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Statistics Quick Overview Copyright by Michael S. Watson, 2012 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% When did you get suspicious of my claim? Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 3 You Formed a Hypothesis…. Proportion of Hersey’s is not 33% Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 4 Hypothesis Testing H0- Null Hypothesis (everything else) Ha- Alternative Hypothesis (what you want to prove) H-0 H-a Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 5 Hypothesis Testing- Candy Example H0- Null Hypothesis (Is 33%) Ha- Alternative Hypothesis (Hershey’s Not 33%) H-0 H-a Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 6 Hypothesis Testing Reject Is 33% H0 Not 33% Ha 1 Not Reject 2 Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 7 Hypothesis Testing What kind of evidence do we need to Reject the Null? Reject Is 33% H0 Not 33% Ha 1 Not Reject 2 Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 8 Hypothesis Testing H0- Not Guilty Ha- Guilty Why this way? “Innocent until proven guilty” Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 9 Hypothesis Testing Does this mean Innocent? Reject H0 1 Not Reject 2 Not Guilty Guilty Ha Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 10 Hypothesis Testing- types of Errors Trial Finds Defendant … Defendant Really is…. Guilty Innocent Guilty Not Guilty What do we do to avoid these errors? Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 11 Basic Statistics– Mean and Standard Deviation Data Point Packaging Example Tire Failure Tire Failure Miles 1 31,603 2 32,586 3 34,394 4 38,954 5 42,503 6 31,754 7 29,459 8 36,157 9 38,559 10 36,478 11 45,809 12 30,981 13 39,355 14 37,406 15 29,545 16 35,975 17 34,867 18 38,878 19 26,031 20 43,564 21 32,852 22 35,589 23 41,458 24 31,989 25 34,576 Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 12 Important Attributes Mean: The average or ‘expected value’ of a distribution. Variance: A measure of dispersion and volatility. Denoted by µ (The Greek letter mu) Denoted by σ2 (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 13 Which Process is More Variable? Case 1 Case 2 Average: 10,000 Standard Deviation: 3,000 Coefficient of Variation (CV) Average: 5,000 Standard Deviation: 2,000 Case 3 Average: 50 Standard Deviation: 25 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 14 What-If With Packing Variability Original Case Less Variability More Variability Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 15 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 16 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 17 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 18 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 19 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 20 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 21 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 22 Normal Distribution: Symmetric So does this half This half has an area = 0.50 Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 23 Three Normal Distributions µ=0 µ=1 σ=1 σ=1 µ=0 σ=2 -4 -3 -2 -1 0 1 2 Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 3 4 24 Shapes of different Normal curves Different Normal Curves M = 50, Std = 4 M = 50, Std = 8 M = 50, Std = 16 0.12 Probability 0.10 0.08 0.06 0.04 0.02 0.00 0 10 20 30 40 50 60 70 80 90 100 Demand Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 25 Normal Distribution Over Time Demand over Time M = 50, Std = 4 Actual Demand 100 80 60 40 20 93 97 93 97 89 85 81 77 73 69 65 61 57 53 49 45 41 37 33 29 25 21 17 13 9 5 1 0 Time Period Demand over Time M = 50, Std = 16 80 60 40 20 89 85 81 77 73 69 65 61 57 53 49 45 41 37 33 29 25 21 17 13 9 5 0 1 Actual Demand 100 Time Period Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 26 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 27 Relationship between demand variability and service level (2) Assume same average demand, with less variation Normal Distribution (Mean = 50, Std Dev = 8) Now you need to hold only 63 for 95% service level 0.06 0.05 Probability 0.04 0.03 0.02 0.01 0.00 - 10 20 30 40 50 60 70 80 90 100 Demand Value Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 28 The average number of items per customer A Normal distribution with µ=10, σ=4 0.12 0.1 Area A measures the probability that the average is greater than 14? 0.08 0.06 0.04 0.02 A 0 -5 0 5 10 15 20 25 Typing =1-normdist(14,10,4,true) in Excel returns this probability Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 29 Using Excel In Excel, you can also click on Insert >>Function>>NORMDIST Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 30 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 0.841 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 (1 - 0.841) or 0.159 Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 31 Inverse Cumulative Normal Distribution A Normal distribution with µ=10, σ=4 0.12 0.1 0.08 0.06 0.04 0.02 Area = 0.3 0 0 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 32 Using Excel: NORMINV In Excel, you can click on Insert >> Function >> NORMINV Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 33 Inverse Cumulative Normal Distribution A Normal distribution with µ=10, σ=4 0.12 0.1 0.08 0.06 0.04 0.02 Area = 0.3 0 0 7.902 When X=7.902 the area to the left equals 0.30 Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book Let’s look at Tire Example 34 The Standard Normal Distribution µ=0 σ=1 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 The Standard Normal (with µ=0 and σ=1) is especially useful. Any normal distribution can be converted into the Standard Normal distribution. Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 35 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 36 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 37 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 Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 38