{"id":266870,"date":"2025-07-23T07:52:38","date_gmt":"2025-07-23T07:52:38","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=266870"},"modified":"2025-07-23T07:52:40","modified_gmt":"2025-07-23T07:52:40","slug":"a-random-sample-of-n-25-iq-scores-is-selected-from-a-population-iq-scores","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/23\/a-random-sample-of-n-25-iq-scores-is-selected-from-a-population-iq-scores\/","title":{"rendered":"A random sample of n = 25 IQ scores is selected from a population (IQ scores"},"content":{"rendered":"\n

A random sample of n = 25 IQ scores is selected from a population (IQ scores: mean = 100, standard deviation = 15). What is the probability that the sample mean will be less than 103?<\/p>\n\n\n\n

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

To calculate the probability that the sample mean will be less than 103 for a random sample of 25 IQ scores, we can use the sampling distribution of the sample mean<\/strong>. The steps are as follows:<\/p>\n\n\n\n

Step 1: Identify the known values<\/h3>\n\n\n\n
    \n
  • Population mean (\u03bc) = 100<\/li>\n\n\n\n
  • Population standard deviation (\u03c3) = 15<\/li>\n\n\n\n
  • Sample size (n) = 25<\/li>\n\n\n\n
  • Sample mean (M) = 103<\/li>\n<\/ul>\n\n\n\n

    Step 2: Calculate the standard error of the sample mean<\/h3>\n\n\n\n

    The standard error (SE) of the sample mean is calculated by the formula:SE=\u03c3nSE = \\frac{\\sigma}{\\sqrt{n}}SE=n\u200b\u03c3\u200b<\/p>\n\n\n\n

    Substitute the known values:SE=1525=155=3SE = \\frac{15}{\\sqrt{25}} = \\frac{15}{5} = 3SE=25\u200b15\u200b=515\u200b=3<\/p>\n\n\n\n

    Step 3: Standardize the sample mean<\/h3>\n\n\n\n

    To find the probability that the sample mean is less than 103, we need to standardize the sample mean (i.e., convert it into a z-score). The formula for the z-score is:z=M\u2212\u03bcSEz = \\frac{M – \\mu}{SE}z=SEM\u2212\u03bc\u200b<\/p>\n\n\n\n

    Substitute the values:z=103\u22121003=33=1z = \\frac{103 – 100}{3} = \\frac{3}{3} = 1z=3103\u2212100\u200b=33\u200b=1<\/p>\n\n\n\n

    Step 4: Find the probability corresponding to the z-score<\/h3>\n\n\n\n

    The z-score of 1 corresponds to a cumulative probability from the standard normal distribution. Using a z-table or a normal distribution calculator, we find:P(z<1)=0.8413P(z < 1) = 0.8413P(z<1)=0.8413<\/p>\n\n\n\n

    Conclusion:<\/h3>\n\n\n\n

    Thus, the probability that the sample mean is less than 103 is 0.8413<\/strong> or 84.13%<\/strong>.<\/p>\n\n\n\n

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

    This means that, given a random sample of 25 IQ scores, there is an 84.13% chance that the sample mean will be less than 103. This is based on the properties of the normal distribution and how the sample mean behaves relative to the population mean, with larger sample sizes resulting in a more accurate estimate of the population mean.<\/p>\n\n\n\n

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

    A random sample of n = 25 IQ scores is selected from a population (IQ scores: mean = 100, standard deviation = 15). What is the probability that the sample mean will be less than 103? The Correct Answer and Explanation is: To calculate the probability that the sample mean will be less than 103 […]<\/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-266870","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/266870","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=266870"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/266870\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=266870"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=266870"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=266870"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}