The Normal Density Normal curves have the following bell shaped symmetric density 2 1 1 y f y e 2 2 Parameters The parameters of a normal curve are the mean and the standard deviation The Normal Distribution Here is an example of a normal curve with 100 and 20 Normal Distribution mu 100 sigma 20 Probability Density Bret Larget 50 3 100 2 1 0 150 1 2 3 Statistics 371 Fall 2004 Department of Statistics 2 University of Wisconsin Madison Standardization The Normal Distribution September 28 2004 The Normal Distribution is the most important distribution All normal curves have the same essential shape Every normal curve can be drawn in exactly the same manner just by changing labels on the axis The standard normal curve is the normal curve with mean 0 and standard deviation 1 The standardization formula is Y Z Every problem that asks for an area under a normal curve is solved by first finding an equivalent problem for the standard normal curve The justification for this comes from calculus An area under a normal curve is a definite integral The integral is simplified by using the standardization formula Statistics 371 Fall 2004 Statistics 371 Fall 2004 3 of continuous random variables The normal density curve is the famous symmetric bellshaped curve The central limit theorem is the reason that the normal curve is so important Essentially many statistics that we calculate from large random samples will have approximate normal distributions or distributions derived from normal distributions even if the distributions of the underlying variables are not normally distributed This fact is the basis of most of the methods of statistical inference we will study in the last half of the course Chapter 4 introduces the normal distribution as a probability distribution Chapter 5 culminates in the central limit theorem the primary theoretical justification for most of the methods of statistical inference in the remainder of the textbook Statistics 371 Fall 2004 1 Example Calculations The 68 95 99 7 Rule Suppose that egg shell thickness is normally distributed with a mean of 0 381 mm and a standard deviation of 0 031 mm Here are a large number of example calculations For every normal curve 68 of the area is within one SD of the mean 95 of the area is within two SDs of the mean and 99 7 of the area is within three SDs of the mean Normal Distribution mu 0 sigma 1 Normal Distribution mu 0 sigma 1 P X 1 0 1587 2 3 6 Example Area Calculation gnorm 0 381 0 031 b 0 34 Probability Density Normal Distribution mu 0 381 sigma 0 031 P X 0 34 0 907 0 30 Statistics 371 Fall 2004 0 35 2 1 0 40 0 0 45 1 0 2 1 2 4 3 P X 2 0 0228 4 2 3 2 0 1 0 2 1 2 4 3 Statistics 371 Fall 2004 P X 3 0 0013 P X 3 0 0013 4 2 3 2 0 1 0 2 1 2 4 3 4 2 The standard normal table lists the area to the left of z under the standard normal curve for each value from 3 49 to 3 49 by 0 01 increments The normal table is on the inside front cover of your textbook Numbers in the margins represent z Numbers in the middle of the table are areas to the left of z R can do this for general values of z and R can do the standardization for you P X 0 34 0 093 3 0 1 P X 2 0 0228 The Standard Normal Table Area to the left Find the proportion of eggs with shell thickness less than 0 34 mm 0 25 2 P 3 X 3 0 9973 Probability Density P X 1 0 1587 4 Statistics 371 Fall 2004 P 2 X 2 0 9545 Probability Density Probability Density P 1 X 1 0 6827 Normal Distribution mu 0 sigma 1 0 50 3 7 Statistics 371 Fall 2004 5 Example Area Calculation Example Area Calculation Area outside two values Find the proportion of eggs with shell thickness smaller than 0 32 mm or greater than 0 40 mm Area to the right Find the proportion of eggs with shell thickness more than 0 36 mm gnorm 0 381 0 031 a 0 32 b 0 4 gnorm 0 381 0 031 a 0 36 Normal Distribution mu 0 381 sigma 0 031 Normal Distribution mu 0 381 sigma 0 031 Probability Density Probability Density P 0 32 X 0 4 0 7055 P X 0 32 0 0245 P X 0 4 0 27 0 25 0 30 3 0 35 2 1 0 40 0 0 45 1 2 0 50 P X 0 36 0 2491 P X 0 36 0 7509 0 25 3 0 30 3 Statistics 371 Fall 2004 7 0 35 2 1 0 40 0 0 45 1 2 0 50 3 Statistics 371 Fall 2004 7 Example Area Calculation Example Area Calculation Central area Find the proportion of eggs with shell thickness within 0 05 mm of the mean Area between two values Find the proportion of eggs with shell thickness between 0 34 and 0 36 mm gnorm 0 381 0 031 a 0 381 0 05 b 0 381 0 05 gnorm 0 381 0 031 a 0 34 b 0 36 Normal Distribution mu 0 381 sigma 0 031 Normal Distribution mu 0 381 sigma 0 031 P X 0 331 0 0534 P X 0 431 0 0534 0 25 0 30 3 Statistics 371 Fall 2004 0 35 2 1 0 40 0 0 45 1 2 P 0 34 X 0 36 0 1561 Probability Density Probability Density P 0 331 X 0 431 0 8932 0 50 P X 0 34 0 093 P X 0 36 0 7509 0 25 3 0 30 3 7 Statistics 371 Fall 2004 0 35 2 1 0 40 0 0 45 1 2 0 50 3 7 Example Quantile Calculations Example Area Calculation Percentile What is the 90th percentile of the egg shell thickness distribution Two tail area Find the proportion of eggs with shell thickness more than 0 07 mm from the mean gnorm 0 381 0 031 quantile 0 9 gnorm 0 381 0 031 a 0 381 0 07 b 0 381 0 07 Normal Distribution mu 0 381 sigma 0 031 P 0 311 X 0 451 0 9761 Probability Density Probability Density Normal Distribution mu 0 381 sigma 0 031 P X 0 4207 0 9 P X 0 311 0 012 P X 0 451 0 012 z 1 28 0 25 0 30 3 0 35 2 1 0 40 0 0 45 1 2 0 50 0 25 3 0 30 3 Statistics 371 Fall 2004 9 Statistics 371 Fall 2004 0 35 2 1 0 40 0 0 45 1 2 0 50 3 7 Example Quantile Calculations Quantiles Upper cut off point What value cuts off the top 15 of …
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