{"id":29935,"date":"2025-06-21T13:22:43","date_gmt":"2025-06-21T13:22:43","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=29935"},"modified":"2025-06-21T13:22:44","modified_gmt":"2025-06-21T13:22:44","slug":"six-randomly-selected-students-took-an-iq-test-aand-the-next-day-they-took-a-very-similar-iq-test-b_-their-scores-are-shown-in-the-table-below","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/six-randomly-selected-students-took-an-iq-test-aand-the-next-day-they-took-a-very-similar-iq-test-b_-their-scores-are-shown-in-the-table-below\/","title":{"rendered":"Six randomly selected students took an IQ test A,and the next day they took a very similar IQ test B_ Their scores are shown in the table below"},"content":{"rendered":"\n

Six randomly selected students took an IQ test A,and the next day they took a very similar IQ test B_ Their scores are shown in the table below: Student Test A Test B 121 121 93 119 104 108 100 103 91 72 122 The difference in the IQ scores (Test A Test B) for this sample results in d Assume that the reading speeds are approximately normally distributed. ~1.50 and $d 2.26 On average; do people score better on the second test than in the first test ithey take? One wishes to test Ho : #d = 0 Ha : pd < 0. Find the rejection region for the above test with & 01. (-oo, 2.575) (-0,3.305) (3.365,0) 0 (-8,-3.365) (2.575,0}<\/p>\n\n\n\n

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

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

To determine if people score better on the second test (Test B), we conduct a left-tailed<\/strong> t-test for the mean difference of paired samples.<\/p>\n\n\n\n

Step 1: Define Hypotheses<\/h3>\n\n\n\n
    \n
  • Null hypothesis: H0:\u03bcd=0H_0: \\mu_d = 0<\/li>\n\n\n\n
  • Alternative hypothesis: Ha:\u03bcd<0H_a: \\mu_d < 0<\/li>\n<\/ul>\n\n\n\n

    This means we suspect that the average difference between Test A and Test B is negative, implying higher scores on the second test.<\/p>\n\n\n\n

    Step 2: Given Data<\/h3>\n\n\n\n
      \n
    • Sample mean difference: d\u02c9=\u22121.50\\bar{d} = -1.50<\/li>\n\n\n\n
    • Standard deviation of differences: sd=2.26s_d = 2.26<\/li>\n\n\n\n
    • Sample size: n=6n = 6<\/li>\n\n\n\n
    • Significance level: \u03b1=0.01\\alpha = 0.01<\/li>\n<\/ul>\n\n\n\n

      Step 3: Calculate Test Statistic<\/h3>\n\n\n\n

      We use the formula for the t-statistic:<\/p>\n\n\n\n

      t=d\u02c9\u2212\u03bc0sd\/n=\u22121.50\u221202.26\/6\u2248\u22121.500.9229\u2248\u22121.625t = \\frac{\\bar{d} – \\mu_0}{s_d \/ \\sqrt{n}} = \\frac{-1.50 – 0}{2.26 \/ \\sqrt{6}} \\approx \\frac{-1.50}{0.9229} \\approx -1.625<\/p>\n\n\n\n

      Step 4: Determine the Rejection Region<\/h3>\n\n\n\n

      Since this is a left-tailed<\/strong> test with \u03b1=0.01\\alpha = 0.01 and degrees of freedom df=5df = 5, we use the t-table to find the critical value:<\/p>\n\n\n\n

      t0.01,5=\u22123.365t_{0.01, 5} = -3.365<\/p>\n\n\n\n

      So, the rejection region<\/strong> is: (\u2212\u221e,\u22123.365)(-\\infty, -3.365)<\/p>\n\n\n\n

      Step 5: Conclusion<\/h3>\n\n\n\n

      The calculated t-value of -1.625 is not<\/em> in the rejection region. Therefore, we fail to reject the null hypothesis<\/strong>. There is insufficient evidence to conclude that people score better on the second test than on the first.<\/p>\n\n\n\n

      Correct answer: (-\u221e, -3.365)<\/strong><\/p>\n\n\n\n

      \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

      Six randomly selected students took an IQ test A,and the next day they took a very similar IQ test B_ Their scores are shown in the table below: Student Test A Test B 121 121 93 119 104 108 100 103 91 72 122 The difference in the IQ scores (Test A Test B) for […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-29935","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/29935","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=29935"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/29935\/revisions"}],"predecessor-version":[{"id":29938,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/29935\/revisions\/29938"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=29935"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=29935"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=29935"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}