VCU STAT 210  Lecture20 (71 pages)
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Lecture20
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 Stat 210  Basic Practice of Statistics
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STAT 210 Lecture 20 Normal Distributions October 11 2017 Practice Problems Pages 162 through 165 Relevant problems VI 4 Recommended problems VI 4 Additional Reading and Examples Read pages 158 through 160 TOP HAT 2 Properties The normal curve is bell shaped The peak of the curve is the population mean m The normal curve is symmetric about m The center and spread are completely specified by specifying the values of the population mean m and the population standard deviation s The total area under the normal curve is 1 or 100 Notation X N m s X is distributed normal with mean m and standard deviation s Standard Normal Distribution Denoted by Z Has population mean m 0 center Has population standard deviation s 1 spread Shape is normal symmetric bell curve No unusual features Z N 0 1 Probabilities are tabled on pages 338 339 Normal Table Gives the probability that the standard normal variable Z falls below some specified value z less than problems Read the value of z down the left most column and across the top row and read the probability from the body of the table Normal Variables A standard normal Z distribution requires that the mean is 0 and that the standard deviation is 1 How many real variables weight height grades on a test etc do you think have a mean of 0 and a standard deviation of 1 Answer I cannot think of any Z Score Transformation Skip to page 147 Suppose X is distributed normal with some mean m not equal to 0 and or some standard deviation s not equal to 1 X N m s Z Score Transformation We convert to a standard normal variable Z N 0 1 Z X m value mean s standard deviation Z Score Transformation Subtraction of m converts the mean to 0 Division by s converts the standard deviation to 1 Z X m s s m s 1 m 0 Z Score Transformation P a X b P a m X m b m s P a m s s s Z b m s Once converted from X to Z the standard normal table on pages 338 and 339 is used to find the probability just as in Examples 34 through 41 Example 47 X N 10 5 P 12 X 20 s 5 12 20 Example 47 X N 10 5 P 12 X 20 P 12 10 Z 20 10 5 5 s 5 12 20 Example 47 X N 10 5 P 12 X 20 P 12 10 Z 20 10 5 5 s 5 P 0 40 Z 2 00 12 20 s 1 0 4 2 Example 47 X N 10 5 P 12 X 20 P 12 10 Z 20 10 5 5 s 5 P 0 40 Z 2 00 12 20 P Z 2 00 P Z 0 40 s 1 0 4 2 00 Example 47 X N 10 5 P 12 X 20 P 12 10 Z 20 10 5 5 s 5 P 0 40 Z 2 00 12 20 P Z 2 00 P Z 0 40 9772 6554 3218 s 1 3218 0 4 2 00 Example 47 X N 10 5 P 12 X 20 P 12 10 Z 20 10 5 5 P 0 40 Z 2 00 P Z 2 00 P Z 0 40 9772 6554 3218 Calculator normalcdf 12 20 10 5 Example 48 Suppose X N 78 12 P X 84 s 12 78 84 Example 48 Suppose X N 78 12 P X 84 P Z 84 78 P Z 0 50 12 s 12 78 84 s 1 0 50 Example 48 Suppose X N 78 12 P X 84 P Z 84 78 P Z 0 50 12 1 P Z 0 50 1 s 1 0 50 s 1 0 50 Example 48 Suppose X N 78 12 P X 84 P Z 84 78 P Z 0 50 12 1 P Z 0 50 1 6915 3085 Example 48 Suppose X N 78 12 P X 84 P Z 84 78 P Z 0 50 12 1 P Z 0 50 1 6915 3085 Calculator normalcdf 84 1E99 78 12 Practice Problems Suppose X N 70 12 1 Find the probability that X is greater than 88 P X 88 2 Find the probability that X is less than 94 P X 94 3 Find the probability that X is between 40 and 64 P 40 X 64 Practice Problem Answers Suppose X N 70 12 1 Find the probability that X is greater than 88 P X 88 P Z 88 70 12 P Z 1 50 1 P Z 1 50 1 9332 0668 Calculator normalcdf 88 1E99 70 12 2 Find the probability that X is less than 94 P X 94 P Z 94 70 12 P Z 2 00 9772 Calculator normalcdf 1E99 94 70 12 3 Find the probability that X is between 40 and 64 P 40 X 64 P 40 70 12 Z 64 70 12 P 2 50 Z 0 50 P Z 0 50 P Z 2 50 3085 0062 3023 Calculator normalcdf 40 64 70 12 Standard Normal Distribution All of the problems from the last lecture asked us to find the probability given a value or values of Z Now suppose the probability or area or proportion or percentage is given and we want to find the corresponding value of Z see page 145 There are three such problems Less Than Problem Suppose you want to find the value z such that the probability of being less than z or less than and equal to z is as specified Less Than Problem Suppose you want to find the value z such that the probability of being less than z or less than and equal to z is as specified To solve 1 Draw a normal curve and mark the information stated in the problem Less Than Problem Suppose you want to find the value z such that the probability of being less than z or less than and equal to z is as specified To solve 1 Draw a normal curve and mark the information stated in the problem 2 In the normal table find the specified less than probability in the body of the table and then read across and up to determine the appropriate z value Finding Values of Z Body of table Values of Z Value Problems on the Calculator See pages 160 and 161 for instructions for using the calculator to determine normal probabilities and values of normal variables 1 Hit 2nd then VARS this gives a list of distributions 2 Choose option 3 invNorm 3 You must enter three numbers the LESS THAN probability then the mean which is currently 0 and then the standard deviation which is currently 1 Example 42 Find the value of z such that the probability of being less than z is 8212 1 z P Z z 8212 Example 42 Find the value of z such that the probability of being less than …
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