{"id":225707,"date":"2025-06-04T11:56:48","date_gmt":"2025-06-04T11:56:48","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=225707"},"modified":"2025-06-04T11:56:58","modified_gmt":"2025-06-04T11:56:58","slug":"the-population-distribution-from-which-the-samples-were-drawn-is-symmetric-and-mound-shaped-with-the-top-of-the-mound-at-32-95-of-the-data-is-between-30-and-34-and-99-796-of-the-data-is-between-29-a","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/04\/the-population-distribution-from-which-the-samples-were-drawn-is-symmetric-and-mound-shaped-with-the-top-of-the-mound-at-32-95-of-the-data-is-between-30-and-34-and-99-796-of-the-data-is-between-29-a\/","title":{"rendered":"The population distribution from which the samples were drawn is symmetric and mound-shaped with the top of the mound at 32. 95% of the data is between 30 and 34, and 99.796% of the data is between 29 and 35"},"content":{"rendered":"\n<p>The population distribution from which the samples were drawn is symmetric and mound-shaped with the top of the mound at 32. 95% of the data is between 30 and 34, and 99.796% of the data is between 29 and 35. How well does each histogram reflect these characteristics? First, consider if the histograms are symmetric and mound-shaped, with the top of the mound at 32. Recall that a mound-shaped distribution has a peak or mound near the middle of the data, and a symmetric distribution means both sides are similar when the graph is folded down the middle: Sample of size 30 Sample of size 50 Sample of size 100 The three histograms show a mound-shaped distribution, and the data gradually reduces on either side of the mound in a select fashion. The highest bar signifies the top of the mound. In histogram (i), the highest bar occurs at a value of x of 32. In histogram (ii), the highest bar occurs at a value of x of units. In histogram (iii), the highest bar occurs at a value of x of units<\/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\/image-119.png\" alt=\"\" class=\"wp-image-225708\"\/><\/figure>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>Let&#8217;s walk through the problem carefully and then provide a thorough explanation of how each histogram reflects the described distribution characteristics.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Correct Answers for the Blanks in the Image:<\/strong><\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>The three histograms show a mound-shaped distribution, and the data gradually reduces on either side of the mound in a <em>symmetric<\/em> fashion.<\/strong><\/li>\n\n\n\n<li><strong>In histogram (i), the highest bar occurs at a value of x of 32.<\/strong><\/li>\n\n\n\n<li><strong>In histogram (ii), the highest bar occurs at a value of x of <em>32<\/em> units.<\/strong><\/li>\n\n\n\n<li><strong>In histogram (iii), the highest bar occurs at a value of x of <em>32<\/em> units.<\/strong><\/li>\n<\/ol>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Thorough 1000-Word Explanation:<\/strong><\/h3>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Understanding the Distribution:<\/strong><\/h4>\n\n\n\n<p>The question is based on a classic symmetric and mound-shaped distribution (a normal distribution). The known characteristics are:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The distribution is symmetric and centered at 32.<\/li>\n\n\n\n<li><strong>95%<\/strong> of the data lies between <strong>30 and 34<\/strong>.<\/li>\n\n\n\n<li><strong>99.7%<\/strong> of the data lies between <strong>29 and 35<\/strong>.<\/li>\n<\/ul>\n\n\n\n<p>This is consistent with the <strong>Empirical Rule<\/strong> (or <strong>68\u201395\u201399.7 rule<\/strong>) in statistics, which states:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>68% of the data falls within 1 standard deviation from the mean.<\/li>\n\n\n\n<li>95% within 2 standard deviations.<\/li>\n\n\n\n<li>99.7% within 3 standard deviations.<\/li>\n<\/ul>\n\n\n\n<p>This implies that:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The mean (\u03bc) is <strong>32<\/strong>.<\/li>\n\n\n\n<li>One standard deviation (\u03c3) is likely <strong>1<\/strong>, because:\n<ul class=\"wp-block-list\">\n<li>95% of data falls between 30 and 34 \u2192 this is \u00b12 units from the mean \u2192 2\u03c3 = 2 \u2192 \u03c3 = 1.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<p>So, we&#8217;re evaluating how well the sample histograms mimic a population that is <strong>normally distributed with \u03bc = 32, \u03c3 = 1.<\/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>Histogram Evaluation:<\/strong><\/h3>\n\n\n\n<p>We now evaluate each histogram in terms of:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Symmetry<\/strong><\/li>\n\n\n\n<li><strong>Mound-shaped nature<\/strong><\/li>\n\n\n\n<li><strong>Central tendency (peak at x = 32)<\/strong><\/li>\n\n\n\n<li><strong>Spread (data concentration within expected intervals)<\/strong><\/li>\n<\/ol>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Histogram (i): Sample Size = 30<\/strong><\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Symmetry<\/strong>: Somewhat symmetric but shows variability due to the small sample size.\n<ul class=\"wp-block-list\">\n<li>Left side of the mound (around 30\u201331) is slightly lower than the right side (33\u201334).<\/li>\n\n\n\n<li>There is a bar at 30 and 34, which supports the 95% coverage between 30 and 34.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Mound-shaped<\/strong>: Yes, there&#8217;s a peak at 32 with frequencies tapering off on either side.<\/li>\n\n\n\n<li><strong>Central Peak<\/strong>: Highest bar is at <strong>32<\/strong>, matching population peak.<\/li>\n\n\n\n<li><strong>Spread<\/strong>: Most data between 30 and 34, with a few data points at 29 and 35, aligning with the empirical rule.<\/li>\n<\/ul>\n\n\n\n<p><strong>Summary<\/strong>: Reasonably good approximation despite sample variability; less smooth due to the small sample.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Histogram (ii): Sample Size = 50<\/strong><\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Symmetry<\/strong>: Stronger symmetry than histogram (i); left and right sides are more balanced.\n<ul class=\"wp-block-list\">\n<li>Peak near 32 is flanked by bars that decrease in frequency as you move away.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Mound-shaped<\/strong>: Clearly mound-shaped.<\/li>\n\n\n\n<li><strong>Central Peak<\/strong>: Highest bar occurs at <strong>32<\/strong>, again matching the population mean.<\/li>\n\n\n\n<li><strong>Spread<\/strong>: Bars stretch from 29 to 35, perfectly in line with 99.7% data rule.\n<ul class=\"wp-block-list\">\n<li>Densest bars fall within 30\u201334, in accordance with the 95% rule.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<p><strong>Summary<\/strong>: This histogram reflects the characteristics very well \u2014 better than histogram (i) due to larger sample size leading to smoother representation.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Histogram (iii): Sample Size = 100<\/strong><\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Symmetry<\/strong>: Excellent symmetry. Almost perfect mirror around x = 32.<\/li>\n\n\n\n<li><strong>Mound-shaped<\/strong>: Very smooth mound; classic bell curve representation.<\/li>\n\n\n\n<li><strong>Central Peak<\/strong>: Highest bar is at <strong>32<\/strong>, directly supporting the population&#8217;s peak.<\/li>\n\n\n\n<li><strong>Spread<\/strong>: Bars extend from 29 to 35 with most of the data between 30 and 34.\n<ul class=\"wp-block-list\">\n<li>Perfectly matches 95% and 99.7% intervals of a normal distribution.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<p><strong>Summary<\/strong>: This histogram best reflects the true nature of the population distribution. Larger sample sizes produce smoother, more accurate approximations of population distributions.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Overall Observations:<\/strong><\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Symmetry Improves with Sample Size<\/strong>:\n<ul class=\"wp-block-list\">\n<li>Histogram (i) shows more irregularities because random variation has more impact in small samples.<\/li>\n\n\n\n<li>As sample size increases, the histogram becomes smoother and more symmetric due to the <strong>Law of Large Numbers<\/strong>, which states that the sample distribution approaches the population distribution as the sample size grows.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Mound Shape Emerges with Larger Samples<\/strong>:\n<ul class=\"wp-block-list\">\n<li>All three histograms are roughly mound-shaped, but the shape becomes more obvious and textbook-like with a sample size of 100.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Peak at 32 is Consistently Captured<\/strong>:\n<ul class=\"wp-block-list\">\n<li>Each histogram correctly identifies the central peak at <strong>x = 32<\/strong>, which reflects the given population mean.<\/li>\n\n\n\n<li>This suggests that even small samples can provide a good sense of central tendency.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Spread Matches Population Expectations<\/strong>:\n<ul class=\"wp-block-list\">\n<li>All histograms show that data mostly lies between <strong>30 and 34<\/strong>, and very few observations fall outside 29 or 35.<\/li>\n\n\n\n<li>This validates the stated <strong>95%<\/strong> and <strong>99.7%<\/strong> intervals.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Conclusion:<\/strong><\/h3>\n\n\n\n<p>Each histogram approximates the population distribution to varying degrees based on sample size:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Histogram (i)<\/strong> provides a basic, less smooth estimate. The peak is correct, but the symmetry and spread are affected by sample variability.<\/li>\n\n\n\n<li><strong>Histogram (ii)<\/strong> shows a much better approximation \u2014 symmetry, peak, and spread all align well with the population characteristics.<\/li>\n\n\n\n<li><strong>Histogram (iii)<\/strong> gives the <strong>best representation<\/strong>, being smooth, symmetric, and centered with a spread that perfectly aligns with the empirical rule.<\/li>\n<\/ul>\n\n\n\n<p>Therefore, <strong>larger sample sizes lead to better approximations<\/strong> of the population distribution, and all three histograms generally support the claim that the underlying distribution is <strong>symmetric and mound-shaped, centered at 32, with most data between 30 and 34<\/strong><\/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-banner6-45.jpeg\" alt=\"\" class=\"wp-image-225710\"\/><\/figure>\n\n\n\n<p>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The population distribution from which the samples were drawn is symmetric and mound-shaped with the top of the mound at 32. 95% of the data is between 30 and 34, and 99.796% of the data is between 29 and 35. How well does each histogram reflect these characteristics? First, consider if the histograms are symmetric [&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-225707","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/225707","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=225707"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/225707\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=225707"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=225707"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=225707"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}