{"id":36219,"date":"2025-06-25T05:24:12","date_gmt":"2025-06-25T05:24:12","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=36219"},"modified":"2025-06-25T05:24:14","modified_gmt":"2025-06-25T05:24:14","slug":"fraction-of-the-iq-scores-would-be-between-97-and-126","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/fraction-of-the-iq-scores-would-be-between-97-and-126\/","title":{"rendered":"fraction of the IQ scores would be between 97 and 126"},"content":{"rendered":"\n

fraction of the IQ scores would be between 97 and 126? 2. What is the 95th percentile of this normal distribution?
The distribution of IQ scores for high school graduates is normally distributed with u=104 and o=16. 1. What fraction of the IQ scores would be between 97 and 126? 2. What is the 95th percentile of this normal distribution?<\/p>\n\n\n\n

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

We are given that IQ scores are normally distributed<\/strong> with a mean (\u03bc) of 104<\/strong> and a standard deviation (\u03c3) of 16<\/strong>. We will use the properties of the normal distribution to answer both questions.<\/p>\n\n\n\n

\n\n\n\n

1. What fraction of the IQ scores would be between 97 and 126?<\/strong><\/h3>\n\n\n\n

To find this, we first convert the raw scores to z-scores<\/strong> using the formula:z=X\u2212\u03bc\u03c3z = \\frac{X – \\mu}{\\sigma}z=\u03c3X\u2212\u03bc\u200b<\/p>\n\n\n\n

For X = 97<\/strong>:z=97\u221210416=\u2212716=\u22120.4375z = \\frac{97 – 104}{16} = \\frac{-7}{16} = -0.4375z=1697\u2212104\u200b=16\u22127\u200b=\u22120.4375<\/p>\n\n\n\n

For X = 126<\/strong>:z=126\u221210416=2216=1.375z = \\frac{126 – 104}{16} = \\frac{22}{16} = 1.375z=16126\u2212104\u200b=1622\u200b=1.375<\/p>\n\n\n\n

Now, using the standard normal distribution table or a calculator:<\/p>\n\n\n\n

    \n
  • The area to the left of z = -0.44<\/strong> is approximately 0.3300<\/strong><\/li>\n\n\n\n
  • The area to the left of z = 1.38<\/strong> is approximately 0.9162<\/strong><\/li>\n<\/ul>\n\n\n\n

    To find the fraction between these z-values, subtract the lower area from the higher:0.9162\u22120.3300=0.58620.9162 – 0.3300 = 0.58620.9162\u22120.3300=0.5862<\/p>\n\n\n\n

    So, about 58.6 percent<\/strong> of IQ scores fall between 97 and 126.<\/p>\n\n\n\n

    \n\n\n\n

    2. What is the 95th percentile of this normal distribution?<\/strong><\/h3>\n\n\n\n

    The 95th percentile corresponds to a z-score where 95 percent<\/strong> of the data lies to the left<\/strong> of that value. Using a standard normal table or calculator:<\/p>\n\n\n\n

      \n
    • The z-score<\/strong> for the 95th percentile is approximately 1.645<\/strong><\/li>\n<\/ul>\n\n\n\n

      Now convert this z-score back to an IQ score using:X=\u03bc+z\u03c3=104+(1.645)(16)=104+26.32=130.32X = \\mu + z\\sigma = 104 + (1.645)(16) = 104 + 26.32 = 130.32X=\u03bc+z\u03c3=104+(1.645)(16)=104+26.32=130.32<\/p>\n\n\n\n

      Thus, the 95th percentile of this distribution is approximately 130.3<\/strong>.<\/p>\n\n\n\n

      \n\n\n\n

      Summary<\/h3>\n\n\n\n
        \n
      1. About 58.6%<\/strong> of IQ scores lie between 97 and 126.<\/li>\n\n\n\n
      2. The 95th percentile<\/strong> IQ score is approximately 130.3<\/strong>.<\/li>\n<\/ol>\n\n\n\n

        This analysis uses the properties of the normal distribution to translate between raw scores and z-scores, allowing us to determine areas under the curve and specific percentiles.<\/p>\n\n\n\n

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

        fraction of the IQ scores would be between 97 and 126? 2. What is the 95th percentile of this normal distribution?The distribution of IQ scores for high school graduates is normally distributed with u=104 and o=16. 1. What fraction of the IQ scores would be between 97 and 126? 2. What is the 95th percentile […]<\/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-36219","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/36219","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=36219"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/36219\/revisions"}],"predecessor-version":[{"id":36221,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/36219\/revisions\/36221"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=36219"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=36219"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=36219"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}