Copyright © 2014, 2011 Pearson Education, Inc. 1 Chapter 12 The Normal Probability Model
Copyright © 2014, 2011 Pearson Education, Inc Normal Random Variable Black Monday (October, 1987) prompted investors to consider insurance against another “accident” in the stock market. How much should an investor expect to pay for this insurance? Insurance costs call for a random variable that can represent a continuum of values (not counts)
Copyright © 2014, 2011 Pearson Education, Inc Normal Random Variable Prices for One-Carat Diamonds
Copyright © 2014, 2011 Pearson Education, Inc Normal Random Variable Percentage Change in Stock Market
Copyright © 2014, 2011 Pearson Education, Inc Normal Random Variable X-ray Measurements of Bone Density
Copyright © 2014, 2011 Pearson Education, Inc Normal Random Variable With the exception of Black Monday, the histogram of market changes is bell-shaped Histograms of diamond prices and bone density measurements are bell-shaped All three involve a continuous range of values; all three can be modeled using normal random variables
Copyright © 2014, 2011 Pearson Education, Inc Normal Random Variable Definition A continuous random variable whose probability distribution defines a standard bell-shaped curve.
Copyright © 2014, 2011 Pearson Education, Inc Normal Random Variable Central Limit Theorem The probability distribution of a sum of independent random variables of comparable variance tends to a normal distribution as the number of summed random variables increases.
Copyright © 2014, 2011 Pearson Education, Inc Normal Random Variable Central Limit Theorem Illustrated
Copyright © 2014, 2011 Pearson Education, Inc Normal Random Variable Central Limit Theorem Explains why bell-shaped distributions are so common Observed data are often the accumulation of many small factors (e.g., the value of the stock market depends on many investors)
Copyright © 2014, 2011 Pearson Education, Inc Normal Random Variable Normal Probability Distribution Defined by the parameters µ and σ 2 The mean µ locates the center The variance σ 2 controls the spread
Copyright © 2014, 2011 Pearson Education, Inc Normal Random Variable Standard Normal Distribution (µ = 0; σ 2 = 1)
Copyright © 2014, 2011 Pearson Education, Inc Normal Random Variable Normal Probability Distribution A normal random variable is continuous and can assume any value in an interval Probability of an interval is area under the distribution over that interval (note: total area under the probability distribution is 1)
Copyright © 2014, 2011 Pearson Education, Inc Normal Random Variable Probabilities are Areas Under the Curve
Copyright © 2014, 2011 Pearson Education, Inc Normal Random Variable Normal Distributions with Different µ’s
Copyright © 2014, 2011 Pearson Education, Inc Normal Random Variable Normal Distributions with Different σ 2 ’s
Copyright © 2014, 2011 Pearson Education, Inc The Normal Model Definition A model in which a normal random variable is used to describe an observable random process with µ set to the mean of the data and σ set to s.
Copyright © 2014, 2011 Pearson Education, Inc The Normal Model Normal Model for Diamond Prices Set µ = $4,066 and σ = $738.
Copyright © 2014, 2011 Pearson Education, Inc The Normal Model Normal Model for Stock Market Changes Set µ = 0.94% and σ = 4.32%.
Copyright © 2014, 2011 Pearson Education, Inc The Normal Model Normal Model for Bone Density Scores Set µ = and σ = 1.3.
Copyright © 2014, 2011 Pearson Education, Inc The Normal Model Standardizing to Find Normal Probabilities Start by converting x into a z-score
Copyright © 2014, 2011 Pearson Education, Inc The Normal Model Standardizing Example: Diamond Prices Normal with µ = $ 4,066 and σ = $738 Want P(X > $5,000)
Copyright © 2014, 2011 Pearson Education, Inc The Normal Model The Empirical Rule, Revisited
Copyright © 2014, 2011 Pearson Education, Inc. 24 4M Example 12.1: 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 600 or more?
Copyright © 2014, 2011 Pearson Education, Inc. 25 4M Example 12.1: SATS AND NORMALITY Method – Use the Normal Model
Copyright © 2014, 2011 Pearson Education, Inc. 26 4M Example 12.1: SATS AND NORMALITY Mechanics A math SAT score of 600 is equivalent z = 1. Using the empirical rule, we find that 15.85% of test takers score 600 or better.
Copyright © 2014, 2011 Pearson Education, Inc. 27 4M Example 12.1: SATS AND NORMALITY 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.
Copyright © 2014, 2011 Pearson Education, Inc The Normal Model Using Normal Tables 27 of 45
Copyright © 2014, 2011 Pearson Education, Inc The Normal Model Example: What is P(-0.5 ≤ Z ≤ 1)? – =
Copyright © 2014, 2011 Pearson Education, Inc Percentiles 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 weight as 16 oz. To what weight should the mean of the process be adjusted so that the chance of an underweight box is only 0.005?
Copyright © 2014, 2011 Pearson Education, Inc Percentiles Quantile of the Standard Normal The p th quantile of the standard normal probability distribution is that value of z such that P(Z ≤ z ) = p. Example: Find z such that P(Z ≤ z ) = z =
Copyright © 2014, 2011 Pearson Education, Inc Percentiles Quantile of the Standard Normal Find new mean weight (µ) for process
Copyright © 2014, 2011 Pearson Education, Inc. 33 4M Example 12.2: VALUE AT RISK Motivation Suppose the $1 million portfolio of an investor is expected to average 10% growth over the next year with a standard deviation of 30%. What is the VaR (value at risk) using the worst 5%?
Copyright © 2014, 2011 Pearson Education, Inc. 34 4M Example 12.2: VALUE AT RISK Method The random variable is percentage change next year in the portfolio. Model it using the normal, specifically N(10, 30 2 ).
Copyright © 2014, 2011 Pearson Education, Inc. 35 4M Example 12.2: VALUE AT RISK Mechanics From the normal table, we find z = for P(Z ≤ z) = 0.05
Copyright © 2014, 2011 Pearson Education, Inc. 36 4M Example 12.2: VALUE AT RISK Mechanics This works out to a change of -39.3% µ σ = 10 – 1.645(30) = -39.3%
Copyright © 2014, 2011 Pearson Education, Inc. 37 4M Example 12.2: VALUE AT RISK Message The annual value at risk for this portfolio is $393,000 at 5% (eliminating the worst 5% of the situations).
Copyright © 2014, 2011 Pearson Education, Inc Departures from Normality Multimodality. More than one mode suggesting data come from distinct groups. Skewness. Lack of symmetry. Outliers. Unusual extreme values.
Copyright © 2014, 2011 Pearson Education, Inc Departures from Normality Normal Quantile Plot Diagnostic scatterplot used to determine the appropriateness of a normal model If data track the diagonal line, the data are normally distributed
Copyright © 2014, 2011 Pearson Education, Inc Departures from Normality Normal Quantile Plot Normal Distributions on Both Axes
Copyright © 2014, 2011 Pearson Education, Inc Departures from Normality Normal Quantile Plot Distribution on y-axis Not Normal
Copyright © 2014, 2011 Pearson Education, Inc Departures from Normality Normal Quantile Plot (Diamond Prices) All points are within dashed curves, normality indicated.
Copyright © 2014, 2011 Pearson Education, Inc Departures from Normality Normal Quantile Plot (Diamonds of Varying Quality) Points outside the dashed curves, normality not indicated.
Copyright © 2014, 2011 Pearson Education, Inc Departures from Normality Skewness Measures lack of symmetry. K 3 = 0 for normal data.
Copyright © 2014, 2011 Pearson Education, Inc Departures from Normality Kurtosis Measures the prevalence of outliers. K 4 = 0 for normal data.
Copyright © 2014, 2011 Pearson Education, Inc Departures from Normality Prices for Diamonds of Varying Quality
Copyright © 2014, 2011 Pearson Education, Inc. 47 Best Practices Recognize that models approximate what will happen. Inspect the histogram and normal quantile plot before using a normal model. Use z–scores when working with normal distributions.
Copyright © 2014, 2011 Pearson Education, Inc. 48 Best Practices (Continued) Estimate normal probabilities using a sketch and the Empirical Rule. Be careful not to confuse the notation for the standard deviation and variance.
Copyright © 2014, 2011 Pearson Education, Inc. 49 Pitfalls Do not use the normal model without checking the distribution of data. Do not think that a normal quantile plot can prove that the data are normally distributed. Do not confuse standardizing with normality.