{"id":166678,"date":"2024-11-14T00:15:17","date_gmt":"2024-11-14T00:15:17","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=166678"},"modified":"2024-11-14T00:15:20","modified_gmt":"2024-11-14T00:15:20","slug":"a-random-sample-of-100-likely-voters-in-a-small-city-produced-59-voters-in-favor-of-candidate-a","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2024\/11\/14\/a-random-sample-of-100-likely-voters-in-a-small-city-produced-59-voters-in-favor-of-candidate-a\/","title":{"rendered":"A random sample of 100 likely voters in a small city produced 59 voters in favor of Candidate A"},"content":{"rendered":"\n

A random sample of 100 likely voters in a small city produced 59 voters in favor of Candidate A. The observed value of the test statistic for testing the null hypothesis H0: p = 0.5 versus the alternative hypothesis Ha: p > 0.5 is\u2026<\/p>\n\n\n\n

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

To find the observed value of the test statistic for testing the null hypothesis (H_0: p = 0.5) versus the alternative hypothesis (H_a: p > 0.5), we use a one-sample proportion z-test<\/strong>.<\/p>\n\n\n\n

Step 1: Define the null hypothesis and alternative hypothesis<\/h3>\n\n\n\n
    \n
  • Null hypothesis<\/strong>: ( H_0: p = 0.5 ) (The proportion of voters in favor of Candidate A is 0.5)<\/li>\n\n\n\n
  • Alternative hypothesis<\/strong>: ( H_a: p > 0.5 ) (The proportion of voters in favor of Candidate A is greater than 0.5)<\/li>\n<\/ul>\n\n\n\n

    Step 2: Find the test statistic formula<\/h3>\n\n\n\n

    The formula for the z-test statistic for proportions is:<\/p>\n\n\n\n

    [
    z = \\frac{\\hat{p} – p_0}{\\sqrt{\\frac{p_0(1 – p_0)}{n}}}
    ]<\/p>\n\n\n\n

    Where:<\/p>\n\n\n\n

      \n
    • ( \\hat{p} ) is the sample proportion (the proportion of voters in favor of Candidate A in the sample),<\/li>\n\n\n\n
    • ( p_0 ) is the hypothesized population proportion (0.5 under the null hypothesis),<\/li>\n\n\n\n
    • ( n ) is the sample size.<\/li>\n<\/ul>\n\n\n\n

      Step 3: Calculate the sample proportion and substitute the values<\/h3>\n\n\n\n
        \n
      • The sample size ( n = 100 ),<\/li>\n\n\n\n
      • The number of voters in favor of Candidate A is 59, so the sample proportion ( \\hat{p} = \\frac{59}{100} = 0.59 ),<\/li>\n\n\n\n
      • The hypothesized population proportion ( p_0 = 0.5 ).<\/li>\n<\/ul>\n\n\n\n

        Now, substitute these values into the formula:<\/p>\n\n\n\n

        [
        z = \\frac{0.59 – 0.5}{\\sqrt{\\frac{0.5(1 – 0.5)}{100}}}
        ]<\/p>\n\n\n\n

        [
        z = \\frac{0.09}{\\sqrt{\\frac{0.25}{100}}}
        ]<\/p>\n\n\n\n

        [
        z = \\frac{0.09}{\\sqrt{0.0025}} = \\frac{0.09}{0.05} = 1.8
        ]<\/p>\n\n\n\n

        Step 4: Interpret the result<\/h3>\n\n\n\n

        The observed value of the test statistic is 1.8<\/strong>.<\/p>\n\n\n\n

        This z-test statistic can be used to compare against critical values from the standard normal distribution (z-table) to make a decision. For a right-tailed test with a significance level of 0.05, the critical z-value is approximately 1.645. Since 1.8 > 1.645, we would reject the null hypothesis at the 0.05 significance level, suggesting that there is sufficient evidence to support the claim that the proportion of voters in favor of Candidate A is greater than 0.5.<\/p>\n","protected":false},"excerpt":{"rendered":"

        A random sample of 100 likely voters in a small city produced 59 voters in favor of Candidate A. The observed value of the test statistic for testing the null hypothesis H0: p = 0.5 versus the alternative hypothesis Ha: p > 0.5 is\u2026 The Correct Answer and Explanation is: To find the observed value […]<\/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-166678","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/166678","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=166678"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/166678\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=166678"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=166678"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=166678"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}