{"id":179300,"date":"2024-12-30T11:27:32","date_gmt":"2024-12-30T11:27:32","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=179300"},"modified":"2024-12-30T11:27:34","modified_gmt":"2024-12-30T11:27:34","slug":"researchers-want-to-determine-whether-all-bags-of-mms-have-the-same-proportion-of-colors-regardless-of-the-flavor-of-mms-2","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-2\/","title":{"rendered":"Researchers want to determine whether all bags of M&amp;Ms have the same proportion of colors regardless of the flavor of M&amp;Ms"},"content":{"rendered":"\n<p>Researchers want to determine whether all bags of M&amp;Ms have the same proportion of colors regardless of the flavor of M&amp;Ms. To test this, they sampled randomly king-size bags of each flavor and recorded their findings in the table.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2024\/12\/image-311.png\" alt=\"\" class=\"wp-image-179301\"\/><\/figure>\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) (10 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<p><strong>Part A: Degrees of Freedom<\/strong><\/p>\n\n\n\n<p>To calculate the degrees of freedom for a chi-squared test, we use the formula:<br>[<br>\\text{Degrees of Freedom} = (r &#8211; 1) \\times (c &#8211; 1)<br>]<br>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>(r) = number of rows in the contingency table<\/li>\n\n\n\n<li>(c) = number of columns in the contingency table<\/li>\n<\/ul>\n\n\n\n<p>Looking at the table from the link, assume there are two flavors of M&amp;Ms (e.g., peanut and regular) and several colors. If the table has 2 rows (one for each flavor) and 6 columns (one for each color), the degrees of freedom would be calculated as:<br>[<br>\\text{Degrees of Freedom} = (2 &#8211; 1) \\times (6 &#8211; 1) = 1 \\times 5 = 5<br>]<\/p>\n\n\n\n<p>So, the degrees of freedom are <strong>5<\/strong>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p><strong>Part B: Expected Count for Green Peanut M&amp;Ms<\/strong><\/p>\n\n\n\n<p>To calculate the expected count for a category, use the formula:<br>[<br>E = \\frac{(R_{\\text{total}} \\times C_{\\text{total}})}{N}<br>]<br>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>(R_{\\text{total}}) = total count for the row (e.g., total number of peanut M&amp;Ms)<\/li>\n\n\n\n<li>(C_{\\text{total}}) = total count for the column (e.g., total number of green M&amp;Ms)<\/li>\n\n\n\n<li>(N) = total number of M&amp;Ms in the entire table (all rows and columns)<\/li>\n<\/ul>\n\n\n\n<p>If we assume:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The total number of peanut M&amp;Ms is (R_{\\text{total}}),<\/li>\n\n\n\n<li>The total number of green M&amp;Ms is (C_{\\text{total}}),<\/li>\n\n\n\n<li>The grand total (N) is the total number of all M&amp;Ms in the table.<\/li>\n<\/ul>\n\n\n\n<p>For instance, if:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The total number of peanut M&amp;Ms ((R_{\\text{total}})) is 200,<\/li>\n\n\n\n<li>The total number of green M&amp;Ms ((C_{\\text{total}})) is 50,<\/li>\n\n\n\n<li>The grand total (N) is 1000,<\/li>\n<\/ul>\n\n\n\n<p>Then the expected count for green peanut M&amp;Ms would be:<br>[<br>E = \\frac{200 \\times 50}{1000} = 10<br>]<br>So, the expected count for green peanut M&amp;Ms would be <strong>10<\/strong>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p><strong>Part C: Testing for Proportional Differences<\/strong><\/p>\n\n\n\n<p>To determine if there is sufficient evidence that the proportion of colors is different across flavors of M&amp;Ms, we perform a chi-squared test for independence. The null hypothesis ((H_0)) is that the proportion of M&amp;M colors is the same across the different flavors. The alternative hypothesis ((H_A)) is that the proportion of M&amp;M colors differs across the flavors.<\/p>\n\n\n\n<p>Steps:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Calculate the chi-squared statistic<\/strong>:<br>The chi-squared statistic is calculated as:<br>[<br>\\chi^2 = \\sum \\frac{(O_i &#8211; E_i)^2}{E_i}<br>]<br>Where:<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>(O_i) = observed count for each category (e.g., the number of green M&amp;Ms in peanut bags),<\/li>\n\n\n\n<li>(E_i) = expected count for each category (calculated as shown above).<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Determine the p-value<\/strong>:<br>Once we have the chi-squared statistic, we compare it to the chi-squared distribution with 5 degrees of freedom to find the p-value.<\/li>\n\n\n\n<li><strong>Make a conclusion<\/strong>:<br>If the p-value is less than the significance level ((\\alpha = 0.05)), we reject the null hypothesis and conclude that there is sufficient evidence to suggest a difference in the proportion of colors across M&amp;M flavors. Otherwise, we fail to reject the null hypothesis.<\/li>\n<\/ol>\n\n\n\n<p>For example, if the chi-squared statistic calculated is 12.5 and the critical value for 5 degrees of freedom at (\\alpha = 0.05) is 11.07, since 12.5 &gt; 11.07, we reject the null hypothesis. This indicates that there is a statistically significant difference in the proportion of colors across the different M&amp;M flavors.<\/p>\n\n\n\n<p>In conclusion, based on the statistical test, we would assess whether there is sufficient evidence to conclude that the color proportions are different across M&amp;M flavors.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Researchers want to determine whether all bags of M&amp;Ms have the same proportion of colors regardless of the flavor of M&amp;Ms. To test this, they sampled randomly king-size bags of each flavor and recorded their findings in the table. Part A: What are the correct degrees of freedom for this table? (2 points) Part B: [&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-179300","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/179300","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=179300"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/179300\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=179300"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=179300"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=179300"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}