{"id":179819,"date":"2025-01-02T12:24:16","date_gmt":"2025-01-02T12:24:16","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=179819"},"modified":"2025-01-02T12:24:18","modified_gmt":"2025-01-02T12:24:18","slug":"in-a-recent-conversation-with-mrs","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/02\/in-a-recent-conversation-with-mrs\/","title":{"rendered":"In a recent conversation with Mrs"},"content":{"rendered":"\n

In a recent conversation with Mrs. ?Cornelius, Mr. ?Wilcox claimed that the weather in Michigan was much cloudier than the weather in Mississippi. To investigate Mr. ?Wilcox took a random sample of 30 ?days from the past year and recorded whether or not it was cloudy in Michigan. Mrs. ?Cornelius took a separate random sample of 30 ?days from the past year and recorded whether or not it was cloudy in Mississippi. Which inference procedure is most appropriate to assess Mr. ?Wilcox’s claim?<\/p>\n\n\n\n

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

The most appropriate inference procedure to assess Mr. Wilcox’s claim would be a two-sample hypothesis test for proportions<\/strong>. This is because the data involves two independent groups (Michigan and Mississippi), and the objective is to compare the proportion of cloudy days between the two states.<\/p>\n\n\n\n

Steps for hypothesis testing:<\/h3>\n\n\n\n
    \n
  1. Formulate Hypotheses:<\/strong><\/li>\n<\/ol>\n\n\n\n
      \n
    • Null Hypothesis (H\u2080):<\/strong> The proportion of cloudy days in Michigan is equal to the proportion of cloudy days in Mississippi.\n
        \n
      • ( H\u2080: p_{\\text{Michigan}} = p_{\\text{Mississippi}} )<\/li>\n<\/ul>\n<\/li>\n\n\n\n
      • Alternative Hypothesis (H\u2081):<\/strong> The proportion of cloudy days in Michigan is greater than the proportion of cloudy days in Mississippi (since Mr. Wilcox claims Michigan is cloudier).\n
          \n
        • ( H\u2081: p_{\\text{Michigan}} > p_{\\text{Mississippi}} )<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n
            \n
          1. Data Collection and Summary:<\/strong><\/li>\n<\/ol>\n\n\n\n
              \n
            • Mrs. Cornelius records the number of cloudy days in Mississippi (say, ( x_{\\text{Mississippi}} ) cloudy days out of 30).<\/li>\n\n\n\n
            • Mr. Wilcox records the number of cloudy days in Michigan (say, ( x_{\\text{Michigan}} ) cloudy days out of 30).<\/li>\n<\/ul>\n\n\n\n
                \n
              1. Test Statistic:<\/strong>
                The test statistic for comparing two proportions is calculated using the formula for the z-score:
                [
                z = \\frac{(p_{\\text{Michigan}} – p_{\\text{Mississippi}})}{\\sqrt{p_{\\text{pooled}}(1 – p_{\\text{pooled}})\\left(\\frac{1}{n_{\\text{Michigan}}} + \\frac{1}{n_{\\text{Mississippi}}}\\right)}}
                ]
                where:<\/li>\n<\/ol>\n\n\n\n
                  \n
                • ( p_{\\text{Michigan}} ) and ( p_{\\text{Mississippi}} ) are the sample proportions of cloudy days in Michigan and Mississippi, respectively.<\/li>\n\n\n\n
                • ( p_{\\text{pooled}} ) is the pooled proportion, combining the data from both samples to estimate the common proportion of cloudy days across both states.<\/li>\n\n\n\n
                • ( n_{\\text{Michigan}} = n_{\\text{Mississippi}} = 30 ) are the sample sizes.<\/li>\n<\/ul>\n\n\n\n
                    \n
                  1. Decision Rule:<\/strong>
                    After calculating the z-score, compare it to the critical value for the chosen significance level (typically ( \\alpha = 0.05 )) using a standard z-table or p-value. If the z-score falls in the rejection region (i.e., the p-value is less than 0.05), reject the null hypothesis and conclude that Michigan is cloudier than Mississippi.<\/li>\n<\/ol>\n\n\n\n

                    Why this procedure is appropriate:<\/h3>\n\n\n\n
                      \n
                    • The data consists of two independent samples (Michigan and Mississippi).<\/li>\n\n\n\n
                    • The outcome (cloudy or not) is categorical, so proportions are the correct measure.<\/li>\n\n\n\n
                    • We are comparing two proportions to see if one is significantly larger than the other.<\/li>\n<\/ul>\n\n\n\n

                      This method allows for a statistical test to evaluate the claim made by Mr. Wilcox, providing evidence based on the sample data.<\/p>\n","protected":false},"excerpt":{"rendered":"

                      In a recent conversation with Mrs. ?Cornelius, Mr. ?Wilcox claimed that the weather in Michigan was much cloudier than the weather in Mississippi. To investigate Mr. ?Wilcox took a random sample of 30 ?days from the past year and recorded whether or not it was cloudy in Michigan. Mrs. ?Cornelius took a separate random sample […]<\/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-179819","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/179819","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=179819"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/179819\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=179819"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=179819"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=179819"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}