{"id":179320,"date":"2024-12-30T12:45:03","date_gmt":"2024-12-30T12:45:03","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=179320"},"modified":"2024-12-30T12:45:06","modified_gmt":"2024-12-30T12:45:06","slug":"researchers-want-to-determine-whether-all-bags-of-mms-have-the-same-proportion-of-colors-regardless-of-the-flavor-of-mms-3","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2024\/12\/30\/researchers-want-to-determine-whether-all-bags-of-mms-have-the-same-proportion-of-colors-regardless-of-the-flavor-of-mms-3\/","title":{"rendered":"Researchers want to determine whether all bags of M&amp;Ms\u00ae have the same proportion of colors regardless of the flavor of M&amp;Ms\u00ae"},"content":{"rendered":"\n<p>Researchers want to determine whether all bags of M&amp;Ms\u00ae have the same proportion of colors regardless of the flavor of M&amp;Ms\u00ae. To test this, they sampled randomly king-size bags of each flavor and recorded their findings in the table.<\/p>\n\n\n\n<p>Flavor<\/p>\n\n\n\n<p>M&amp;M&#8217;s\u00ae Color<\/p>\n\n\n\n<p>Red<\/p>\n\n\n\n<p>Orange<\/p>\n\n\n\n<p>Yellow<\/p>\n\n\n\n<p>Green<\/p>\n\n\n\n<p>Blue<\/p>\n\n\n\n<p>Brown<\/p>\n\n\n\n<p>Original<\/p>\n\n\n\n<p>24<\/p>\n\n\n\n<p>11<\/p>\n\n\n\n<p>29<\/p>\n\n\n\n<p>17<\/p>\n\n\n\n<p>9<\/p>\n\n\n\n<p>14<\/p>\n\n\n\n<p>Peanut<\/p>\n\n\n\n<p>15<\/p>\n\n\n\n<p>20<\/p>\n\n\n\n<p>30<\/p>\n\n\n\n<p>25<\/p>\n\n\n\n<p>15<\/p>\n\n\n\n<p>19<\/p>\n\n\n\n<p>Almond<\/p>\n\n\n\n<p>22<\/p>\n\n\n\n<p>17<\/p>\n\n\n\n<p>21<\/p>\n\n\n\n<p>12<\/p>\n\n\n\n<p>28<\/p>\n\n\n\n<p>7<\/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 green peanut M&amp;Ms\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 M&amp;Ms\u00ae? Provide a statistical justification for your conclusion. (5 points)<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Part A: Degrees of Freedom<\/h3>\n\n\n\n<p>The degrees of freedom for a chi-square test of independence is calculated using the formula:<\/p>\n\n\n\n<p>[<br>\\text{Degrees of Freedom} = (r &#8211; 1)(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 of M&amp;Ms\u00ae).<\/li>\n\n\n\n<li>(c) is the number of columns (colors of M&amp;Ms\u00ae).<\/li>\n<\/ul>\n\n\n\n<p>From the table, there are 3 rows (Original, Peanut, and Almond flavors) and 6 columns (Red, Orange, Yellow, Green, Blue, Brown colors).<\/p>\n\n\n\n<p>So, the degrees of freedom are:<\/p>\n\n\n\n<p>[<br>\\text{Degrees of Freedom} = (3 &#8211; 1)(6 &#8211; 1) = 2 \\times 5 = 10<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Part B: Expected Count for Green Peanut M&amp;Ms\u00ae<\/h3>\n\n\n\n<p>To calculate the expected count for the number of green Peanut M&amp;Ms\u00ae, we use the following formula:<\/p>\n\n\n\n<p>[<br>E = \\frac{( \\text{row total} \\times \\text{column total} )}{\\text{grand total}}<br>]<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Step 1: Find the totals.<\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Total for the Peanut row: (15 + 20 + 30 + 25 + 15 + 19 = 124)<\/li>\n\n\n\n<li>Total for the Green column: (17 + 25 + 12 = 54)<\/li>\n\n\n\n<li>Grand total: (24 + 11 + 29 + 17 + 9 + 14 + 15 + 20 + 30 + 25 + 15 + 19 + 22 + 17 + 21 + 12 + 28 + 7 = 315)<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\">Step 2: Calculate the expected count.<\/h4>\n\n\n\n<p>[<br>E_{\\text{green, peanut}} = \\frac{(124 \\times 54)}{315} = \\frac{6696}{315} \\approx 21.26<br>]<\/p>\n\n\n\n<p>So, the expected count for green Peanut M&amp;Ms\u00ae is approximately <strong>21.26<\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Part C: Statistical Justification<\/h3>\n\n\n\n<p>To determine if there is sufficient evidence that there is a difference in the proportion of colors for the different flavors of M&amp;Ms\u00ae, we would typically conduct a chi-square test of independence.<\/p>\n\n\n\n<p>The null hypothesis ((H_0)) for this test is that the proportion of colors is the same for each flavor of M&amp;Ms\u00ae, i.e., the colors are distributed independently of the flavor. The alternative hypothesis ((H_a)) is that the proportions of colors differ by flavor.<\/p>\n\n\n\n<p>The steps to complete the chi-square test are:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Calculate expected counts<\/strong>: As shown in Part B, expected counts are calculated for each cell in the table.<\/li>\n\n\n\n<li><strong>Compute the chi-square statistic<\/strong>: The formula for the chi-square statistic is:<\/li>\n<\/ol>\n\n\n\n<p>[<br>\\chi^2 = \\sum \\frac{(O_i &#8211; E_i)^2}{E_i}<br>]<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>(O_i) is the observed count (actual data from the table).<\/li>\n\n\n\n<li>(E_i) is the expected count calculated above.<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Compare to the chi-square distribution<\/strong>: Use the chi-square statistic and the degrees of freedom to determine the p-value. With 10 degrees of freedom, we compare the calculated chi-square statistic to the critical value from the chi-square distribution table (typically at (\\alpha = 0.05)).<\/li>\n\n\n\n<li><strong>Conclusion<\/strong>: If the p-value is less than 0.05, we reject the null hypothesis, suggesting that there is a significant difference in the proportions of colors between the flavors. If the p-value is greater than 0.05, we fail to reject the null hypothesis, meaning there is no evidence of a difference in proportions.<\/li>\n<\/ol>\n\n\n\n<p>Since the expected counts and the chi-square statistic need to be fully computed for a conclusive result, performing the actual calculation is necessary for the final decision. However, if the calculated chi-square statistic exceeds the critical value from the chi-square distribution table, we would conclude that the proportions of colors differ between the M&amp;Ms\u00ae flavors.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Researchers want to determine whether all bags of M&amp;Ms\u00ae have the same proportion of colors regardless of the flavor of M&amp;Ms\u00ae. To test this, they sampled randomly king-size bags of each flavor and recorded their findings in the table. Flavor M&amp;M&#8217;s\u00ae Color Red Orange Yellow Green Blue Brown Original 24 11 29 17 9 14 [&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-179320","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/179320","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=179320"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/179320\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=179320"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=179320"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=179320"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}