1 - BUAD 310: Applied Business Statistics September 1, 2010...

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September 1, 2010 BUAD 310: Applied Business Statistics 1
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Outline for Today Normal model (distribution) The “68-95-99.7 rule” (empirical rule for normal populations) Standardizing Finding normal probabilities (going from Z values to areas under the curve) 2
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The Normal Model Definition A model in which a normal random variable is used to describe an observable random process with μ estimated as the mean of the data and σ estimated as s X 3
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Normal Model for Stock Market Changes Estimate μ as 0.972% and estimate σ as 4.49% The Normal Model Normal Model for Diamond Prices Estimate μ as $4,066 and σ as $738 4
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Standardizing to Find Normal Probabilities Start by converting x into a z-score σ μ - = x z The Normal Model 5
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Standardizing Example: Diamond Prices Normal with μ = $4,066 and σ = $738 Want P(X > $5,000) ( 29 - = - - = 27 . 1 738 066 , 4 000 , 5 000 , 5 000 , 5 $ Z P X P X P σ μ The Normal Model 6
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The Empirical Rule, Revisited Standard normal density curve The Normal Model 7
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Tolerance Interval Definition: an interval of numbers that contains a specified percentage of the individual measurements in a population The empirical rule says that if a population has mean μ and standard deviation σ and is described by a normal curve , then 68.26% of the population measurements lie within one standard deviation of the mean: [ μ - , μ + ] 95.44% of the population measurements lie within two standard deviations of the mean: [ μ -2 , μ +2 ] 99.73% of the population measurements lie within three standard deviations of the mean: [ μ -3 , μ +3 ] These 3 intervals [ μ - , μ + ], [ μ - , μ +2 ], and [ μ - , μ +3 ] are all tolerance intervals 8
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Tolerance Interval Empirical Rule for Normal Populations 9
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Example: SATS and Normality Motivation Math SAT scores are normally distributed with a mean of 500 and standard deviation of 100. What is the probability of a company hiring someone with a math SAT score of at least 600? Method – Use the Normal Model 10
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11 68.3 % 15.85 % 15.85 % Answer = 15.85% 600 500 400 Example: SATS and Normality Mechanics A math SAT score of 600 is equivalent to z = 1. Using the empirical rule, we find that 15.85% of test takers score 600 or better
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Message About one-sixth of those who take the math SAT score 600 or above. Although not that common, a company can expect to find candidates who meet this requirement Example: SATS and Normality 12
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Using Normal Tables Example: SATS and Normality 13
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Example: What is P(-0.5 Z 1)? ≤ ≤ From the table: P(X 1)=0.8413 and P(X<-0.5)=0.3085 P(-0.5 X 1)= ≤ ≤ 0.8413 – 0.3085 = 0.5328 Example 14
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Percentiles Motivating Example : Suppose a packaging system fills boxes such that the weights are normally distributed with a μ = 16.3 oz. and σ = 0.2 oz. The package label states the
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This note was uploaded on 11/30/2010 for the course BUAD 310 at USC.

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1 - BUAD 310: Applied Business Statistics September 1, 2010...

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