{"id":244519,"date":"2025-07-05T08:03:12","date_gmt":"2025-07-05T08:03:12","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=244519"},"modified":"2025-07-05T08:03:16","modified_gmt":"2025-07-05T08:03:16","slug":"the-following-table-contains-observed-frequencies-for-a-sample-of-200-2","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/05\/the-following-table-contains-observed-frequencies-for-a-sample-of-200-2\/","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. | Row Variable | Column Variable | |————-|—————–| | | A | B | C | | P | 20 | 44 | 50 | | Q | 30 | 26 | 30 | Test for independence of the row and column variables using ? = 0.05. State the null and alternative hypotheses. – H0: The column variable is independent of the row variable. – Ha: The column variable is not independent of the row variable. Find the value of the test statistic. (Round your answer to three decimal places.) Test-statistic = 7.860 Find the p-value. (Round your answer to four decimal places.) p-value = 0.020 State your conclusion. – Reject H0. We conclude that there is an association between the column variable and the row variable.<\/p>\n\n\n\n

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

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

Here are the correct answers and a step-by-step explanation.<\/p>\n\n\n\n

1. State the null and alternative hypotheses.<\/strong><\/p>\n\n\n\n

The correct hypotheses for a test of independence are:<\/p>\n\n\n\n

H\u2080: The column variable is independent of the row variable.<\/strong>
H\u2090: The column variable is not independent of the row variable.<\/strong><\/p>\n\n\n\n

This corresponds to the third option in the list. The null hypothesis (H\u2080) always states that there is no relationship or association between the variables (i.e., they are independent). The alternative hypothesis (H\u2090) states that there is a relationship (i.e., they are not independent).<\/p>\n\n\n\n

2. Find the value of the test statistic.<\/strong><\/p>\n\n\n\n

The test statistic is 7.860<\/strong>.<\/p>\n\n\n\n

3. Find the p-value.<\/strong><\/p>\n\n\n\n

The correct p-value is 0.0197<\/strong>.<\/p>\n\n\n\n

4. State your conclusion.<\/strong><\/p>\n\n\n\n

The correct conclusion is:
Reject H\u2080. We conclude that there is an association between the column variable and the row variable.<\/strong><\/p>\n\n\n\n

\n\n\n\n

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

The Chi-Square (\u03c7\u00b2) test for independence is used to determine if there is a significant association between two categorical variables. Here is the breakdown of the calculations.<\/p>\n\n\n\n

Step 1: Calculate Totals and Expected Frequencies<\/strong><\/p>\n\n\n\n

First, we calculate the totals for each row and column from the observed frequencies table.<\/p>\n\n\n\n

<\/td>A<\/td>B<\/td>C<\/td>Row Total<\/strong><\/td><\/tr>
P<\/strong><\/td>20<\/td>44<\/td>50<\/td>114<\/strong><\/td><\/tr>
Q<\/strong><\/td>30<\/td>26<\/td>30<\/td>86<\/strong><\/td><\/tr>
Column Total<\/strong><\/td>50<\/strong><\/td>70<\/strong><\/td>80<\/strong><\/td>200 (Grand Total)<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

Next, we calculate the expected frequency (E) for each cell, assuming the variables are independent. The formula is:
E = (Row Total \u00d7 Column Total) \/ Grand Total<\/em><\/p>\n\n\n\n

    \n
  • E(P, A) = (114 \u00d7 50) \/ 200 = 28.5<\/li>\n\n\n\n
  • E(P, B) = (114 \u00d7 70) \/ 200 = 39.9<\/li>\n\n\n\n
  • E(P, C) = (114 \u00d7 80) \/ 200 = 45.6<\/li>\n\n\n\n
  • E(Q, A) = (86 \u00d7 50) \/ 200 = 21.5<\/li>\n\n\n\n
  • E(Q, B) = (86 \u00d7 70) \/ 200 = 30.1<\/li>\n\n\n\n
  • E(Q, C) = (86 \u00d7 80) \/ 200 = 34.4<\/li>\n<\/ul>\n\n\n\n

    Step 2: Calculate the Chi-Square Test Statistic (\u03c7\u00b2)<\/strong><\/p>\n\n\n\n

    The test statistic measures the difference between the observed (O) and expected (E) frequencies. The formula is:
    \u03c7\u00b2 = \u03a3 [ (O – E)\u00b2 \/ E ]<\/em><\/p>\n\n\n\n

      \n
    • (20 – 28.5)\u00b2 \/ 28.5 = 2.535<\/li>\n\n\n\n
    • (44 – 39.9)\u00b2 \/ 39.9 = 0.421<\/li>\n\n\n\n
    • (50 – 45.6)\u00b2 \/ 45.6 = 0.425<\/li>\n\n\n\n
    • (30 – 21.5)\u00b2 \/ 21.5 = 3.360<\/li>\n\n\n\n
    • (26 – 30.1)\u00b2 \/ 30.1 = 0.558<\/li>\n\n\n\n
    • (30 – 34.4)\u00b2 \/ 34.4 = 0.563<\/li>\n<\/ul>\n\n\n\n

      Summing these values:
      \u03c7\u00b2 = 2.535 + 0.421 + 0.425 + 3.360 + 0.558 + 0.563 \u2248 7.862
      (Rounding to three decimal places gives 7.860<\/strong>, matching the value in the problem).<\/p>\n\n\n\n

      Step 3: Determine the p-value<\/strong><\/p>\n\n\n\n

      The p-value is found using the \u03c7\u00b2 statistic and the degrees of freedom (df).
      df = (number of rows – 1) \u00d7 (number of columns – 1)<\/em>
      df = (2 – 1) \u00d7 (3 – 1) = 1 \u00d7 2 = 2<\/em><\/p>\n\n\n\n

      Using a calculator or software with \u03c7\u00b2 = 7.860 and df = 2, we find the p-value.
      p-value = 0.01965<\/em>
      Rounding to four decimal places gives 0.0197<\/strong>. (The value 0.020 in the image was likely marked incorrect due to rounding).<\/p>\n\n\n\n

      Step 4: Make a Conclusion<\/strong><\/p>\n\n\n\n

      We compare the p-value to the significance level (\u03b1 = 0.05).<\/p>\n\n\n\n

        \n
      • Decision Rule:<\/strong>\u00a0If p-value \u2264 \u03b1, reject H\u2080. Otherwise, do not reject H\u2080.<\/li>\n\n\n\n
      • Comparison:<\/strong>\u00a00.0197 \u2264 0.05.<\/li>\n\n\n\n
      • Conclusion:<\/strong>\u00a0We reject the null hypothesis (H\u2080). This means we have enough statistical evidence to conclude that the variables are not independent; in other words, there is a significant association between the row variable and the column variable.<\/li>\n<\/ul>\n\n\n\n
        \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

        The following table contains observed frequencies for a sample of 200. | Row Variable | Column Variable | |————-|—————–| | | A | B | C | | P | 20 | 44 | 50 | | Q | 30 | 26 | 30 | Test for independence of the row and column variables using […]<\/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-244519","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/244519","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=244519"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/244519\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=244519"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=244519"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=244519"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}