{"id":44579,"date":"2025-06-30T17:34:46","date_gmt":"2025-06-30T17:34:46","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=44579"},"modified":"2025-06-30T17:34:48","modified_gmt":"2025-06-30T17:34:48","slug":"the-heights-of-adult-men-in-america-are-normally-distributed-with-a-mean-of-69-1-inches-and-a-standard-deviation-of-2-65-inches","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/the-heights-of-adult-men-in-america-are-normally-distributed-with-a-mean-of-69-1-inches-and-a-standard-deviation-of-2-65-inches\/","title":{"rendered":"The heights of adult men in America are normally distributed, with a mean of 69.1 inches and a standard deviation of 2.65 inches"},"content":{"rendered":"\n

The heights of adult men in America are normally distributed, with a mean of 69.1 inches and a standard deviation of 2.65 inches. The heights of adult women in America are also normally distributed, but with a mean of 64.3 inches and a standard deviation of 2.51 inches. a) If a man is 6 feet 3 inches tall, what is his z-score (to two decimal places)? z = b) If a woman is 5 feet 11 inches tall, what is her z-score (to two decimal places)? z =<\/p>\n\n\n\n

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

Problem Breakdown:<\/h3>\n\n\n\n

We are given two normal distributions: one for the heights of adult men and one for the heights of adult women in America. The task is to calculate the z-scores for a man who is 6 feet 3 inches tall and a woman who is 5 feet 11 inches tall.<\/p>\n\n\n\n

Step 1: Z-Score Formula<\/h4>\n\n\n\n

The formula to calculate a z-score is: z=X\u2212\u03bc\u03c3z = \\frac{{X – \\mu}}{{\\sigma}}z=\u03c3X\u2212\u03bc\u200b<\/p>\n\n\n\n

Where:<\/p>\n\n\n\n

    \n
  • XXX is the value for which you want to find the z-score.<\/li>\n\n\n\n
  • \u03bc\\mu\u03bc is the mean of the distribution.<\/li>\n\n\n\n
  • \u03c3\\sigma\u03c3 is the standard deviation of the distribution.<\/li>\n<\/ul>\n\n\n\n

    a) Z-Score for the Man (6 feet 3 inches)<\/h4>\n\n\n\n

    First, we need to convert the man’s height from feet and inches into inches only:<\/p>\n\n\n\n

      \n
    • 6 feet = 72 inches<\/li>\n\n\n\n
    • 3 inches = 3 inches<\/li>\n\n\n\n
    • Therefore, the man’s total height is 72+3=7572 + 3 = 7572+3=75 inches.<\/li>\n<\/ul>\n\n\n\n

      Now, we apply the z-score formula for the distribution of men’s heights:<\/p>\n\n\n\n

        \n
      • Mean height for men \u03bc=69.1\\mu = 69.1\u03bc=69.1 inches<\/li>\n\n\n\n
      • Standard deviation for men \u03c3=2.65\\sigma = 2.65\u03c3=2.65 inches<\/li>\n\n\n\n
      • The man’s height X=75X = 75X=75 inches<\/li>\n<\/ul>\n\n\n\n

        Substitute these values into the z-score formula: z=75\u221269.12.65=5.92.65\u22482.23z = \\frac{{75 – 69.1}}{{2.65}} = \\frac{{5.9}}{{2.65}} \\approx 2.23z=2.6575\u221269.1\u200b=2.655.9\u200b\u22482.23<\/p>\n\n\n\n

        So, the z-score for the man is 2.23<\/strong>.<\/p>\n\n\n\n

        b) Z-Score for the Woman (5 feet 11 inches)<\/h4>\n\n\n\n

        Next, we need to convert the woman’s height from feet and inches into inches only:<\/p>\n\n\n\n

          \n
        • 5 feet = 60 inches<\/li>\n\n\n\n
        • 11 inches = 11 inches<\/li>\n\n\n\n
        • Therefore, the woman’s total height is 60+11=7160 + 11 = 7160+11=71 inches.<\/li>\n<\/ul>\n\n\n\n

          Now, we apply the z-score formula for the distribution of women’s heights:<\/p>\n\n\n\n

            \n
          • Mean height for women \u03bc=64.3\\mu = 64.3\u03bc=64.3 inches<\/li>\n\n\n\n
          • Standard deviation for women \u03c3=2.51\\sigma = 2.51\u03c3=2.51 inches<\/li>\n\n\n\n
          • The woman’s height X=71X = 71X=71 inches<\/li>\n<\/ul>\n\n\n\n

            Substitute these values into the z-score formula: z=71\u221264.32.51=6.72.51\u22482.67z = \\frac{{71 – 64.3}}{{2.51}} = \\frac{{6.7}}{{2.51}} \\approx 2.67z=2.5171\u221264.3\u200b=2.516.7\u200b\u22482.67<\/p>\n\n\n\n

            So, the z-score for the woman is 2.67<\/strong>.<\/p>\n\n\n\n

            Summary of Results:<\/h3>\n\n\n\n
              \n
            • The z-score for the man who is 6 feet 3 inches tall is 2.23<\/strong>.<\/li>\n\n\n\n
            • The z-score for the woman who is 5 feet 11 inches tall is 2.67<\/strong>.<\/li>\n<\/ul>\n\n\n\n

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

              The z-score tells us how many standard deviations a particular data point (height in this case) is away from the mean of the distribution. A positive z-score indicates that the value is above the mean, while a negative z-score indicates it is below the mean.<\/p>\n\n\n\n

              For the man, a z-score of 2.23 means that his height is 2.23 standard deviations above the average height of men in America. Similarly, for the woman, a z-score of 2.67 means her height is 2.67 standard deviations above the average height of women in America.<\/p>\n\n\n\n

              These z-scores give us a way to compare how unusual or typical their heights are within the context of the two different height distributions for men and women.<\/p>\n\n\n\n

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

              The heights of adult men in America are normally distributed, with a mean of 69.1 inches and a standard deviation of 2.65 inches. The heights of adult women in America are also normally distributed, but with a mean of 64.3 inches and a standard deviation of 2.51 inches. a) If a man is 6 feet […]<\/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-44579","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/44579","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=44579"}],"version-history":[{"count":2,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/44579\/revisions"}],"predecessor-version":[{"id":44585,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/44579\/revisions\/44585"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=44579"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=44579"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=44579"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}