{"id":511,"date":"2022-07-11T19:47:18","date_gmt":"2022-07-11T19:47:18","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/alphamodule\/?post_type=chapter&p=511"},"modified":"2022-07-11T19:47:18","modified_gmt":"2022-07-11T19:47:18","slug":"z-score-and-the-empirical-rule-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/alphamodule\/chapter\/z-score-and-the-empirical-rule-learn-it-3\/","title":{"raw":"Z-Score and the Empirical Rule: Learn It 3","rendered":"Z-Score and the Empirical Rule: Learn It 3"},"content":{"raw":"

Standardizing a Score<\/h3>\r\nAt this point, students will be presented with two datasets. They will be able to choose which one they would like to use to answer example questions.<\/span>\r\n\r\nNow that you have obtained the standard deviation of the data set Runtimes using technology, you can calculate any observation's z-score to locate it in the data set relative to the mean.\r\n
\r\n

calculating z-scores<\/h3>\r\n[Worked example video - a 3-instructor video that works through an example like questions 6 - 9]<\/span>\r\n\r\n<\/div>\r\n
\r\n

interactive Example<\/h3>\r\nRecall, to calculate a z-score given an observation, use the formula [latex]z=\\dfrac{x-\\mu}{\\sigma}[\/latex], where [latex]x[\/latex] represents the value of the observation, [latex]\\mu[\/latex] represents the population mean, [latex]\\sigma[\/latex] represents the population standard deviation, and [latex]z[\/latex] represents the standardized value, or z-score.\r\n\r\nWe'll use data set Sleep Study: Average Sleep, you saw in\u00a0\u00a0Comparing Variability of Data Sets: What to Know<\/a><\/em>\u00a0to demonstrate how to calculate z-scores for individual observations in the data set. The mean [latex]\\mu[\/latex] and standard deviation [latex]\\sigma[\/latex] in the formula represent the population the sample came from. Since we don't know these, we'll use the sample mean and standard deviation in our calculations.\r\n\r\nThe mean of the data is 7.97 hours with a standard deviation of 0.965. Calculate the z-scores for each of the following observations\u00a0and indicate if the given value lies above or below the mean. Round your calculations to two decimal places.\r\n
    \r\n \t
  1. [latex]6.93\\text{ hours}[\/latex]<\/li>\r\n \t
  2. [latex]9.87\\text{ hours}[\/latex]<\/li>\r\n \t
  3. [latex]7.97\\text{ hours}[\/latex]<\/li>\r\n \t
  4. [latex]4.95\\text{ hours}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"172851\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"172851\"]\r\n
      \r\n \t
    1. [latex]z=\\dfrac{6.93-7.97}{0.965}\\approx -1.08[\/latex]. This value is [latex]1.08[\/latex] standard deviations\u00a0below<\/strong> the mean.<\/li>\r\n \t
    2. [latex]z=\\dfrac{9.87-7.97}{0.965}\\approx 1.97[\/latex]. This value is\u00a0[latex]1.97[\/latex] standard deviations\u00a0above\u00a0<\/strong>the mean.<\/li>\r\n \t
    3. [latex]z=\\dfrac{7.97-7.97}{0.965}=0[\/latex]. This value is equal to the\u00a0mean.<\/li>\r\n \t
    4. [latex]z=\\dfrac{4.95-7.97}{0.965}\\approx -3.13[\/latex]. This value is [latex]-3.13[\/latex] standard deviations\u00a0below<\/strong> the mean.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nUse the mean and the standard deviation you calculated in Questions 2 and 3 to answer Questions 6 - 9.\r\n
      \r\n

      question 6<\/h3>\r\n[ohm_question hide_question_numbers=1]241215[\/ohm_question]\r\n\r\n[reveal-answer q=\"563409\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"563409\"]Use the formula [latex]z=\\dfrac{x-\\mu}{\\sigma}[\/latex], where [latex]x= 103[\/latex], [latex]\\mu=91[\/latex], and [latex]\\sigma=11.8[\/latex], and then use the order of operations.[\/hidden-answer]\r\n\r\n<\/div>\r\n
      \r\n

      question 7<\/h3>\r\n[ohm_question hide_question_numbers=1]241216[\/ohm_question]\r\n\r\n[reveal-answer q=\"914105\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"914105\"]Use the formula [latex]z=\\dfrac{x-\\mu}{\\sigma}[\/latex], where [latex]x= 127[\/latex], [latex]\\mu=91[\/latex], and [latex]\\sigma=11.8[\/latex], and then use the order of operations.[\/hidden-answer]\r\n\r\n<\/div>\r\n
      \r\n

      question 8<\/h3>\r\n[ohm_question hide_question_numbers=1]241217[\/ohm_question]\r\n\r\n[reveal-answer q=\"84618\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"84618\"]Use the formula [latex]z=\\dfrac{x-\\mu}{\\sigma}[\/latex], where [latex]x=73[\/latex], [latex]\\mu=91[\/latex], and [latex]\\sigma=11.8[\/latex], and then use the order of operations.[\/hidden-answer]\r\n\r\n<\/div>\r\n
      \r\n

      question 9<\/h3>\r\n[ohm_question hide_question_numbers=1]241218[\/ohm_question]\r\n\r\n[reveal-answer q=\"503665\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"503665\"]Use the formula [latex]z=\\dfrac{x-\\mu}{\\sigma}[\/latex], where [latex]x=91[\/latex], [latex]\\mu=91[\/latex], and [latex]\\sigma=11.8[\/latex], and then use the order of operations.[\/hidden-answer]\r\n\r\n<\/div>","rendered":"

      Standardizing a Score<\/h3>\n

      At this point, students will be presented with two datasets. They will be able to choose which one they would like to use to answer example questions.<\/span><\/p>\n

      Now that you have obtained the standard deviation of the data set Runtimes using technology, you can calculate any observation’s z-score to locate it in the data set relative to the mean.<\/p>\n

      \n

      calculating z-scores<\/h3>\n

      [Worked example video – a 3-instructor video that works through an example like questions 6 – 9]<\/span><\/p>\n<\/div>\n

      \n

      interactive Example<\/h3>\n

      Recall, to calculate a z-score given an observation, use the formula [latex]z=\\dfrac{x-\\mu}{\\sigma}[\/latex], where [latex]x[\/latex] represents the value of the observation, [latex]\\mu[\/latex] represents the population mean, [latex]\\sigma[\/latex] represents the population standard deviation, and [latex]z[\/latex] represents the standardized value, or z-score.<\/p>\n

      We’ll use data set Sleep Study: Average Sleep, you saw in\u00a0\u00a0Comparing Variability of Data Sets: What to Know<\/a><\/em>\u00a0to demonstrate how to calculate z-scores for individual observations in the data set. The mean [latex]\\mu[\/latex] and standard deviation [latex]\\sigma[\/latex] in the formula represent the population the sample came from. Since we don’t know these, we’ll use the sample mean and standard deviation in our calculations.<\/p>\n

      The mean of the data is 7.97 hours with a standard deviation of 0.965. Calculate the z-scores for each of the following observations\u00a0and indicate if the given value lies above or below the mean. Round your calculations to two decimal places.<\/p>\n

        \n
      1. [latex]6.93\\text{ hours}[\/latex]<\/li>\n
      2. [latex]9.87\\text{ hours}[\/latex]<\/li>\n
      3. [latex]7.97\\text{ hours}[\/latex]<\/li>\n
      4. [latex]4.95\\text{ hours}[\/latex]<\/li>\n<\/ol>\n
        Show Solution<\/span><\/p>\n
        \n
          \n
        1. [latex]z=\\dfrac{6.93-7.97}{0.965}\\approx -1.08[\/latex]. This value is [latex]1.08[\/latex] standard deviations\u00a0below<\/strong> the mean.<\/li>\n
        2. [latex]z=\\dfrac{9.87-7.97}{0.965}\\approx 1.97[\/latex]. This value is\u00a0[latex]1.97[\/latex] standard deviations\u00a0above\u00a0<\/strong>the mean.<\/li>\n
        3. [latex]z=\\dfrac{7.97-7.97}{0.965}=0[\/latex]. This value is equal to the\u00a0mean.<\/li>\n
        4. [latex]z=\\dfrac{4.95-7.97}{0.965}\\approx -3.13[\/latex]. This value is [latex]-3.13[\/latex] standard deviations\u00a0below<\/strong> the mean.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n

          Use the mean and the standard deviation you calculated in Questions 2 and 3 to answer Questions 6 – 9.<\/p>\n

          \n

          question 6<\/h3>\n