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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?
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Copyright by Michael S. Watson, 2012
Statistics Quick Overview
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 2
LET’S START WITH CANDY
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 3
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 4
You Formed a Hypothesis….
Proportion of Hersey’s is not 33%
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 5
Hypothesis Testing
H0- Null Hypothesis (everything else)
Ha- Alternative Hypothesis (what you want to prove)
H-0H-a
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 6
Hypothesis Testing- Candy Example
H0- Null Hypothesis (Is 33%)
Ha- Alternative Hypothesis (Hershey’s Not 33%)
H-0H-a
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 7
Hypothesis Testing
H0
Ha
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
H0
Ha
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
H0- Not Guilty
Ha- Guilty
Why this way? “Innocent until proven guilty”
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 10
Hypothesis Testing
H0
Ha
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
Tria
l Fin
ds D
efen
dant
…
Defendant Really is….
What do we do to avoid these errors?
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 12
Basic Statistics– Mean and Standard Deviation
Data Point Tire Failure Miles1 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
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 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 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 15
What-If With Packing VariabilityOriginal Case
Less Variability More 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
-4 -3 -2 -1 0 1 2 3 4
µ=0σ=1
µ=0σ=2
µ=1σ=1
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 25
Shapes of different Normal curves
Different Normal Curves
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0 10 20 30 40 50 60 70 80 90 100
Demand
Prob
abili
ty
M = 50, Std = 4M = 50, Std = 8M = 50, Std = 16
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 26
Normal Distribution Over Time
Demand over Time
0
20
40
60
80
1001 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97
Time Period
Act
ual D
eman
d
M = 50, Std = 4
Demand over Time
0
20
40
60
80
100
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97
Time Period
Act
ual D
eman
d
M = 50, Std = 16
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
Normal Distribution (Mean = 50, Std Dev = 8)
0.00
0.01
0.02
0.03
0.04
0.05
0.06
- 10 20 30 40 50 60 70 80 90 100
Demand Value
Prob
abili
ty
Relationship between demand variability and service level (2)
Assume same average demand, with less variation
Now you need to holdonly 63 for 95% service level
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 29
The average number of items per customer
0
0.02
0.04
0.06
0.08
0.1
0.12
-5 0 5 10 15 20 25
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 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 32
Inverse Cumulative Normal Distribution
A Normal distribution with µ=10, σ=4
0
0.02
0.04
0.06
0.08
0.1
0.12
0
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
0
0.02
0.04
0.06
0.08
0.1
0.12
0
Area = 0.3
7.902
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=0σ=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
normalz 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