{"id":190845,"date":"2025-02-13T09:35:54","date_gmt":"2025-02-13T09:35:54","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=190845"},"modified":"2025-02-13T09:35:56","modified_gmt":"2025-02-13T09:35:56","slug":"the-following-table-shows-the-observed-frequencies-of-the-quarterly-returns-for-a-sample-of-60-hedge-funds","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/13\/the-following-table-shows-the-observed-frequencies-of-the-quarterly-returns-for-a-sample-of-60-hedge-funds\/","title":{"rendered":"The following table shows the observed frequencies of the quarterly returns for a sample of 60 hedge funds"},"content":{"rendered":"\n

The following table shows the observed frequencies of the quarterly returns for a sample of 60 hedge funds. The table also contains the hypothesized proportions of each class assuming the quarterly returns have a normal distribution. The sample mean and standard deviation are 3.6% and 7.4% respectively.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a. Set up the competing hypotheses for the goodness-of-fit test of normality for the quarterly returns.
b. Calculate the value of the test statistic and determine the degrees of freedom.
c. Compute the p-value. Does the evidence suggest that the quarterly returns do not have a normal distribution at the 10% significance level?<\/p>\n\n\n\n

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

I’ll go step by step and solve each part of your question.<\/p>\n\n\n\n

a. Setting Up the Hypotheses<\/strong><\/h3>\n\n\n\n

For a goodness-of-fit test<\/strong> to assess normality, we use the Chi-Square goodness-of-fit test<\/strong>.<\/p>\n\n\n\n

    \n
  • Null Hypothesis ((H_0))<\/strong>: The observed quarterly returns follow<\/strong> a normal distribution.<\/li>\n\n\n\n
  • Alternative Hypothesis ((H_1))<\/strong>: The observed quarterly returns do not follow<\/strong> a normal distribution.<\/li>\n<\/ul>\n\n\n\n

    b. Calculating the Test Statistic and Degrees of Freedom<\/strong><\/h3>\n\n\n\n

    The test statistic for a Chi-Square goodness-of-fit test<\/strong> is given by:<\/p>\n\n\n\n

    [
    \\chi^2 = \\sum \\frac{(O_i – E_i)^2}{E_i}
    ]<\/p>\n\n\n\n

    where:<\/p>\n\n\n\n

      \n
    • (O_i) = observed frequency for each category<\/li>\n\n\n\n
    • (E_i) = expected frequency for each category (calculated from hypothesized proportions)<\/li>\n<\/ul>\n\n\n\n

      The degrees of freedom (df)<\/strong> for this test are:<\/p>\n\n\n\n

      [
      df = \\text{(number of categories)} – 1 – \\text{(number of estimated parameters)}
      ]<\/p>\n\n\n\n

      Since the normal distribution parameters (mean and standard deviation) were estimated from the sample, we subtract 2<\/strong> parameters.<\/p>\n\n\n\n

      [
      df = k – 1 – 2
      ]<\/p>\n\n\n\n

      where ( k ) is the number of categories.<\/p>\n\n\n\n

      I’ll compute the test statistic and degrees of freedom now.<\/p>\n\n\n\n

      c. Compute the p-value and Conclusion<\/strong><\/h3>\n\n\n\n

      From our calculations:<\/p>\n\n\n\n

        \n
      • Chi-Square test statistic<\/strong> = 2.78<\/strong><\/li>\n\n\n\n
      • Degrees of freedom (df)<\/strong> = 2<\/strong><\/li>\n\n\n\n
      • p-value<\/strong> = 0.2494<\/strong><\/li>\n<\/ul>\n\n\n\n

        Decision at 10% Significance Level ((\\alpha = 0.10))<\/strong><\/h4>\n\n\n\n

        Since the p-value (0.2494) is greater than 0.10<\/strong>, we fail to reject the null hypothesis<\/strong>. This means that we do not have enough statistical evidence to conclude that the quarterly returns do not follow a normal distribution.<\/p>\n\n\n\n

        \n\n\n\n

        Final Explanation (300 Words)<\/strong><\/h3>\n\n\n\n

        The Chi-Square goodness-of-fit test<\/strong> is used to determine if the observed distribution of data matches an expected (theoretical) distribution. In this case, we tested whether the quarterly returns of 60 hedge funds follow a normal distribution.<\/p>\n\n\n\n

        First, we set up the hypotheses:<\/p>\n\n\n\n

          \n
        • Null Hypothesis ((H_0))<\/strong>: The quarterly returns follow a normal distribution.<\/li>\n\n\n\n
        • Alternative Hypothesis ((H_1))<\/strong>: The quarterly returns do not follow a normal distribution.<\/li>\n<\/ul>\n\n\n\n

          To conduct the test, we calculated the expected frequencies<\/strong> based on the hypothesized proportions and then used the Chi-Square formula<\/strong>:<\/p>\n\n\n\n

          [
          \\chi^2 = \\sum \\frac{(O_i – E_i)^2}{E_i}
          ]<\/p>\n\n\n\n

          where (O_i) and (E_i) represent the observed and expected frequencies for each category.<\/p>\n\n\n\n

          The computed Chi-Square test statistic was 2.78<\/strong>, with 2 degrees of freedom<\/strong>. Using the Chi-Square distribution, we found the p-value to be 0.2494<\/strong>.<\/p>\n\n\n\n

          At a 10% significance level ((\\alpha = 0.10))<\/strong>, we compare the p-value:<\/p>\n\n\n\n

            \n
          • If p-value < 0.10<\/strong>, we reject (H_0) (evidence against normality).<\/li>\n\n\n\n
          • If p-value \u2265 0.10<\/strong>, we fail to reject (H_0) (data is consistent with normality).<\/li>\n<\/ul>\n\n\n\n

            Since 0.2494 > 0.10<\/strong>, we fail to reject (H_0). Thus, we do not<\/strong> have strong enough evidence to conclude that the quarterly returns deviate significantly from a normal distribution.<\/p>\n\n\n\n

            In conclusion, at the 10% significance level<\/strong>, the observed data is consistent<\/strong> with normality. However, failing to reject (H_0) does not prove normality; it simply means there is no strong evidence against it based on this test.<\/p>\n","protected":false},"excerpt":{"rendered":"

            The following table shows the observed frequencies of the quarterly returns for a sample of 60 hedge funds. The table also contains the hypothesized proportions of each class assuming the quarterly returns have a normal distribution. The sample mean and standard deviation are 3.6% and 7.4% respectively. a. Set up the competing hypotheses for the […]<\/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-190845","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/190845","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=190845"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/190845\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=190845"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=190845"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=190845"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}