The Normal Distribution Lecture 22 Section 6 3 1 Tue Oct 2 2007 The 68 95 99 7 Rule For any normal distribution Approximately 68 of the values lie within one standard deviation of the mean Approximately 95 of the values lie within two standard deviations of the mean Approximately 99 7 of the values lie within three standard deviations of the mean The Empirical Rule The well known Empirical Rule is similar but more general If a distribution has a mound shape then Approximately 68 lie within one standard deviation of the mean Approximately 95 lie within two standard deviations of the mean Nearly all lie within three standard deviations of the mean The Standard Normal Distribution The standard normal distribution It is denoted by the letter Z That is Z is N 0 1 The Standard Normal Distribution N 0 1 z 3 2 1 0 1 2 3 Areas Under the Standard Normal Curve Easy questions What is the total area under the curve What proportion of values of Z will fall below 0 What proportion of values of Z will fall above 0 Areas Under the Standard Normal Curve Harder questions What proportion of values will fall below 1 What proportion of values will fall above 1 What proportion of values will fall below 1 What proportion of values will fall between 1 and 1 Areas Under the Standard Normal Curve It turns out that the area to the left of 1 is 0 8413 0 8413 z 3 2 1 0 1 2 3 Areas Under the Standard Normal Curve So what is the area to the right of 1 Area 0 8413 z 3 2 1 0 1 2 3 Areas Under the Standard Normal Curve So what is the area to the left of 1 Area 0 8413 z 3 2 1 0 1 2 3 Areas Under the Standard Normal Curve So what is the area between 1 and 1 Area 0 8413 0 8413 z 3 2 1 0 1 2 3 Areas Under the Standard Normal Curve There are two methods to finding standard normal areas The TI 83 function normalcdf Standard normal table I will show you how to use the table but we will normally use the TI 83 TI 83 Standard Normal Areas Press 2nd DISTR Select normalcdf Item 2 Enter the lower and upper bounds of the interval If the interval is infinite to the left enter E99 as the lower bound If the interval is infinite to the right enter E99 as the upper bound Press ENTER Standard Normal Areas Use the TI 83 to find the following The area between 1 and 1 The area to the left of 1 The area to the right of 1 Other Normal Curves If we are working with a different normal distribution say N 30 5 then how can we find areas under the curve TI 83 Area Under Normal Curves Use the same procedure as before except enter the mean and standard deviation as the 3rd and 4th parameters of the normalcdf function Find area between 25 and 38 in the distribution N 30 5 IQ Scores Intelligence Quotient Understanding and Interpreting IQ IQ scores are standardized to have a mean of 100 and a standard deviation of 15 Psychologists often assume a normal distribution of IQ scores as well IQ Scores What percentage of the population has an IQ above 120 above 140 What percentage of the population has an IQ between 75 and 125 The Standard Normal Table See pages 406 407 or pages A 4 and A5 in Appendix A The table is designed for the standard normal distribution The entries in the table are the areas to the left of the z value The Standard Normal Table To find the area to the left of 1 locate 1 00 in the table and read the entry z 00 01 02 0 9 0 8159 0 8186 0 8212 1 0 0 8413 0 8438 0 8461 1 1 0 8643 0 8665 0 8686 The Standard Normal Table To find the area to the left of 2 31 locate 2 31 in the table and read the entry z 00 01 02 2 2 0 9861 0 9864 0 9868 2 3 0 9893 0 9896 0 9898 2 4 0 9918 0 9920 0 9922 The Standard Normal Table The area to the left of 1 00 is 0 8413 That means that 84 13 of the population is below 1 00 0 8413 3 2 1 0 1 2 3 The Three Basic Problems Find the area to the left of a Look up the value for a a Find the area to the right of a Look up the value for a subtract it from 1 Find the area between a and b a Look up the values for a and b subtract the smaller value from the larger a b Standard Normal Areas Use the Standard Normal Tables to find the following The area between 2 14 and 1 36 The area to the left of 1 42 The area to the right of 1 42 Tables Area Under Normal Curves If X is N 30 5 what is the area to the left of 35 15 20 25 30 35 40 45 Tables Area Under Normal Curves If X is N 30 5 what is the area to the left of 35 15 20 25 30 35 40 45 Tables Area Under Normal Curves If X is N 30 5 what is the area to the left of 35 15 20 25 30 35 40 45 Tables Area Under Normal Curves If X is N 30 5 what is the area to the left of 35 X 15 20 25 30 35 40 45 3 2 1 0 1 2 3 Z Tables Area Under Normal Curves If X is N 30 5 what is the area to the left of 35 0 8413 X 15 20 25 30 35 40 45 3 2 1 0 1 2 3 Z Z Scores Z score or standard score Compute the z score of x as x x or x z z s Equivalently x x zs or x z Areas Under Other Normal Curves If a variable X has a normal distribution then the z scores of X have a standard normal distribution X If X is N then is N 0 1 Example Let X be N 30 5 What proportion of values of X are below 38 Compute z 38 30 5 8 5 1 6 Find the area to the left of 1 6 under the standard normal curve Answer 0 9452 Therefore 94 52 of the values of X are below 38 Bag A vs Bag B Suppose we have two bags Bag A and Bag B Each bag contains millions of vouchers In Bag A the values of the vouchers have distribution N 50 10 In Bag B the values of the vouchers have distribution N 80 15 Bag A vs Bag B H0 Bag A H1 Bag B 30 40 50 60 …
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