#### Transcript Slides 1-17 Normal and CLT

BA 275 Quantitative Business Methods Agenda Quiz #1 Experiencing Random Behavior Normal Probability Distribution Normal Probability Table 1 Review Question: Warranty Level Mean = 30,000 miles STD = 5,000 miles Q1: If the level of warranty is set at 15,000 miles, about what % of tires will be returned under the warranty? Q2: If we can accept that up to 2.5% of tires can be returned under warranty, what should be the warranty level? 0.04 0.03 0.02 0.01 0 0 10 20 30 40 50 60 2 The Empirical Rule is not Enough Mean = 30,000 miles STD = 5,000 miles Q1: If the level of warranty is set at 12,000 miles, about what % of tires will be returned under the warranty? Q2: If we can accept that up to 3% of tires can be returned under warranty, what should be the warranty level? 0.04 0.03 0.02 0.01 0 0 10 20 30 40 50 60 3 The Normal Probability Distribution A specific curve that is symmetric and bell-shaped with two parameters m and s2. It has been used to describe variables that are too cumbersome to be consider as discrete (i.e., continuous variable). For example, Physical measurements of members of a biological population (e.g., heights and weights), IQ and exam scores, amounts of rainfall, scientific measurements, etc. It can be used to describe the outcome of a binomial experiment when the number of trials is large. It is the foundation of classical statistics. Central Limit Theorem 4 Standard Normal Probabilities (Table A) 5 Standard Normal Probabilities (Table A) 6 Example 1 m =0 s=1 a = 1.96 A Prob = ??? a 7 Example 2 C m=0 s=1 a = ????? Prob = 0.0793 a 8 Example 3 D m=0 s=2 a = 2.00 b = ?????? Prob = 0.1005 a b 9 Sampling Distribution (Section 4.4) A sampling distribution describes the distribution of all possible values of a statistic over all possible random samples of a specific size that can be taken from a population. 45 3,169,000,000,000 25 10 Central Limit Theorem (CLT) The CLT applied to Means If X ~ N ( m , s 2 ) , then X ~ N ( m , s2 ). n If X ~ any distribution with a mean m, and variance s2, then X ~ N ( m , CLT demo s2 n ) given that n is large. With a sample of size n = 25, can we predict the value of the sample mean? Example 1: X ~ a normal distribution with the mean 16, and variance 25. Example 2: X ~ a distribution with the mean 8.08, and variance 38.6884. 11 Answer: Review Question: Warranty Level Mean = 30,000 miles STD = 5,000 miles Q1: If the level of warranty is set at 15,000 miles, about what % of tires will be returned under the warranty? => 0.15% Q2: If we can accept that up to 2.5% of tires can be returned under warranty, what should be the warranty level? => 20,000 miles 0.04 0.03 0.02 0.01 0 0 10 20 30 40 50 60 12 Answer: The Empirical Rule is not Enough Mean = 30,000 miles STD = 5,000 miles Q1: If the level of warranty is set at 12,000 miles, about what % of tires will be returned under the warranty? => almost 0.0000 Q2: If we can accept that up to 3% of tires can be returned under warranty, what should be the warranty level? => 20,600 miles 0.04 0.03 0.02 0.01 0 0 10 20 30 40 50 60 13 Answer: Example 1 m =0 s=1 a = 1.96 A Prob = ??? a Prob = 0.025 14 Answer: Example 2 C m=0 s=1 a = ????? Prob = 0.0793 a a = -1.41 15 Answer: Example 3 D m=0 s=2 a = 2.00 b = ?????? Prob = 0.1005 a b b = 3.14 16