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Notes: The Normal Distribution Density Curves Normal Distribution Normal Distribution notation Empirical Rule An IQ test is normally distributed Heights of American males are with mean 100 and standard normally distributed with mean 69 in deviation 15. and standard deviation 2.5 in. An IQ test is normally distributed with mean 100 and standard deviation 15. Just by knowing the mean and standard deviation, we can answer the Average American adult male height is 69 inches (5’ 9”) tall with a standard deviation of 2.5 inches. following questions: What is the likelihood that a randomly selected (1) Between what two values do approximately 68% of IQ scores lie? (2) Between what two values do approximately 95% of IQ scores lie? adult male would have a height less than 69 inches? What is the likelihood that a randomly selected adult male would have a height of less than 66.5 (3) Between what two values do approximately 99.7% of IQ scores lie? (4) What percent of IQ scores are less than 115? inches? What is the likelihood that a randomly selected adult male would have a height of greater than 74 (5) What percent of IQ scores are greater than 130? inches? (6) What percent of IQ scores are less than 85? What is the probability that a randomly selected adult male would have a height between 64 and 76.5 inches? Illiana’s Grade After 5 weeks of class Iliana must transfer from a stat class at Taylor to this class. Last week was the chapter 2 test in both classes. Iliana scored a 61 out of 70. Our test was out of 100 points. What score should she be given? Standard Normal Distribution Standardizing and z-scores Suppose a student has taken two quizzes in a statistics course. On the first quiz the mean score was 32, the standard deviation was 8, and the student received a 44. The student obtained a 28 on the second quiz, for which the mean was 23 and the standard deviation was 3. If test scores are approximately normal, on which quiz did the student perform better relative to the rest of the class? 4 skills you must have 1. 2. 3. 4. P(Z < z*) = .80 P(Z < z*) = .77 A married couple is employed by the same company. The husband works in a department for which the mean hourly rate is $12.80 and the standard deviation is $1.20. His wife is employed in a department where the mean rate is $13.50 and the standard deviation is $1.80. Both department salaries are normally distributed. • Relative to their departments, which is better paid if the husband earns $14.60 and the wife earns $15.75? Finding Normal Proportions 1. 2. 3. 4. What percentile is the husband located in his department? What percent of employees in the wife’s department earn better than her? What percent of employees in the wife’s department earn between $12.00 and $17.00? What would the wife need to earn to match her husband’s relative position? If the husband wanted to earn in the 95th percentile, how much should he earn per hour? METHODS OF ASSESSING NORMALITY Stemplot/Histogram Boxplot Normal Probability Plot Data Set 1 – Ages of the presidents at inauguration 57 61 57 57 58 57 61 54 68 51 49 64 50 48 65 52 56 46 54 49 51 47 55 55 54 42 51 56 55 51 54 51 60 61 43 55 56 61 52 69 64 46 54 47 Stemplot/Histogram Boxplot Normal Probability Plot Data Set 2 – The number of frost days during April at Greenwich, England, over a 65-year period. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 5 5 5 5 5 5 6 6 7 7 7 8 8 9 10 10 Stemplot/Histogram Boxplot Normal Probability Plot