{"id":213043,"date":"2025-05-09T08:23:48","date_gmt":"2025-05-09T08:23:48","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=213043"},"modified":"2025-05-09T08:23:50","modified_gmt":"2025-05-09T08:23:50","slug":"a-newspaper-reporter-asked-an-srs-of-100-residents-in-a-large-city-for-their-opinion-about-the-mayors-job-performance","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/09\/a-newspaper-reporter-asked-an-srs-of-100-residents-in-a-large-city-for-their-opinion-about-the-mayors-job-performance\/","title":{"rendered":"A newspaper reporter asked an SRS of 100 residents in a large city for their opinion about the mayor\u2019s job performance"},"content":{"rendered":"\n

A newspaper reporter asked an SRS of 100 residents in a large city for their opinion about the mayor\u2019s job performance. Using the results from the sample, the C% confidence interval for the proportion of all residents in the city who approve of the mayor\u2019s job performance is 0.565 to 0.695. What is the value of C?<\/p>\n\n\n\n

Do your best to show me an equation that you used to determine the value of C.<\/p>\n\n\n\n

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

To determine the value of C<\/strong>, which represents the confidence level<\/strong> of the given interval, we need to understand how confidence intervals are constructed and how we can work backward from a confidence interval to find C<\/strong>.<\/p>\n\n\n\n

\n\n\n\n

Given:<\/strong><\/h3>\n\n\n\n
    \n
  • Sample size n=100n = 100<\/li>\n\n\n\n
  • Confidence interval for population proportion pp: [0.565, 0.695]<\/li>\n\n\n\n
  • Sample proportion p^=0.565+0.6952=0.63\\hat{p} = \\frac{0.565 + 0.695}{2} = 0.63<\/li>\n\n\n\n
  • Margin of error ME=0.695\u22120.63=0.065ME = 0.695 – 0.63 = 0.065<\/li>\n<\/ul>\n\n\n\n
    \n\n\n\n

    Step 1: Use the formula for Margin of Error<\/strong><\/h3>\n\n\n\n

    The formula for the margin of error for a population proportion is: ME=z\u2217\u22c5p^(1\u2212p^)nME = z^* \\cdot \\sqrt{\\frac{\\hat{p}(1 – \\hat{p})}{n}}<\/p>\n\n\n\n

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

      \n
    • z\u2217z^* is the critical value<\/strong> (z-score) corresponding to the confidence level CC,<\/li>\n\n\n\n
    • p^=0.63\\hat{p} = 0.63 (sample proportion),<\/li>\n\n\n\n
    • n=100n = 100,<\/li>\n\n\n\n
    • ME=0.065ME = 0.065<\/li>\n<\/ul>\n\n\n\n
      \n\n\n\n

      Step 2: Plug in known values and solve for z\u2217z^*<\/strong><\/h3>\n\n\n\n

      0.065=z\u2217\u22c50.63(1\u22120.63)1000.065 = z^* \\cdot \\sqrt{\\frac{0.63(1 – 0.63)}{100}} 0.065=z\u2217\u22c50.2331100=z\u2217\u22c50.0023310.065 = z^* \\cdot \\sqrt{\\frac{0.2331}{100}} = z^* \\cdot \\sqrt{0.002331} 0.065=z\u2217\u22c50.048280.065 = z^* \\cdot 0.04828 z\u2217=0.0650.04828\u22481.346z^* = \\frac{0.065}{0.04828} \\approx 1.346<\/p>\n\n\n\n

      \n\n\n\n

      Step 3: Find the confidence level corresponding to z\u2217=1.346z^* = 1.346<\/strong><\/h3>\n\n\n\n

      A z-score of 1.346<\/strong> corresponds to a cumulative probability of approximately 0.910<\/strong> (91.0%) on one side. Since confidence intervals are two-tailed: C=2\u22c50.910\u22121=0.82 or 82%C = 2 \\cdot 0.910 – 1 = 0.82 \\text{ or } 82\\%<\/p>\n\n\n\n

      \n\n\n\n

      \u2705 Correct Answer: 82% Confidence Level<\/strong><\/h3>\n\n\n\n
      \n\n\n\n

      Explanation (300+ words)<\/strong><\/h3>\n\n\n\n

      A confidence interval is used in statistics to estimate a range within which a population parameter (like a proportion or mean) is likely to fall. The confidence level (C)<\/strong> indicates how certain we are that the true parameter lies within this interval. In this problem, a simple random sample (SRS) of 100 residents was surveyed, and the confidence interval for the proportion who approve of the mayor\u2019s performance was calculated to be between 0.565 and 0.695<\/strong>.<\/p>\n\n\n\n

      To find the confidence level C<\/strong>, we first determine the sample proportion<\/strong> (p^\\hat{p}) by averaging the endpoints of the interval: p^=0.565+0.6952=0.63\\hat{p} = \\frac{0.565 + 0.695}{2} = 0.63<\/p>\n\n\n\n

      Next, we calculate the margin of error (ME)<\/strong>, which is the distance from the sample proportion to either endpoint of the interval: ME=0.695\u22120.63=0.065ME = 0.695 – 0.63 = 0.065<\/p>\n\n\n\n

      The formula for the margin of error for a proportion is: ME=z\u2217\u22c5p^(1\u2212p^)nME = z^* \\cdot \\sqrt{\\frac{\\hat{p}(1 – \\hat{p})}{n}}<\/p>\n\n\n\n

      Using the values of p^\\hat{p} and n=100n = 100, we plug in and solve for the critical value z\u2217z^*, which turns out to be approximately 1.346<\/strong>. A z-score of 1.346 corresponds to a cumulative probability of about 91.0%. Because this is a two-tailed interval, the total confidence level is: C=2\u22c50.91\u22121=0.82C = 2 \\cdot 0.91 – 1 = 0.82<\/p>\n\n\n\n

      Thus, the confidence level for this interval is 82%<\/strong>, meaning that if we repeated this sampling method many times, about 82% of the intervals we compute would contain the true proportion of city residents who approve of the mayor\u2019s performance.<\/p>\n","protected":false},"excerpt":{"rendered":"

      A newspaper reporter asked an SRS of 100 residents in a large city for their opinion about the mayor\u2019s job performance. Using the results from the sample, the C% confidence interval for the proportion of all residents in the city who approve of the mayor\u2019s job performance is 0.565 to 0.695. What is the value […]<\/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-213043","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/213043","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=213043"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/213043\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=213043"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=213043"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=213043"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}