# VCU STAT 210 - Lecture21(2) (1) (95 pages)

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## Lecture21(2) (1)

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- Pages:
- 95
- School:
- Virginia Commonwealth University
- Course:
- Stat 210 - Basic Practice of Statistics

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STAT 210 Lecture 21 Normal Distributions October 13 2016 Practice Problems Pages 162 through 165 Relevant problems VI 6 through VI 15 Recommended problems VI 14 and VI 15 Additional Reading and Examples Read pages 158 through 160 Test 4 Wednesday October 19 Questions for the first 10 minutes then test papers due promptly at the end of class Covers chapter 6 pages 139 168 Combination of multiple choice questions and written short answer questions and problems Formulas and Z table provided Bring a calculator Practice Tests and Formula Sheet on Blackboard Clicker Notation X N m s X is distributed normal with mean m and standard deviation s Clicker 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 Probability The 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 Z Score Transformation 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 For a problem that asks to find a probability we convert from X to Z using the following Z Score Transformation Z X m value mean s standard deviation This was used in examples 47 and 48 Review Problems Suppose X N 450 85 1 Find the probability that X is greater than 500 P X 500 2 Find the probability that X is between 400 and 640 P 400 X 640 Review Problem Answers Suppose X N 450 85 1 Find the probability that X is greater than 500 P X 500 P Z 500 450 85 P Z 0 59 1 P Z 0 59 1 7224 2776 Calculator normalcdf 500 1E99 450 85 2 Find the probability that X is between 400 and 640 P 400 X 640 P 400 450 85 Z 640 450 85 P 0 59 Z 2 24 P Z 2 24 P Z 0 59 9875 2776 7099 Calculator normalcdf 400 640 450 85 Standard Normal Distribution All of the problems from the last lecture asked us to find the probability given a value or values 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 Optional 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 Optional 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 TI 83 84 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 z is 8212 1 z P Z z 8212 2 In the normal table find 8212 in the body of the table Page 339 Example 42 Example 42 Find the value of z such that the probability of being less than z is 8212 1 z P Z z 8212 2 In the normal table find 8212 in the body of the table This corresponds to z 0 92 Calculator invNorm 8212 0 1 Example 43 Find the value of z such that the probability of being less than z is 10 1 z P Z z 10 Example 43 Find the value of z such that the probability of being less than z is 10 1 z P Z z 10 1 10 z 0 Example 43 Find the value of z such that the probability of being less than z is 10 1 z P Z z 10 1 10 z 0 2 In the normal table find 10 in the body of the table Page 338 Example 43 Find the value of z such that the probability of being less than z is 10 1 z P Z z 10 1 10 z 0 2 In the normal table find 10 in the body of the table Closest is 1003 corresponding to z 1 28 Calculator invNorm 10 0 1 Greater Than Problem Suppose you want to find the value z such that the probability of being greater than z or greater than and equal to z is as specified Greater Than Problem Suppose you want to find the value z such that the probability of being greater than z or greater than and equal to z is as specified To solve 1 Optional Draw a normal curve and mark the information stated in the problem Greater Than Problem Suppose you want to find the value z such that the probability of being greater than z or greater than and equal to z is as specified To solve 1 Optional Draw a normal curve and mark the information stated in the problem 2 Convert the problem from a greater than problem to a less than problem by subtracting the given probability from 1 Greater Than Problem Suppose you want to find the value z such that the probability of being greater than z or greater than and equal to z is as specified To solve 1 Optional Draw a normal curve and mark the information stated in the problem 2 Convert the problem from a greater than problem to a less than problem by subtracting the given probability from 1 3 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 Example 44 Find the value of z such that the probability of being greater than z is 33 1 z P Z z 33 Example 44 Find the value …

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