{"id":236799,"date":"2025-06-16T14:12:09","date_gmt":"2025-06-16T14:12:09","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=236799"},"modified":"2025-06-16T14:12:11","modified_gmt":"2025-06-16T14:12:11","slug":"the-average-income-tax-refund-for-the-2009-tax-year-was-52876_-assume-the-refund-per-person-follows-the-norma-probability-distribution-with-standard-deviation-of-5971","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/16\/the-average-income-tax-refund-for-the-2009-tax-year-was-52876_-assume-the-refund-per-person-follows-the-norma-probability-distribution-with-standard-deviation-of-5971\/","title":{"rendered":"The average income tax refund for the 2009 tax year was 52876_ Assume the refund per person follows the norma probability distribution with standard deviation of 5971 ."},"content":{"rendered":"\n<p>The average income tax refund for the 2009 tax year was 52876_ Assume the refund per person follows the norma probability distribution with standard deviation of 5971 . Complete parts through below<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To solve this problem, we are working with a normal distribution. The average (mean) income tax refund for the 2009 tax year is <strong>$5,287.60<\/strong>, and the <strong>standard deviation<\/strong> is <strong>$597.10<\/strong>. We\u2019ll interpret the parts you are referring to as likely involving calculations with the normal distribution, such as finding probabilities or values corresponding to certain percentiles.<\/p>\n\n\n\n<p>Let\u2019s walk through three typical parts that often come with a normal distribution problem:<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>a) Find the probability that a randomly selected person received a refund greater than $6,000.<\/strong><\/h3>\n\n\n\n<p>We use the <strong>z-score formula<\/strong>:z=X\u2212\u03bc\u03c3z = \\frac{X &#8211; \\mu}{\\sigma}z=\u03c3X\u2212\u03bc\u200b<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>X=6000X = 6000X=6000<\/li>\n\n\n\n<li>\u03bc=5287.60\\mu = 5287.60\u03bc=5287.60<\/li>\n\n\n\n<li>\u03c3=597.10\\sigma = 597.10\u03c3=597.10<\/li>\n<\/ul>\n\n\n\n<p>z=6000\u22125287.60597.10\u2248712.40597.10\u22481.193z = \\frac{6000 &#8211; 5287.60}{597.10} \\approx \\frac{712.40}{597.10} \\approx 1.193z=597.106000\u22125287.60\u200b\u2248597.10712.40\u200b\u22481.193<\/p>\n\n\n\n<p>Now, we look up z=1.19z = 1.19z=1.19 on the standard normal distribution table (or use a calculator). The area to the <strong>left<\/strong> of z=1.19z = 1.19z=1.19 is about <strong>0.8830<\/strong>.<\/p>\n\n\n\n<p>So, the probability of receiving <strong>more than $6,000<\/strong> is:P(X&gt;6000)=1\u2212P(Z&lt;1.19)=1\u22120.8830=0.1170P(X &gt; 6000) = 1 &#8211; P(Z &lt; 1.19) = 1 &#8211; 0.8830 = 0.1170P(X&gt;6000)=1\u2212P(Z&lt;1.19)=1\u22120.8830=0.1170<\/p>\n\n\n\n<p><strong>Answer: 0.1170 or 11.70%<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>b) Find the probability that a refund is less than $5,000.<\/strong><\/h3>\n\n\n\n<p>z=5000\u22125287.60597.10=\u2212287.60597.10\u2248\u22120.4817z = \\frac{5000 &#8211; 5287.60}{597.10} = \\frac{-287.60}{597.10} \\approx -0.4817z=597.105000\u22125287.60\u200b=597.10\u2212287.60\u200b\u2248\u22120.4817<\/p>\n\n\n\n<p>Using the z-table, the area to the left of z=\u22120.48z = -0.48z=\u22120.48 is about <strong>0.3156<\/strong>.<\/p>\n\n\n\n<p><strong>Answer: 0.3156 or 31.56%<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>c) What refund amount represents the top 10% of refunds?<\/strong><\/h3>\n\n\n\n<p>We are looking for XXX such that P(X&gt;x)=0.10P(X &gt; x) = 0.10P(X&gt;x)=0.10, or P(Z&lt;z)=0.90P(Z &lt; z) = 0.90P(Z&lt;z)=0.90. From the z-table, the z-value for the <strong>90th percentile<\/strong> is approximately <strong>1.28<\/strong>.<\/p>\n\n\n\n<p>Now solve for XXX:X=\u03bc+z\u03c3=5287.60+1.28(597.10)\u22485287.60+763.29=6050.89X = \\mu + z\\sigma = 5287.60 + 1.28(597.10) \\approx 5287.60 + 763.29 = 6050.89X=\u03bc+z\u03c3=5287.60+1.28(597.10)\u22485287.60+763.29=6050.89<\/p>\n\n\n\n<p><strong>Answer: Approximately $6,050.89<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Explanation (300 words)<\/strong><\/h3>\n\n\n\n<p>This problem is a classic example of applying the normal distribution, a core concept in statistics used to model data that clusters around a mean. The normal distribution is symmetric and defined by two parameters: the <strong>mean (average)<\/strong> and the <strong>standard deviation (a measure of spread)<\/strong>.<\/p>\n\n\n\n<p>The <strong>mean refund<\/strong> is $5,287.60, which represents the central point of the distribution. The <strong>standard deviation<\/strong> of $597.10 indicates how much variation exists from the mean. Larger values indicate more variability in refund amounts.<\/p>\n\n\n\n<p>To find the probability that a person receives more than a specific amount (like $6,000), we compute the <strong>z-score<\/strong>, which measures how many standard deviations a value is from the mean. This score is then used to look up the corresponding probability from the standard normal distribution table. Subtracting this probability from 1 gives the proportion of people with refunds above that amount.<\/p>\n\n\n\n<p>Similarly, to find the chance of receiving less than a certain amount, we use the same formula and interpret the area under the curve to the left of the z-score.<\/p>\n\n\n\n<p>To identify refund amounts for the <strong>top 10%<\/strong>, we reverse the process: find the z-score that corresponds to the 90th percentile (since only 10% of refunds are higher), then convert it back into a dollar amount using the mean and standard deviation. This technique helps policymakers and analysts understand income patterns and prepare for future tax seasons.<\/p>\n\n\n\n<p>These tools are widely used in fields like finance, healthcare, and education to make data-driven decisions.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-733.jpeg\" alt=\"\" class=\"wp-image-236800\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>The average income tax refund for the 2009 tax year was 52876_ Assume the refund per person follows the norma probability distribution with standard deviation of 5971 . Complete parts through below The Correct Answer and Explanation is: To solve this problem, we are working with a normal distribution. The average (mean) income tax refund [&hellip;]<\/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-236799","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/236799","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=236799"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/236799\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=236799"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=236799"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=236799"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}