.<\/p>\n\n\n\n Since the sample size is 16, degrees of freedom (df) = ( n – 1 = 15 ).<\/p>\n\n\n\n From the t-distribution table<\/strong>, the critical value for a 95% confidence interval and 15 degrees of freedom is approximately:<\/p>\n\n\n\n [ [ [ So the width<\/strong> of the confidence interval is:<\/p>\n\n\n\n [ A. \u00b1 0.639<\/strong><\/p>\n\n\n\n A confidence interval<\/strong> provides a range of values within which we believe the true population mean<\/strong> lies, with a certain level of confidence (95% in this case). The formula for the confidence interval when the population standard deviation is unknown uses the t-distribution<\/strong>, which adjusts for sample variability.<\/p>\n\n\n\n The width<\/strong> of this interval is determined by multiplying the standard error<\/strong> by the appropriate t-critical value<\/strong>, which depends on both the confidence level and the degrees of freedom. A smaller sample size results in a larger t-value, reflecting increased uncertainty.<\/p>\n\n\n\n The result ( \\pm 0.639 ) means we are 95% confident that the true mean ATM transaction time lies within 0.639 minutes above or below the sample mean of 2.8 minutes.<\/p>\n","protected":false},"excerpt":{"rendered":" A random sample of 16 ATM transactions at the Last National Bank of Flat Rock revealed a mean transaction time of 2.8 minutes with a standard deviation of 1.2 minutes. The width (in minutes) of the 95 percent confidence interval for the true mean transaction time is: A. \u00b1 0.639B. \u00b1 0.588C. \u00b1 0.300D. \u00b1 […]<\/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-210415","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/210415","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=210415"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/210415\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=210415"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=210415"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=210415"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n\n\n\nStep 1: Given Data<\/strong><\/h3>\n\n\n\n
\n
\n\n\n\nStep 2: Determine the t-score<\/strong><\/h3>\n\n\n\n
t_{0.025, 15} = 2.131
]<\/p>\n\n\n\n
\n\n\n\nStep 3: Compute the Standard Error (SE)<\/strong><\/h3>\n\n\n\n
SE = \\frac{s}{\\sqrt{n}} = \\frac{1.2}{\\sqrt{16}} = \\frac{1.2}{4} = 0.3
]<\/p>\n\n\n\n
\n\n\n\nStep 4: Compute the Margin of Error (ME)<\/strong><\/h3>\n\n\n\n
ME = t \\times SE = 2.131 \\times 0.3 = 0.6393 \\approx 0.639
]<\/p>\n\n\n\n
\\pm 0.639 \\text{ minutes}
]<\/p>\n\n\n\n
\n\n\n\nFinal Answer:<\/strong><\/h3>\n\n\n\n
\n\n\n\nExplanation:<\/strong><\/h3>\n\n\n\n