Business analytics data analysis and decision making 5th by wayne l winston chapter 05

May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.The Normal Distribution  The single most important distribution in statistics

DECISION MAKING

Normal, Binomial, Poisson, and Exponential Distributions

5

 Several specific distributions commonly occur in

a variety of business situations:

 Normal distribution—a continuous distribution

characterized by a symmetric bell-shaped curve

 Binomial distribution—a discrete distribution that is relevant when we sample from a population with

only two types of members or when we perform a series of independent, identical experiments with

only two possible outcomes

 Poisson distribution—a discrete distribution that

describes the number of events in any period of time

 Exponential distributions—a continuous distribution

that describes the times between events

© 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

The Normal Distribution

 The single most important distribution in

statistics is the normal distribution

 It is a continuous distribution and is the basis of the familiar symmetric bell-shaped curve

 Any particular normal distribution is specified by its mean and standard deviation

 By changing the mean, the normal curve shifts to the right

 The normal distribution is a two-parameter family, where

the two parameters are the mean and standard deviation.

Continuous Distributions and

Density Functions (slide 1 of 2)

 For continuous distributions, instead of a list

of possible values, there is a continuum of

possible values, such as all values between

0 and 100 or all values greater than 0

 Instead of assigning probabilities to each

individual value in the continuum, the total

probability of 1 is spread over this continuum

 The key to this spreading is called a density

function, which acts like a histogram

 The higher the value of the density function, the

more likely this region of the continuum is.

© 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Continuous Distributions and

Density Functions (slide 2 of 2)

 A density function , usually denoted by f(x), specifies the probability distribution of a continuous random variable X

 The higher f(x) is, the more likely x is

 The total area between the graph of f(x) and the horizontal

axis, which represents the total probability, is equal to 1

 f(x) is nonnegative for all possible values of X.

 Probabilities are found from a density function as areas under the curve.

The Normal Density

 The normal distribution is a continuous distribution

with possible values ranging over the entire number

line—from “minus infinity” to “plus infinity.”

 Only a relatively small range has much chance of occurring

 The normal density function is actually quite complex, in

spite of its “nice” bell-shaped appearance.

 The formula for the normal density function, where μ

and σ are the mean and standard deviation, is:

© 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Standardizing: Z-Values

 The standard normal distribution has mean 0 and standard deviation 1, so it is denoted by

N(0,1)

 It is also referred to as the Z distribution.

 To standardize a variable, subtract its mean

and then divide the difference by the standard deviation:

 A Z-value is the number of standard deviations to the right or left of the mean.

 If Z is positive, the original value is to the right of the

mean.

 If Z is negative, the original value is the left of the mean.

Example 5.1:

Standardizing.xlsx

 Objective: To use Excel® to standardize annual returns of

various mutual funds.

 Solution: Data set includes the annual returns of 30 mutual

© 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Normal Tables and Z-Values

 A common use for Z-values and the standard normal

distribution is in calculating probabilities and percentiles

by the traditional method.

 This method is based on a table of the standard normal

distribution found in many statistics textbooks An example of such a table is given below.

 The body of the table contains probabilities.

 The left and top margins contain possible values.

Normal Calculations in Excel

 Two types of calculations are typically made with

normal distributions: finding probabilities and

finding percentiles

 The functions used for normal probability calculations

are NORMDIST and NORMSDIST

 The main difference between these is that the one with the

“S” (for standardized) applies only to N(0, 1) calculations, whereas NORMDIST applies to any normal distribution.

 Percentile calculations that take a probability and return

a value are often called inverse calculations.

 The Excel functions for these are named NORMINV and

NORMSINV

 Again, the “S” in the second of these indicates that it

applies to the standard normal distribution.

© 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Example 5.2:

Normal Calculations.xlsx (slide 1 of 2)

 Objective: To calculate probabilities and

percentiles for standard normal and general

normal distributions in Excel.

 Solution: For “less than” probabilities, use

NORMDIST or NORMSDIST directly.

 For “greater than” probabilities, subtract the

NORMDIST or NORMSDIST function from 1.

 For “between” probabilities, subtract the two

NORMDIST or NORMSDIST functions.

 For percentile calculations, use the NORMINV or

NORMSINV function with the specified probability

as the first argument.

Example 5.2:

Normal Calculations.xlsx (slide 2 of 2)

© 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Empirical Rules Revisited

 Three empirical rules apply to many data sets:

 About 68% of the data fall within one

standard deviation of the mean.

 About 95% fall within two standard

deviations of the mean.

 Almost all fall within three standard

deviations of the mean.

 For these rules to hold with real data, the distribution of the data must be at least approximately symmetric and bell-

shaped.

Weighted Sums of Normal

Random Variables

 One very attractive property of the normal

distribution is that if you create a weighted sum of normally distributed random variables, the weighted sum is also normally distributed.

 This is true even if the random variables are not

independent.

 If X 1 through X n are n independent and normally

distributed random variables with common mean μ and

common standard deviation σ, then the sum X 1 + … +

X n is normally distributed with mean nμ, variance nσ 2 ,

and standard deviation √nσ.

 If a 1 through a n are any constants, then the weighted

sum a 1 X 1 + … + a n X n is normally distributed with mean

a 1 μ 1 + … + a n μ n and variance a 21 σ 21 + … + a 2n σ 2n

© 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Example 5.3:

Personnel Decisions.xlsx

 Objective: To determine test scores that can be used to accept or

reject job applicants at ZTel.

 Solution: Scores of all applicants are approximately normally

distributed with mean 525 and standard deviation 55.

 Calculate the percentage of applicants who are automatic accepts or rejects, given the current standards of 600 for automatic accept and

425 for automatic reject.

 Find new cutoff values that reject 10% and accept 15% of applicants.

Example 5.4:

Paper Machine Settings.xlsx

 Objective: To determine the machine settings that result in paper of

acceptable quality at PaperStock Company.

 Solution: A given roll of paper must be rejected if its actual fiber

content is less than 19.8 pounds or greater than 20.3 pounds.

 The variability in fiber content is 0.10 pound when the process is

“good,” but increases to 0.15 pound when the machine goes “bad.”

 Calculate the probability that a given roll is rejected, for a setting of μ

= 20, when the machine is “good” and when it is “bad.”

© 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Example 5.5:

Tax on Stock Returns.xlsx

 Objective: To determine the after-tax profit Howard Davis can be

90% certain of earning.

 Solution: Howard is in the 33% tax bracket, so his after-tax profit

is 67% of his before-tax profit He invests $10,000 in a certain

stock, whose annual return is normally distributed with mean 5% and standard deviation 14%.

 Calculate the dollar amount such that Howard’s after-tax profit is 90% certain to be less than this amount; that is, calculate the

90th percentile of his after-tax profit.

Example 5.6:

 Objective: To construct and analyze a spreadsheet

model for microwave oven demand over the next 12

years using Excel’s NORMINV function, and to show

how models using the normal distribution can lead to nonsensical outcomes unless they are modified

appropriately.

 Solution: Using historical data, the company assumes

that demand in year 1 is normally distributed with

mean 5000 and standard deviation 1500

 It also assumes that demand in each subsequent year

is normally distributed with mean equal to the actual

demand from the previous year and standard

deviation 1500.

© 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Example 5.6:

 Using this model may lead to nonsensical

results as shown below:

Example 5.6:

 One way to modify the model is to let the standard deviation and

mean move together That is, if the mean is low, then the standard deviation will also be low.

 To be even safer, it is possible to truncate the demand distribution

at some nonnegative value such as 250, as shown below.

© 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

The Binomial Distribution

can occur in two situations:

 When sampling from a population with only two types of

members (males and females, for example)

 When performing a sequence of identical experiments, each

of which has only two possible outcomes

 Consider a situation where there are n independent,

identical trials, where the probability of a success on

each trial is p and the probability of a failure is 1 – p.

 Define X to be the random number of successes in the n

trials.

 Then X has a binominal distribution with parameters n and p.

 In Excel, calculate binomial probabilities with the

BINOMDIST function.

Example 5.7:

Binomial Calculations.xlsx

 Objective: To use Excel’s BINOMDIST and CRITBINOM functions

for calculating binomial probabilities and percentiles in the

context of flashlight batteries.

 Solution: Let X be the number of successes in 100 trials of

flashlight batteries, where a success means that the battery is still functioning after eight hours.

 Find the probabilities of various events, using the BINOMDIST

function, as shown in the spreadsheet below.

 Find the 95th percentile of the distribution of X, using the

CRITBINOM function.

© 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Mean and Standard Deviation of the

Binomial Distribution

 It can be shown that the mean and standard

deviation of a binomial distribution with parameters n and p are given by the following equations

 The empirical rules discussed in Chapter 2 also apply,

at least approximately, to the binomial distribution

 There is about a 95% chance that the actual number of successes will be within two standard deviations of the mean.

 There is almost no chance that the number of successes will be more than three standard deviations from the

mean.

The Binomial Distribution in the

Context of Sampling

 If sampling is done without replacement , each

member of the population can be sampled only once

 That is, once a person is sampled, his or her name is struck from the list and cannot be sampled again

 If sampling is done with replacement , then it is

possible, although maybe not likely, to select a given member of the population more than once

 Most real-world sampling is performed without

replacement.

 The binomial model applies only to sampling with

replacement.

 However, if no more than 10% of the population is

sampled, the binomial model can be used safely even if

sampling is performed without replacement.

© 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

The Normal Approximation

to the Binomial

 If you graph the binomial probabilities, you will see an interesting phenomenon: the graph begins to look

symmetric and bell-shaped when n is fairly large and p

is not too close to 0 or 1

 The normal distribution provides a very good approximation

to the binomial under these conditions.

 One practical consequence of the normal approximation to the binomial is that the empirical rules apply very well to binomial distributions.

Example 5.8:

Beating the Market.xlsx

 Objective: To determine the probability of a mutual fund outperforming a

standard market index at least 37 out of 52 weeks.

 Solution: The number of weeks where a given fund outperforms the market

index is binomially distributed with n = 52 and p = 0.5 This probability is

quite small (0.00159).

 Now let Y be the number of the 400 best mutual funds that beat the market

at least 37 of 52 weeks Y is also binomially distributed, with parameters n

= 400 and p = 0.00159 The resulting probability is nearly 0.5.

© 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Example 5.9:

Supermarket Spending.xlsx

 Objective: To use the normal and binomial distributions to calculate the

typical number of customers who spend at least $100 per day and the

probability that at least 30% of all 500 daily customers spend at least $100.

 Solution: Historical data indicate that the amount spent per customer is

normally distributed with mean $85 and standard deviation $30.

 If 500 customers shop in a given day, calculate the mean and standard

deviation of the number who spend at least $100.

 Then calculate the probability that at least 30% of the 500 customers spend

at least $100 This is the probability that a binomially distributed random

variable, with n = 500 and p = 0.309, is at least 150.

Example 5.10:

Airline Overbooking.xlsx (slide 1 of 2)

 Objective: To assess the benefits and

drawbacks of airline overbooking.

 Solution: Assume that the no-show rate is 10%

—that is, each ticketed passenger shows up with probability 0.90.

 For a flight with 200 seats, calculate the

probability that more than 205 passengers show up; that more than 200 passengers show up;

that at least 195 seats are filled; and that at

least 190 seats are filled.

 Use the BINOMDIST function and a data table to

determine the probabilities.