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Unit 5 Probability Distribution MM207 JoEllen Green, MS Normal Distribution Characteristics • Most Statistical test are based on the Normal Curve. You have already hear about the normal curve in school when some students would ask their instructor, “Are you grading on a curve?” What does grading on a curve mean? 0.2 0.15 0.1 0.05 F 0 -∞ -9 -8 -7 -6 -5 -4 -3 C D -2 -1 0 B 1 A 2 3 P(x) 4 5 6 7 8 P(x) 9 ∞ Characteristics of a Normal Distribution It is a continuous distribution. 0.2 0.15 0.1 0.05 0 -∞ -9 -8 -7 -6 -5 -4 -3 -2 -1 P(x) 0 1 2 3 4 5 6 7 8 P(x) 9 ∞ Normal Curve Characteristics It is symmetric on both sides of the mean. 0.2 0.15 0.1 0.05 0 -∞ -9 -8 -7 -6 -5 -4 -3 -2 -1 P(x) 0 1 2 3 4 5 6 7 8 P(x) 9 ∞ Normal Curve Characteristics Values range from -∞ to ∞ 0.2 0.15 0.1 0.05 0 -∞ -9 -8 -7 -6 -5 -4 -3 -2 -1 P(x) 0 1 2 3 4 5 6 7 8 P(x) 9 ∞ Normal Curve Characteristics Mean = Median = Mode 0.2 0.15 0.1 0.05 0 -∞ -9 -8 -7 -6 -5 -4 -3 -2 -1 P(x) 0 1 2 3 4 5 6 7 8 P(x) 9 ∞ Normal Curve Characteristics Follows the Empirical Rule 34% 13.5% 0.015% 2.35% 34% 13.5% 2.35% 0.015% Standard Normal Curve • Mean = µ = 0 • Standard Deviation= σ = 1 • We have tables to find the probability for this distribution Finding Probabilities with Standard Normal • Turn to page A16 and A17 in your text book or take the chart that folds out in the back of the book. • We will use Table 4. Finding the Probabilities • Find the probability that Z < -1.35. • Word Phrases: to the left of -1.35 less than -1.35 • P(Z < -1.35) • When I show you how to find the P(Z < -1.35), You can find the P(Z < -1.96). To Calculate P( Z> 1.25) • What is the P(Z < 1.25) = 0.8944 • Subtract it from 1? • What is your answer? 1 – 0.8944 0.1056 Finding other Probabilities • Find P(Z > -1.35) Other phrases: more than -1.35 greater than -1.35 Finding other Probabilities • P(Z > -1.96) = • P(Z < 1.23) = • P(Z > 1.65) = Computing P( -1.50 < Z < 1.25) = ? A. What is P(Z < 1.25) = ? B. What is the P(Z < -1.50)? C. Subtract B from A? Finding the Probabilities • P( -1.35 < Z < 1.23) = • P( 0.53 < Z < 1.88 )= • P(-0.15 < Z < 2.53) = P( Z< -1.20 or Z > 1.30) • Do the ranges overlap? Computing P( Z< -1.20 or Z > 1.30) • Compute P( Z < -1.20) • Compute P( Z > 1.30) • Add them together Compute the following probabilities • P( Z < 0 or Z > 2.33) = • P( Z < -2.00 or Z > 1.30) How to compute a Z score • If your mean is not 0 and the standard deviation is not 1, can you use the tables to compute the probabilities? Z Score Find the Probability • The Mean µ = 86 and standard deviation σ = 5. • Find the probability P(X < 80) • • • • What is the Value? What is the Mean? What is the Standard Deviation? Compute Z Finish the Problem • P(X < 80) = P(Z < -1.20) = • Find P(X < 100) Calculating Probabilities • Find P( 70 < X < 80) A) Find Z Score for 70 B) Find the Probability for Find the Z Score for 80 c) Find the Probability for Subtract A from B Calculating Probabilities • Mean µ = 86 and Standard Deviation σ = 5. • P(x < 75) • P( 85 < X < 95) Page 257 #14 The lengths of Atlantic croaker fish are normally distributed, with a mean of 10 inches and a standard deviation of 2 inches. An Atlantic croaker fish is randomly selected. a) Find the probability that his height is less than 66 inches? Central Limit Theorem What if the distribution is not normal? We need only gather a sample of size, n > 30 or more. We can assume the sample is normal with mean µ and standard deviation σ/√n Find the Probability Page 279 Problem #13 For a sample of n=36, find the probability of a sample mean being less than 12.2 if µ = 12 and σ = 0.95. What is the value? What is the mean? What is the standard deviation? Compute the Z score? What is the probability? Find the Probability Page 279 Problem #14 For a sample of n=100, find the probability of a sample mean being greater than 12.2 if µ = 12 and σ = 0.95. What is the value? What is the mean? What is the standard deviation? Compute the Z score? What is the probability? Thanks for Attending!