MATH 225N Understanding the Empirical Rule: Week 5 (summer 2020) complete answers solution guide.

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Understanding the Empirical Rule :week5 1. A random sample of CO2 levels in a school has a sample mean of x¯=598.4 ppm and sample standard deviation of s=86.7 ppm. Use the Empirical Rule to determi ... ne the approximate percentage of CO2 levels that lie between 338.3 and 858.5 ppm. To use the Empirical Rule, we need to know how many standard deviations from the mean are the given values 338.3 and 858.5. Since the mean is 598.4, we see that the value 338.3 is 598.4−338.3=260.1 ppm below the mean. This is 3 standard deviations, since 260.1=3×86.7, so 338.3 is 3 standard deviations less than the mean. Similarly, the value 858.5 is 858.5−598.4=260.1 ppm above the mean. Again, this is 3 standard deviations, since 260.1=3×86.7, so 858.5 is 3 standard deviations greater than the mean. The Empirical Rule states that approximately 99.7% of the data is within 3 standard deviations of the mean. So by the Empirical Rule, we can say that approximately 99.7% of CO2 levels in the school are between 338.3 and 858.5 ppm. 2. Suppose that a random sample of redwood trees has a sample mean diameter of x¯=24.1 feet, with a sample standard deviation of s=3.7 feet. Since the diameters of redwood trees are generally symmetric and bellshaped, we can apply the Empirical Rule. Between what two diameters are approximately 68% of the data? he value 1 standard deviation below the mean is x¯−s=24.1−3.7=20.4. The value 1 standard deviation above the mean is x¯+s=24.1+3.7=27.8. So by the Empirical Rule, we can say that approximately 68% of the redwood diameters are between 20.4 and 27.8 feet. This study source was downloaded by 100000831988016 from CourseHero.com on 01-19-2022 14:21:10 GMT -06:00 https://www.coursehero.com/file/58247108/Understanding-the-Empirical-Rule-week5docx/ 3. Suppose a random sample of monthly rainfalls in a given area has a sample mean of x¯=22.2 inches, with a sample standard deviation of s=3.5 inches. Since rainfall amounts in this area are generally symmetric and bell-shaped, we can apply the Empirical Rule. Between what two amounts are approximately 99.7% of the data? The value 3 standard deviations below the mean is x¯−3s=22.2−3(3.5)=11.7. The value 3 standard deviations above the mean is x¯+3s=22.2+3(3.5)=32.7. So by the Empirical Rule, we can say that approximately 99.7% of the rainfall amounts are between 11.7 and 32.7 inches. 4. Suppose a random sample of adult women has a sample mean height of x¯=64.3 inches, with a sample standard deviation of s=2.4 inches. Since height distribution are generally symmetric and bell-shaped, we can apply the Empirical Rule. Between what two heights are approximately 99.7% of the data? The Empirical Rule states that approximately 99.7% of the data is within three standard deviations of the mean. In terms of the sample mean x¯ and sample standard deviation s, that is between x¯−3s and x¯+3s. The value three standard deviations below the mean is x¯−3s=64.3−3(2.4)=57.1 The value three standard deviations above the mean is x¯+3s=64.3+3(2.4)=71.5 So by the Empirical Rule, we can say that Approximately 99.7% of the women's heights are between 57.1 and 71.5 inches. This study source was downloaded by 100000831988016 from CourseHero.com on 01-19-2022 14:21:10 GMT -06:00 https://www.coursehero.com/file/58247108/Understanding-the-Empirical-Rule-week5docx/ 5. For the same random sample of adult women, with a sample mean height of x¯=64.3 inches and sample standard deviation of s=2.4 inches, use the Empirical Rule to determine the approximate percent of heights that lie between 59.5 inches and 69.1 inches. To use the Empirical Rule, we need to know how many standard deviations from the mean are the given values 59.5 and 69.1, in inches. The value 59.5 is less than x¯=64.3 by 64.3−59.5=4.8 inches. This is twice the standard deviation, 4.8=2×2.4, so 59.5 is two standard deviations less than the mean. The value 69.1 is greater than x¯=64.3 by 69.1−64.3=4.8 inches. Again, this is twice the standard deviation, 4.8=2×2.4, so 69.1 is two standard deviations greater than the mean. The Empirical Rule states that approximately 95% of the data is within two standard deviations of the mean. So by the Empirical Rule, we can say that Approximately 95% of the women's heights are between 59.5 and 69.1 inches. [Show More]

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