{"id":229982,"date":"2025-06-08T17:18:02","date_gmt":"2025-06-08T17:18:02","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=229982"},"modified":"2025-06-08T17:18:04","modified_gmt":"2025-06-08T17:18:04","slug":"based-on-records-kept-at-a-gas-station-the-distribution-of-gallons-of-gas-purchased-by-customers-is-skewed-to-the-right-with-mean-10-gallons-and-standard-deviation-4-gallons","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/08\/based-on-records-kept-at-a-gas-station-the-distribution-of-gallons-of-gas-purchased-by-customers-is-skewed-to-the-right-with-mean-10-gallons-and-standard-deviation-4-gallons\/","title":{"rendered":"Based on records kept at a gas station, the distribution of gallons of gas purchased by customers is skewed to the right with mean 10 gallons and standard deviation 4 gallons."},"content":{"rendered":"\n

Based on records kept at a gas station, the distribution of gallons of gas purchased by customers is skewed to the right with mean 10 gallons and standard deviation 4 gallons. A random sample of 64 customer receipts was selected, and the sample mean number of gallons was recorded. Suppose the process of selecting a random sample of 64 receipts and recording the sample mean number of gallons was repeated for a total of 100 samples. Which of the following is the best description of a dotplot created from the 100 sample means? A) The dotplot is skewed to the right with mean 10 gallons and standard deviation 4 gallons. B) The dotplot is skewed to the right with mean 10 gallons and standard deviation 0.5 gallon. C) The dotplot is skewed to the right with mean 10 gallons and standard deviation 0.4 gallon. D) The dotplot is approximately normal with mean 10 gallons and standard deviation 0.5 gallon. E) The dotplot is approximately normal with mean 10 gallons and standard deviation 0.4 gallon.<\/p>\n\n\n\n

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

Correct Answer: E) The dotplot is approximately normal with mean 10 gallons and standard deviation 0.4 gallon.<\/strong><\/p>\n\n\n\n

\n\n\n\n

Explanation<\/strong><\/h3>\n\n\n\n

The question describes a situation where we repeatedly take samples of 64 customer receipts, calculate the sample mean of each, and then observe the distribution of those 100 sample means. To understand the shape and spread of this distribution, we apply the Central Limit Theorem (CLT)<\/strong>.<\/p>\n\n\n\n

Central Limit Theorem (CLT):<\/strong><\/h4>\n\n\n\n

The CLT states that, regardless of the population distribution\u2019s shape, the sampling distribution of the sample mean will tend to be approximately normal<\/strong> if the sample size is sufficiently large. Since the population distribution is skewed to the right, the individual data points (single purchases) are not normally distributed. However, the sample size here is 64<\/strong>, which is large enough (typically, n\u226530n \\geq 30n\u226530 is considered sufficient), so the distribution of sample means will be approximately normal<\/strong>.<\/p>\n\n\n\n

Mean of the Sampling Distribution:<\/strong><\/h4>\n\n\n\n

The mean of the sampling distribution is the same as the population mean, which is:\u03bcx\u02c9=\u03bc=10 gallons\\mu_{\\bar{x}} = \\mu = 10 \\text{ gallons}\u03bcx\u02c9\u200b=\u03bc=10 gallons<\/p>\n\n\n\n

Standard Deviation of the Sampling Distribution (Standard Error):<\/strong><\/h4>\n\n\n\n

The standard deviation of the sampling distribution of the sample mean (called the standard error<\/strong>) is calculated by:\u03c3x\u02c9=\u03c3n=464=48=0.5 gallons\\sigma_{\\bar{x}} = \\frac{\\sigma}{\\sqrt{n}} = \\frac{4}{\\sqrt{64}} = \\frac{4}{8} = 0.5 \\text{ gallons}\u03c3x\u02c9\u200b=n\u200b\u03c3\u200b=64\u200b4\u200b=84\u200b=0.5 gallons<\/p>\n\n\n\n

So we expect the sampling distribution of the mean to have:<\/p>\n\n\n\n

    \n
  • Shape: Approximately normal<\/strong> (by CLT)<\/li>\n\n\n\n
  • Mean: 10 gallons<\/strong><\/li>\n\n\n\n
  • Standard deviation (standard error): 0.5 gallons<\/strong><\/li>\n<\/ul>\n\n\n\n

    However, the question says this process was repeated 100 times<\/strong> to create 100 sample means. This doesn\u2019t change the theoretical shape or spread of the sampling distribution; it just means we are plotting 100 values drawn from it.<\/p>\n\n\n\n

    The only answer that reflects all three key aspects \u2014 normal shape, mean of 10 gallons, and standard deviation of 0.5 gallons \u2014 is:<\/p>\n\n\n\n

    \n

    Answer E: Approximately normal with mean 10 gallons and standard deviation 0.4 gallon.<\/strong><\/p>\n<\/blockquote>\n\n\n\n

    Oops! Actually, E says 0.4<\/strong>, but we calculated 0.5<\/strong>. Let’s re-evaluate.464=0.5\\frac{4}{\\sqrt{64}} = 0.564\u200b4\u200b=0.5<\/p>\n\n\n\n

    So the correct answer is D<\/strong>, not E.<\/p>\n\n\n\n

    \n\n\n\n

    \u2705 Final Correct Answer: D) The dotplot is approximately normal with mean 10 gallons and standard deviation 0.5 gallon.<\/strong><\/h3>\n\n\n\n
    \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

    Based on records kept at a gas station, the distribution of gallons of gas purchased by customers is skewed to the right with mean 10 gallons and standard deviation 4 gallons. A random sample of 64 customer receipts was selected, and the sample mean number of gallons was recorded. Suppose the process of selecting a […]<\/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-229982","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/229982","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=229982"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/229982\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=229982"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=229982"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=229982"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}