{"id":244512,"date":"2025-07-05T07:58:34","date_gmt":"2025-07-05T07:58:34","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=244512"},"modified":"2025-07-05T07:58:36","modified_gmt":"2025-07-05T07:58:36","slug":"the-following-table-contains-observed-frequencies-for-a-sample-of-200","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/05\/the-following-table-contains-observed-frequencies-for-a-sample-of-200\/","title":{"rendered":"The following table contains observed frequencies for a sample of 200."},"content":{"rendered":"\n

The following table contains observed frequencies for a sample of 200. Column Variable Row Variable 30 58 55 20 12 25 Test for independence of the row and column variables using 0 = .05 Compute the value of the x2 test statistic (to 2 decimals): The p-value is Select your answer What is your conclusion? Select your answer<\/p>\n\n\n\n

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

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

Correct Answer:<\/strong><\/p>\n\n\n\n

Compute the value of the \u03c7\u00b2 test statistic (to 2 decimals):<\/strong>
7.97<\/p>\n\n\n\n

The p-value is:<\/strong>
between .01 and .025<\/p>\n\n\n\n

What is your conclusion?<\/strong>
Reject H\u2080. The variables are dependent.<\/p>\n\n\n\n

Explanation<\/h3>\n\n\n\n

This problem requires a chi-square (\u03c7\u00b2) test for independence to determine if there is a statistically significant relationship between the row and column variables.<\/p>\n\n\n\n

1. State the Hypotheses<\/strong><\/p>\n\n\n\n

    \n
  • Null Hypothesis (H\u2080):<\/strong>\u00a0The row and column variables are independent.<\/li>\n\n\n\n
  • Alternative Hypothesis (H\u2081):<\/strong>\u00a0The row and column variables are dependent.<\/li>\n<\/ul>\n\n\n\n

    2. Calculate Row and Column Totals<\/strong>
    First, we find the totals for each row and column from the observed frequencies table. The grand total (n) is 200.<\/p>\n\n\n\n

    <\/td>A<\/strong><\/td>B<\/strong><\/td>C<\/strong><\/td>Row Total<\/strong><\/td><\/tr>
    P<\/strong><\/td>30<\/td>58<\/td>55<\/td>143<\/strong><\/td><\/tr>
    Q<\/strong><\/td>20<\/td>12<\/td>25<\/td>57<\/strong><\/td><\/tr>
    Col Total<\/strong><\/td>50<\/strong><\/td>70<\/strong><\/td>80<\/strong><\/td>200<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

    3. Calculate Expected Frequencies (E)<\/strong>
    The expected frequency for each cell is calculated assuming the null hypothesis is true, using the formula:
    E = (Row Total \u00d7 Column Total) \/ Grand Total<\/em><\/p>\n\n\n\n

      \n
    • E(P, A) = (143 \u00d7 50) \/ 200 = 35.75<\/li>\n\n\n\n
    • E(P, B) = (143 \u00d7 70) \/ 200 = 50.05<\/li>\n\n\n\n
    • E(P, C) = (143 \u00d7 80) \/ 200 = 57.20<\/li>\n\n\n\n
    • E(Q, A) = (57 \u00d7 50) \/ 200 = 14.25<\/li>\n\n\n\n
    • E(Q, B) = (57 \u00d7 70) \/ 200 = 19.95<\/li>\n\n\n\n
    • E(Q, C) = (57 \u00d7 80) \/ 200 = 22.80<\/li>\n<\/ul>\n\n\n\n

      4. Compute the \u03c7\u00b2 Test Statistic<\/strong>
      The chi-square test statistic is calculated using the formula:
      \u03c7\u00b2 = \u03a3 [ (Observed – Expected)\u00b2 \/ Expected ]<\/em><\/p>\n\n\n\n

        \n
      • \u03c7\u00b2 = (30 – 35.75)\u00b2\/35.75 + (58 – 50.05)\u00b2\/50.05 + (55 – 57.20)\u00b2\/57.20 + (20 – 14.25)\u00b2\/14.25 + (12 – 19.95)\u00b2\/19.95 + (25 – 22.80)\u00b2\/22.80<\/li>\n\n\n\n
      • \u03c7\u00b2 = 0.9248 + 1.2628 + 0.0846 + 2.3202 + 3.1680 + 0.2123<\/li>\n\n\n\n
      • \u03c7\u00b2 \u2248 7.9727<\/li>\n<\/ul>\n\n\n\n

        Rounding to two decimals, the \u03c7\u00b2 test statistic is 7.97<\/strong>.<\/p>\n\n\n\n

        5. Determine the p-value and Conclusion<\/strong><\/p>\n\n\n\n

          \n
        • Degrees of Freedom (df):<\/strong>\u00a0df = (rows – 1)(columns – 1) = (2 – 1)(3 – 1) = 2.<\/li>\n\n\n\n
        • p-value:<\/strong>\u00a0Using a chi-square distribution table or calculator with df = 2 and \u03c7\u00b2 = 7.97, we find the p-value is between 0.01 and 0.025.<\/li>\n\n\n\n
        • Conclusion:<\/strong>\u00a0We compare the p-value to the significance level (\u03b1 = 0.05). Since the p-value (between .01 and .025) is less than \u03b1 (0.05), we\u00a0reject the null hypothesis (H\u2080)<\/strong>. There is sufficient statistical evidence to conclude that the row and column variables are dependent.<\/li>\n<\/ul>\n\n\n\n
          \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

          The following table contains observed frequencies for a sample of 200. Column Variable Row Variable 30 58 55 20 12 25 Test for independence of the row and column variables using 0 = .05 Compute the value of the x2 test statistic (to 2 decimals): The p-value is Select your answer What is your conclusion? […]<\/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-244512","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/244512","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=244512"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/244512\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=244512"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=244512"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=244512"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}