Copyright by Michael S. Watson, 2012 Statistics Quick Overview

Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 2

Given That 1/3 of the Bag is Of Each Type, What is the Probability Of…… Getting 1: 33.3% Getting 2: 33.3% x 33.3% = 11.1% Getting 3: 33.3% x 33.3% x 33.3% = 3.7% Getting 4: 1.2% Getting 5: 0.4% Getting 6: 0.1% 3 When did you get suspicious of my claim?

Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book You Formed a Hypothesis…. 4 Proportion of Hersey’s is not 33%

Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 5 Hypothesis Testing H 0 - Null Hypothesis (everything else) H a - Alternative Hypothesis (what you want to prove)

Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 6 Hypothesis Testing- Candy Example H 0 - Null Hypothesis (Is 33%) H a - Alternative Hypothesis (Hershey’s Not 33%)

Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 7 Hypothesis Testing H0H0 HaHa Reject Not Reject Get this for Free 1 2 Is 33% Not 33%

Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 8 Hypothesis Testing H0H0 HaHa Reject Not Reject Get this for Free 1 2 What kind of evidence do we need to Reject the Null? Is 33% Not 33%

Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 9 Hypothesis Testing H 0 - Not Guilty H a - Guilty Why this way? “Innocent until proven guilty”

Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 10 Hypothesis Testing H0H0 HaHa Reject Not Reject Get this for Free 1 2 Does this mean Innocent? Not Guilty Guilty

Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 11 Hypothesis Testing- types of Errors Guilty Not Guilty Guilty Innocent Trial Finds Defendant … Defendant Really is…. What do we do to avoid these errors?

Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book Basic Statistics– Mean and Standard Deviation 12 Packaging Example Tire Failure

Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 13 Important Attributes Mean: The average or ‘expected value’ of a distribution. Denoted by µ (The Greek letter mu) Variance: A measure of dispersion and volatility. Denoted by σ 2 (Sigma Squared) Standard deviation: A related measure of dispersion computed as the square root of the variance. Denoted by σ (The Greek letter sigma)

Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 14 Which Process is More Variable? Case 1 Average: 50 Standard Deviation: 25 Case 2 Average: 5,000 Standard Deviation: 2,000 Case 3 Average: 10,000 Standard Deviation: 3,000 Coefficient of Variation (CV) CV = (Standard Deviation) / (Average) The CV allows you to compare relative variations Case 1: 50% Case 2: 40% Case 3: 30% Let’s take a look at spreadsheet

Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book What-If With Packing Variability 15 Original Case Less VariabilityMore Variability

Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 16 Strategic Importance of Understanding Variability (From GE) 1998 GE Letter to Shareholders Six Sigma program is uncovering “hidden factory” after “hidden factory” Now realize that “Variability is evil in any customer-touching process.” 2001 Book “Jack” − “We got away from averages and focused on variation by tightening what we call ‘span’”

Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 17 Probability Distributions Many things a firm deals with involves quantities that fluctuate Sales Returned items Items bought by a customer Time spent by sales clerk with customer Machine failures Etc… One way to summarize these fluctuations is with a probability distribution Although “demand” or some variable is random, it still follows a “Distribution” A Distribution is a mathematical equation that defines the shape of the curve that the distribution follows

Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 18 Probability Distributions A probability distribution allows us to compute the chance that a variable lies within a given range Examples: Probability sales are between 10,000 and 50,000 Probability that a customer buys 2 items Probability that a machine will break down and probability that it will take more than 2 hours to fix Probability that lead time will be more than 2 weeks

Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 19 Probability Distributions: Types Probability distributions can be Discrete: only taking on certain values Continuous: taking on any value within a range or set of ranges Examples: The number of items that a customer buys follows a discrete probability distribution The daily sales at a store follows a continuous probability distribution

Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 20 Continuous Distributions This area represents the probability that Sales will be between 20,000 and 30,000

Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 21 Normal Distribution One of the most common distributions in statistics is the normal distribution There are actually innumerable normal distributions each characterized by two parameters: The mean The standard deviation The standard normal has a mean of zero and a standard deviation of one Why the Normal? Many random variables follow this pattern When you are doing many samples from unknown distributions, the output of the samples follow the Normal distribution When you are dealing with forecast error, it only matters that the forecast error is normally distributed, not the underlying distribution Normal is mathematically less complex than others − Easily expressed in terms of the mean and standard deviation

Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 22 The Normal Distribution: A Bell Curve The area under this curve (and all continuous distributions) is equal to one

Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 23 Normal Distribution: Symmetric This half has an area = 0.50 So does this half

Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 24 Three Normal Distributions µ=0 σ=1 µ=0 σ=2 µ=1 σ=1

Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 25 Shapes of different Normal curves

Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 26 Normal Distribution Over Time

Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 27 Relationship between demand variability and service level (1) Assume that demand for a week has an equal chance of being any number between 0 and 100. Is this a Normal distribution? Average is 50, standard deviation is approximately 30 How much inventory do you need at the beginning of the week to ensure that you will meet demand 95% of time, on average

Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 28 Relationship between demand variability and service level (2) Assume same average demand, with less variation Now you need to hold only 63 for 95% service level

Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 29 The average number of items per customer A A Normal distribution with µ=10, σ=4 Area A measures the probability that the average is greater than 14? Typing =1-normdist(14,10,4, true ) in Excel returns this probability

Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 30 Using Excel In Excel, you can also click on Insert >>Function>>NORMDIST

Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 31 Using Excel (continued) NORMDIST function provides the area to the LEFT of the value that you input for “X” In this case (X=14) that area equals We want to measure A which is an area to the right of “X” Since the total area is equal to one, we know that the area A equals ( ) or 0.159

Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 32 Inverse Cumulative Normal Distribution A Normal distribution with µ=10, σ=4 Area = 0.3 X What value of X gives an area of 0.3 to its left ? We’ll use Excel’s NORMINV function to find out.

Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 33 Using Excel: NORMINV In Excel, you can click on Insert >> Function >> NORMINV

Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 34 Inverse Cumulative Normal Distribution A Normal distribution with µ=10, σ=4 Area = When X=7.902 the area to the left equals 0.30 Let’s look at Tire Example

Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 35 The Standard Normal Distribution µ=0 σ=1 The Standard Normal (with µ=0 and σ=1) is especially useful. Any normal distribution can be converted into the Standard Normal distribution. If X is a Normal Distribution, z = (X- µ)/ σ standardizes X and z follows a standard normal z measures the number of standard deviation away from the mean

Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 36 Using Excel Computations for the standard normal distribution in Excel can be done using the same NORMDIST and NORMINV functions as before (with µ=0, σ=1) You can also use the direct functions: NORMSDIST(z) NORMSINV(prob)

Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 37 Standard Normal in Excel This function determines the Area under a standard normal distribution to the left of -0.75

Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 38 Inverse Standard Normal in Excel This function determines the value of z needed to have an area under a standard normal of.2266 to the left of z Let’s look at Tire Example