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Check for Understanding The Empirical Rule 1. What does the Empirical Rule tell us? 2. Graduate School Exam (GRE) Verbal Scores are Normally distributed with a mean of 544 and a standard deviation of 103. a. Sketch this Normal distribution and label the scores -3, -2, -1, 0, 1, 2, and 3 standard deviations away from the mean. b. Using your sketch, give an interval that contains the middle 95% of GRE scores. c. What percent of scores fall above 647? 3. MCAT scores (the medical school entrance exam) are Normally distributed with a mean of 28.1. The top 2.5% of students scored above 32.1. What is the standard deviation of this distribution? Answers: 1. What does the Empirical Rule tell us? In a Normal distribution, about 68% of observations fall within one standard deviation of the mean, about 95% fall within two standard deviations of the mean, and about 99.7% of observations fall within three standard deviations of the mean. 2. Graduate School Exam (GRE) Verbal Scores are Normally distributed with a mean of 544 and a standard deviation of 103. a. Sketch this Normal distribution and label the scores -3, -2, -1, 0, 1, 2, and 3 standard deviations away from the mean. 235 338 441 544 647 750 853 GRE scores b. Using your sketch, give an interval that contains the middle 95% of GRE scores. The middle 95% of scores fell between 338 and 750. c. What percent of scores fall above 647? About 16% of scores fell above 647. If 68% of scores are in the middle, that leaves out 32%, or 16% on either side of the first standard deviation away. 3. MCAT scores (the medical school entrance exam) are Normally distributed with a mean of 28.1. The top 2.5% of students scored above 32.1. What is the standard deviation of this distribution? The standard deviation is about 2.0. With 95% of the data within two standard deviations of the mean, only 2.5% of the data lies more than two standard deviations higher than the mean and more than two standard deviations lower than the mean, because 2.5% + 2.5% = 5%. Since the top 2.5% scored 4 points better than the mean, two standard deviations must equal 4 points, and one standard deviation must equal 2 points.