This preview shows pages 1–16. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
Standardizing to Find Normal Probabilities Start by converting x into a z-score σ μ - = x z The Normal Model 5

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
The Empirical Rule, Revisited Standard normal density curve The Normal Model 7

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
Tolerance Interval Empirical Rule for Normal Populations 9

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
Using Normal Tables Example: SATS and Normality 13

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.