A civil engineer is analyzing the compressive strength of concrete. Compressive strength is approximately normally distributed with variance \u03c32 = 1000 psi2. A random sample of 12 specimens has a mean compressive strength of<\/p>\n\n\n\n
(a) Test the hypothesis that mean compressive strength is 3500 psi. Use a fixed-level test with \u03b1 = 0.01.<\/p>\n\n\n\n
(b) What is the smallest level of significance at which you would be willing to reject the null hypothesis?<\/p>\n\n\n\n
(c) Construct a 95% two-sided CI on mean compressive strength.<\/p>\n\n\n\n
(d) Construct a 99% two-sided CI on mean compressive strength. Compare the width of this confidence interval with the width of the one found in part (c). Comment.<\/p>\n\n\n\n
The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n To address the given statistical problems, we need the sample mean ((\\bar{X})) of the 12 concrete specimens. However, since it is not provided, we will express the solutions in terms of (\\bar{X}).<\/p>\n\n\n\n (a) Hypothesis Testing at (\\alpha = 0.01):<\/strong><\/p>\n\n\n\n We aim to test the null hypothesis (H_0: \\mu = 3500) psi against the alternative (H_1: \\mu \\neq 3500) psi. Given the population variance (\\sigma^2 = 1000) psi\u00b2 and sample size (n = 12), the standard error (SE) is:<\/p>\n\n\n\n [ The test statistic (z) is calculated as:<\/p>\n\n\n\n [ For a two-tailed test at (\\alpha = 0.01), the critical z-values are (\\pm 2.576). If (z) falls outside this range, we reject (H_0). Thus, we reject (H_0) if:<\/p>\n\n\n\n [ (b) Smallest Significance Level ((p)-value):<\/strong><\/p>\n\n\n\n The (p)-value is calculated as:<\/p>\n\n\n\n [ Where (Z) follows the standard normal distribution. The smallest significance level at which (H_0) would be rejected corresponds to this (p)-value.<\/p>\n\n\n\n (c) 95% Confidence Interval (CI) for (\\mu):<\/strong><\/p>\n\n\n\n A 95% CI for the mean is given by:<\/p>\n\n\n\n [ This simplifies to:<\/p>\n\n\n\n [ (d) 99% Confidence Interval (CI) for (\\mu):<\/strong><\/p>\n\n\n\n A 99% CI is:<\/p>\n\n\n\n [ This simplifies to:<\/p>\n\n\n\n [ Comparison of CI Widths:<\/strong><\/p>\n\n\n\n The width of the 95% CI is (2 \\times 17.89 = 35.78) psi, while the 99% CI width is (2 \\times 23.53 = 47.06) psi. The 99% CI is wider, reflecting increased uncertainty due to the higher confidence level. This trade-off between confidence level and interval precision is typical in statistical analyses.<\/p>\n\n\n\n Conclusion:<\/strong><\/p>\n\n\n\n Without the sample mean, we cannot compute exact numerical results. However, the provided formulas allow for substitution of (\\bar{X}) to obtain specific outcomes.<\/p>\n","protected":false},"excerpt":{"rendered":" A civil engineer is analyzing the compressive strength of concrete. Compressive strength is approximately normally distributed with variance \u03c32 = 1000 psi2. A random sample of 12 specimens has a mean compressive strength of (a) Test the hypothesis that mean compressive strength is 3500 psi. Use a fixed-level test with \u03b1 = 0.01. (b) What […]<\/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-185328","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/185328","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=185328"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/185328\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=185328"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=185328"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=185328"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\\text{SE} = \\sqrt{\\frac{\\sigma^2}{n}} = \\sqrt{\\frac{1000}{12}} \\approx 9.13 \\text{ psi}
]<\/p>\n\n\n\n
z = \\frac{\\bar{X} – 3500}{\\text{SE}} = \\frac{\\bar{X} – 3500}{9.13}
]<\/p>\n\n\n\n
|\\bar{X} – 3500| > 2.576 \\times 9.13 \\approx 23.53 \\text{ psi}
]<\/p>\n\n\n\n
p\\text{-value} = 2 \\times P(Z > |z|)
]<\/p>\n\n\n\n
\\bar{X} \\pm z_{\\alpha\/2} \\times \\text{SE} = \\bar{X} \\pm 1.96 \\times 9.13
]<\/p>\n\n\n\n
\\bar{X} \\pm 17.89 \\text{ psi}
]<\/p>\n\n\n\n
\\bar{X} \\pm z_{\\alpha\/2} \\times \\text{SE} = \\bar{X} \\pm 2.576 \\times 9.13
]<\/p>\n\n\n\n
\\bar{X} \\pm 23.53 \\text{ psi}
]<\/p>\n\n\n\n