Slide 1Practice ProblemsAdditional Reading and ExamplesTest 4Slide 5NotationSlide 7Standard Normal DistributionProbabilityZ-Score TransformationZ-Score TransformationReview ProblemsReview Problem AnswersStandard Normal DistributionLess Than ProblemLess Than ProblemLess Than ProblemFinding Values of ZTI-83/84 CalculatorExample 42Example 42Example 42Example 42Example 43Example 43Example 43Example 43Greater Than ProblemGreater Than ProblemGreater Than ProblemGreater Than ProblemExample 44Example 44Example 44Example 44Example 45Example 45Example 45Example 45Between ProblemBetween ProblemBetween ProblemBetween ProblemBetween ProblemBetween ProblemBetween ProblemBetween ProblemExample 46Example 46Example 46Example 46Example 46Example 46Example 46Example 46Normal VariablesZ-Score TransformationZ-Score TransformationZ-Score TransformationZ-Score TransformationExampleZ-Score TransformationExample 49Example 49Example 49Example 49Example 49Example 49Example 49Example 49Example 50Example 50Example 50Example 50Example 50Example 50Example 50Example 50Steps for SolvingExample 51Example 51Example 51Example 51Example 51Example 51Example 52Example 52Example 52Example 52Example 52Example 52Example 52Practice ProblemsPractice Problem AnswersSlide 95STAT 210Lecture 21Normal DistributionsOctober 13, 2016Practice ProblemsPages 162 through 165Relevant problems: VI.6 through VI.15Recommended problems: VI.14 and VI.15Additional Reading and ExamplesRead pages 158 through 160Test 4Wednesday, October 19Questions 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.ClickerNotationX ~ N (m , s)“X is distributed normal with mean m and standard deviation s”ClickerStandard 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 - 339ProbabilityThe 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 TransformationSuppose X is distributed normal with some mean m not equal to 0 and/or some standarddeviation s not equal to 1:X ~ N(m, s)Z-Score TransformationFor a problem that asks to find a probabi l i ty, we convert from X to Z using the following Z-Score Transformation:Z = X - m = value - mean s standard deviationThis was used in examples 47 and 48.Review ProblemsSuppose 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 AnswersSuppose 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 = .2776Calculator: 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 = .7099Calculator: normalcdf(400, 640, 450, 85)Standard Normal DistributionAll 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 ProblemSuppose 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 ProblemSuppose 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 ProblemSuppose 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 ZBody of tableValues of ZTI-83/84 CalculatorSee pages 160 and 161 for instructions for using thecalculator to determine normal probabilities and valuesof normal variables.1. Hit 2nd, then VARS – this gives a list of distributions2. Choose option 3: invNorm3. 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 42Find the value of z such that the probability of being less than z is .8212.1. z: P(Z < z) = .8212Example 42Find 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 42Example 42Find 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 = 0.92Calculator: invNorm(.8212, 0, 1)Example 43Find the value of z such that the probability of being less than z is .10.1. z: P(Z < z) = .10Example 43Find the value of z such that the probability of being less than z is .10.1. z: P(Z < z) = .10z= 10.10Example 43Find the value of z such that the probability of being less than z is .10.1. z: P(Z < z) = .10 2. In the normal table, find .10 in the body of the table. Page 338: z= 10.10Example 43Find the value of z such that the probability of being less than z is .10.1. z: P(Z < z) = .10 2. In the normal table, find .10 in the body of the table. Closest is .1003, corresponding to z = -1.28z= 10.10Calculator: invNorm(.10, 0, 1)Greater Than ProblemSuppose 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 ProblemSuppose you want to find the value z such that the probability of being greater than z (or greater than and equal to
View Full Document
Unlocking...