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Section 5.9 The 68-95-99.7 Rule Chapter 5: Exploring Data: Distributions The 68-95-99.7 Rule Because any particular normal distribution is completely determined by its mean and standard deviation, it is not surprising that all normal distributions are the same in terms of what proportion of observations are any given number of standard deviations from the mean. Here is an important rule based on this fact. The 68-95-99.7 Rule for Normal Distributions Chapter 5: Exploring Data: Distributions The 68-95-99.7 Rule According to the 68-95-99.7 rule, in any normal distribution: 1. 68% of the observations fall within 1 standard deviation of the mean. 2. 95% of the observations fall within 2 standard deviations of the mean. 3. 99.7% of the observations fall within 3 standard deviations of the mean. Chapter 5: Exploring Data: Distributions The 68-95-99.7 Rule Example 1 Chapter 5: Exploring Data: Distributions The 68-95-99.7 Rule The heights of women between the ages of 18 and 24 are roughly normally distributed, with mean 64.5 inches and standard deviation 2.5 inches. Using the 68-95-99.7 Rule find the 2.5% of the tallest women and the 2.5% of the shortest women. Chapter 5: Exploring Data: Distributions The 68-95-99.7 Rule 2 standard deviations = 2(2.5) = 5 inches 95% of the women are between (64.5 – 5) inches and (64.5 + 5) inches or between 59.5 inches and 69.5 inches. So the other 5% of the women have outside the range of 59.5 inches to 69.5 inches. Chapter 5: Exploring Data: Distributions The 68-95-99.7 Rule Since normal distributions are symmetric we know that 2.5% of these women will be above 69.5 inches and 2.5% of these women will be below 59.5 inches. This is our answer. Example 2 Chapter 5: Exploring Data: Distributions The 68-95-99.7 Rule The distribution of scores on tests such as the SAT college entrance exams is close to normal. Scores on each of the three sections (math, critical reading, writing) of the SAT are adjusted so that the mean score is about 500 and the standard deviation is about 100. Using this information answer the following questions. Chapter 5: Exploring Data: Distributions The 68-95-99.7 Rule A. How high must a student score to fall in the top 25%? This question is asking for the third quartile. Q3 = (.67)(standard deviation) = (.67)(100) = 67 points above the mean = 500 + 67 = 567 Scores above 567 are in the top 25%. Chapter 5: Exploring Data: Distributions The 68-95-99.7 Rule B. What percent of scores fall between 200 and 800? These scores are 3 standard deviations from the mean. 200 = 500 – 3(100) 800 = 500 + 3(100) The percent of scores are 99.7%. Chapter 5: Exploring Data: Distributions The 68-95-99.7 Rule C. What percent of the scores are above 700? 700 is 2 standard deviations above the mean. 700 = 500 + 2(100) 95% of the scores are between 500 – 2(100) and 500 + 2(100) or 300 to 700. 2.5% of scores are above 700 points. Chapter 5: Exploring Data: Distributions The 68-95-99.7 Rule SAT scores have Normal distribution Page 180 37, 38, 41