Chapter 6 Introduction to Continuous Probability Distributions Business Statistics: A Decision-Making Approach 8th Edition Chapter 6 Introduction to Continuous Probability Distributions Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

Chapter Goals After completing this chapter, you should be able to: Convert values from any normal distribution to a standardized z-score Find probabilities using a normal distribution table Apply the normal distribution to business problems Recognize when to apply the uniform and exponential distributions Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

Continuous Probability Distributions A continuous random variable is a variable that can assume any value on a defined continuum (can assume an uncountable number of values) – see Chapter 5 thickness of an item time required to complete a task These can potentially take on any value, depending only on the ability to measure accurately. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

A B Types of Continuous Distributions Three types Normal Uniform Exponential A B Involves determining the probability for a RANGE of values rather than 1 particular incident or outcome Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

The Normal Distribution ‘Bell Shaped’ Symmetrical Mean=Median=Mode Location is determined by the mean, μ Spread is determined by the standard deviation, σ The random variable has an infinite theoretical range: +  to   f(x) σ x μ Mean Median Mode Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

Many Normal Distributions By varying the parameters μ and σ, we obtain different normal distributions Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

The Normal Distribution Shape f(x) Changing μ shifts the distribution left or right. Changing σ increases or decreases the spread. σ μ x Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

Finding Normal Probabilities Probability is measured by the area under the curve f(x) P ( a  x  b ) a b x Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

Probability as Area Under the Curve The total area under the curve is 1.0, and the curve is symmetric, so half is above the mean, half is below f(x) 0.5 0.5 μ x Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

The Standard Normal Distribution Also known as the “z” distribution Mean is defined to be 0 Standard Deviation is 1 f(z) 1 z Values above the mean have positive z-values Values below the mean have negative z-values Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

The Standard Normal Any normal distribution (with any mean and standard deviation combination) can be transformed into the standard normal distribution (z) Need to transform x units into z units Where x is any point of interest Can use the z value to determine probabilities Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

Translation to the Standard Normal Distribution Translate from x to the standard normal (the “z” distribution) by subtracting the mean of x and dividing by its standard deviation: z is the number of standard deviations units that x is away from the population mean Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

Example If x is distributed normally with mean of 100 and standard deviation of 50, the z value for x = 250 is This says that x = 250 is three standard deviations (3 increments of 50 units) above the mean of 100. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

Comparing x and z units 100 250 x 3.0 z μ = 100 σ = 50 3.0 z Note that the distribution is the same, only the scale has changed. We can express the problem in original units (x) or in standardized units (z) Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

The Standard Normal Table The Standard Normal table in the textbook (Appendix D) Gives the probability from the mean (zero) up to a desired value for z 0.4772 Example: P(0 < z < 2.00) = 0.4772 z 2.00 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

The Standard Normal Table (continued) The Standard Normal Table gives the probability between the mean and a certain z value The z value ALWAYS refers to the area between some value (-z or +z) and the mean Since the distribution is symmetrical, the Standard Normal Table only displays probabilities for ½ of the full distribution Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

The Standard Normal Table (continued) The column gives the value of z to the second decimal point The row shows the value of z to the first decimal point The value within the table gives the probability from z = 0 up to the desired z value . 2.0 .4772 P(0 < z < 2.00) = 0.4772 2.0 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

General Procedure for Finding Probabilities Determine m and s Define the event of interest e.g., P(x > x1) Convert to standard normal Use the table to find the probability Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

z Table Example Suppose x is normal with mean 8.0 and standard deviation 5.0. Find P(8 < x < 8.6) Calculate z-values: 8 8.6 x 0.12 Z P(8 < x < 8.6) = P(0 < z < 0.12) Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

z Table Example (continued) Suppose x is normal with mean 8.0 and standard deviation 5.0. Find P(8 < x < 8.6) = 8  = 5 = 0  = 1 x z 8 8.6 0.12 P(8 < x < 8.6) P(0 < z < 0.12) Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

Solution: Finding P(0 < z < 0.12) Standard Normal Probability Table (Portion) P(8 < x < 8.6) = P(0 < z < 0.12) .02 z .00 .01 0.0478 0.0 .0000 .0040 .0080 0.1 .0398 .0438 .0478 0.2 .0793 .0832 .0871 Z 0.3 .1179 .1217 .1255 0.00 0.12 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

Finding Normal Probabilities Suppose x is normal with mean 8.0 and standard deviation 5.0. Now Find P(x < 8.6) The probability of obtaining a value less than 8.6 P = 0.5 Z 8.0 8.6 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

Finding Normal Probabilities (continued) Suppose x is normal with mean 8.0 and standard deviation 5.0. Now Find P(x < 8.6) 0.0478 0.5000 P(x < 8.6) = P(z < 0.12) = P(z < 0) + P(0 < z < 0.12) = 0.5000 + 0.0478 = 0.5478 Z 0.00 0.12 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

Upper Tail Probabilities Suppose x is normal with mean 8.0 and standard deviation 5.0. Now Find P(x > 8.6) Z 8.0 8.6 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

Upper Tail Probabilities (continued) Now Find P(x > 8.6)… P(x > 8.6) = P(z > 0.12) = P(z > 0) - P(0 < z < 0.12) = 0.5000 - 0.0478 = 0.4522 0.0478 0.5000 0.4522 Z Z 0.12 0.12 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

Lower Tail Probabilities Suppose x is normal with mean 8.0 and standard deviation 5.0. Now Find P(7.4 < x < 8) Z 8.0 7.4 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

Lower Tail Probabilities (continued) Now Find P(7.4 < x < 8)…the probability between 7.4 and the mean of 8 The Normal distribution is symmetric, so we use the same table even if z-values are negative: P(7.4 < x < 8) = P(-0.12 < z < 0) = 0.0478 0.0478 Z 8.0 7.4 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

Normal Probabilities in PHStat We can use Excel and PHStat to quickly generate probabilities for any normal distribution We will find P(8 < x < 8.6) when x is normally distributed with mean 8 and standard deviation 5 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

Select desired options and enter values PHStat Dialogue Box Select desired options and enter values Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

PHStat Output Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

Recall Tchebyshev from Chpt. 3 Empirical Rules What can we say about the distribution of values around the mean if the distribution is normal? f(x) μ ± 1σ covers about 68% of x’s σ σ Recall Tchebyshev from Chpt. 3 x μ-1σ μ μ+1σ 68.26% Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

μ ± 2σ covers about 95% of x’s μ ± 3σ covers about 99.7% of x’s The Empirical Rule (continued) μ ± 2σ covers about 95% of x’s μ ± 3σ covers about 99.7% of x’s 3σ 3σ 2σ 2σ μ x μ x 95.44% 99.72% Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

Importance of the Rule If a value is about 2 or more standard deviations away from the mean in a normal distribution, then it is far from the mean The chance that a value that far or farther away from the mean is highly unlikely, given that particular mean and standard deviation Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

The Uniform Distribution The uniform distribution is a probability distribution that has equal probabilities for all possible outcomes of the random variable Referred to as the distribution of “little information” Probability is the same for ANY interval of the same width Useful when you have limited information about how the data “behaves” (e.g., is it skewed left?) Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

The Uniform Distribution (continued) The Continuous Uniform Distribution: f(x) = where f(x) = value of the density function at any x value a = lower limit of the interval of interest b = upper limit of the interval of interest Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

The Mean and Standard Deviation for the Uniform Distribution The mean (expected value) is: The standard deviation is where a = lower limit of the interval from a to b b = upper limit of the interval from a to b Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

Steps for Using the Uniform Distribution Define the density function Define the event of interest Calculate the required probability f(x) x Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

Uniform Distribution Example: Uniform Probability Distribution Over the range 2 ≤ x ≤ 6: 1 f(x) = = .25 for 2 ≤ x ≤ 6 6 - 2 f(x) .25 x 2 6 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

Uniform Distribution Example: Uniform Probability Distribution Over the range 2 ≤ x ≤ 6: Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

The Exponential Distribution Used to measure the time that elapses between two occurrences of an event (the time between arrivals) Examples: Time between trucks arriving at a dock Time between transactions at an ATM Machine Time between phone calls to the main operator Recall l = mean for Poisson (see Chpt. 5) Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

The Exponential Distribution The probability that an arrival time is equal to or less than some specified time a is where 1/ is the mean time between events and e = 2.7183 NOTE: If the number of occurrences per time period is Poisson with mean , then the time between occurrences is exponential with mean time 1/  and the standard deviation also is 1/l Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

Exponential Distribution (continued) Shape of the exponential distribution f(x)  = 3.0 (mean = .333)  = 1.0 (mean = 1.0) = 0.5 (mean = 2.0) x Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

Example Example: Customers arrive at the claims counter at the rate of 15 per hour (Poisson distributed). What is the probability that the arrival time between consecutive customers is less than five minutes? Time between arrivals is exponentially distributed with mean time between arrivals of 4 minutes (15 per 60 minutes, on average) 1/ = 4.0, so  = .25 P(x < 5) = 1 - e-a = 1 – e-(.25)(5) = 0.7135 There is a 71.35% chance that the arrival time between consecutive customers is less than 5 minutes Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

Using PHStat Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

Chapter Summary Reviewed key continuous distributions normal uniform exponential Found probabilities using formulas and tables Recognized when to apply different distributions Applied distributions to decision problems Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

Printed in the United States of America. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall