{"id":270199,"date":"2025-07-25T05:24:08","date_gmt":"2025-07-25T05:24:08","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=270199"},"modified":"2025-07-25T05:24:10","modified_gmt":"2025-07-25T05:24:10","slug":"the-state-transportation-department-is-conducting-a-study-on-the-duration-of-green-lights-in-a-certain-city","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/25\/the-state-transportation-department-is-conducting-a-study-on-the-duration-of-green-lights-in-a-certain-city\/","title":{"rendered":"The state transportation department is conducting a study on the duration of green lights in a certain city"},"content":{"rendered":"\n

The state transportation department is conducting a study on the duration of green lights in a certain city. The green lights’ durations are normally distributed with a mean of 45 seconds and a standard deviation of 15 seconds. Use a calculator to find the duration that separates the lowest 24% of the means from the highest 76% in a sampling distribution of 75 traffic lights. Round \u00afx to two decimal places.<\/p>\n\n\n\n

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

To find the duration that separates the lowest 24% from the highest 76% in the sampling distribution of 75 traffic lights, we are looking for a certain percentile in a sampling distribution<\/strong>. In this case, we need to find the 24th percentile<\/strong> (because the lowest 24% corresponds to this point).<\/p>\n\n\n\n

Given:<\/h3>\n\n\n\n
    \n
  • Mean (\u03bc\\mu\u03bc) = 45 seconds<\/li>\n\n\n\n
  • Standard deviation (\u03c3\\sigma\u03c3) = 15 seconds<\/li>\n\n\n\n
  • Sample size (n) = 75<\/li>\n\n\n\n
  • We are looking for the point where 24% of the sample means lie below it (i.e., the 24th percentile).<\/li>\n<\/ul>\n\n\n\n

    Step 1: Understand the Sampling Distribution<\/h3>\n\n\n\n

    The sampling distribution<\/strong> of the sample mean (x\u203e\\overline{x}x) is normally distributed with:<\/p>\n\n\n\n

      \n
    • Mean of the sample means: \u03bcx\u203e=\u03bc=45\\mu_{\\overline{x}} = \\mu = 45\u03bcx\u200b=\u03bc=45 seconds.<\/li>\n\n\n\n
    • Standard deviation of the sample means (also called the standard error):<\/li>\n<\/ul>\n\n\n\n

      \u03c3x\u203e=\u03c3n=1575\u22481.732\\sigma_{\\overline{x}} = \\frac{\\sigma}{\\sqrt{n}} = \\frac{15}{\\sqrt{75}} \\approx 1.732\u03c3x\u200b=n\u200b\u03c3\u200b=75\u200b15\u200b\u22481.732<\/p>\n\n\n\n

      Step 2: Find the Z-Score for the 24th Percentile<\/h3>\n\n\n\n

      To find the point that separates the lowest 24% from the highest 76%, we need to find the Z-score corresponding to the 24th percentile<\/strong>. Using a Z-table or a calculator, the Z-score for the 24th percentile is approximately:Z=\u22120.705Z = -0.705Z=\u22120.705<\/p>\n\n\n\n

      Step 3: Convert the Z-Score to the Sample Mean<\/h3>\n\n\n\n

      The formula to convert a Z-score to a sample mean (x\u203e\\overline{x}x) is:x\u203e=\u03bcx\u203e+Z\u22c5\u03c3x\u203e\\overline{x} = \\mu_{\\overline{x}} + Z \\cdot \\sigma_{\\overline{x}}x=\u03bcx\u200b+Z\u22c5\u03c3x\u200b<\/p>\n\n\n\n

      Substituting the known values:x\u203e=45+(\u22120.705)\u22c51.732\\overline{x} = 45 + (-0.705) \\cdot 1.732x=45+(\u22120.705)\u22c51.732x\u203e=45\u22121.220\\overline{x} = 45 – 1.220x=45\u22121.220x\u203e\u224843.78\\overline{x} \\approx 43.78x\u224843.78<\/p>\n\n\n\n

      Final Answer:<\/h3>\n\n\n\n

      The duration that separates the lowest 24% from the highest 76% in the sampling distribution of 75 traffic lights is approximately 43.78 seconds<\/strong>.<\/p>\n\n\n\n

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

      This is the point in the distribution of sample means where 24% of all sample means will fall below. Since the sampling distribution is normal, we use the Z-score corresponding to the 24th percentile and apply it to the sampling distribution parameters (mean and standard deviation). This approach helps us identify thresholds for the duration of the green lights based on sampling statistics.<\/p>\n\n\n\n

      \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

      The state transportation department is conducting a study on the duration of green lights in a certain city. The green lights’ durations are normally distributed with a mean of 45 seconds and a standard deviation of 15 seconds. Use a calculator to find the duration that separates the lowest 24% of the means from the […]<\/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-270199","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/270199","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=270199"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/270199\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=270199"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=270199"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=270199"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}