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National Guard
Black Belt Training
Module 33
Hypothesis Testing Basics
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CPI Roadmap – Analyze
8-STEP PROCESS
6. See
1.Validate 2. Identify 3. Set 4. Determine 5. Develop 7. Confirm 8. Standardize
Counter-
the Performance Improvement Root Counter- Results Successful
Measures
Problem Gaps Targets Cause Measures & Process Processes
Through
Define Measure Analyze Improve Control
ACTIVITIES TOOLS
• Value Stream Analysis
• Identify Potential Root Causes • Process Constraint ID
• Reduce List of Potential Root • Takt Time Analysis
Causes • Cause and Effect Analysis
• Brainstorming
• Confirm Root Cause to Output
• 5 Whys
Relationship
• Affinity Diagram
• Estimate Impact of Root Causes • Pareto
on Key Outputs • Cause and Effect Matrix
• FMEA
• Prioritize Root Causes
• Hypothesis Tests
• Complete Analyze Tollgate • ANOVA
• Chi Square
• Simple and Multiple
Regression
Note: Activities and tools vary by project. Lists provided here are not necessarily all-inclusive. UNCLASSIFIED / FOUO
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Learning Objectives
Review the terms “Parameters” and “Statistics” as
they relate to Populations and Samples.
Introduce Confidence Intervals for expressing the
uncertainty when predicting a population parameter
using a sample statistic, and how to calculate CI’s
for some common situations for different sample
sizes.
Show how the Central Limit Theorem and the
Standard Error of the Mean applies to the use of
Confidence Intervals and Tests
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Learning Objectives (Cont.)
Introduce statistical tests for some common tests and
introduce the t-distribution with testing
Learn about Hypothesis Testing to prove a statistical
difference in process performance in applications of
Minitab
Understand the tradeoffs and influences of sample
sizes on statistical tests.
Apply knowledge of different classes of statistical
errors to the decisions used in process improvement
to minimize risk.
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Application Examples
Transactional – A Black Belt has just finished a pilot of a new
process for handling blanket Purchase Orders and wants to
know if it has a statistically significant: a) shorter cycle time
and b) increased accuracy over the old process.
Administrative – The manager of an AAFES1 order entry
department wants to compare two order entry procedures to
see if one is faster than the other.
Service – Medical diagnostic imaging services are provided
from two different medical treatment facilities to a central
hospital which wants to know if there are differences in the
quality of service, particularly: a) the number of lost records
and re-takes, and b) average waiting time for MRIs and X-rays.
1AAFES, Army and Air Force Exchange System
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 5
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Population vs. Sample
Population Sample
All U.S. registered voters 10,000 people are asked
who they will vote for
President
All sufferers of a certain 3,000 people are given a
disease that might be new treatment in a clinical
given the new treatment study
All appraisals completed 25 appraisals chosen at
that month random from a given
month
Since it is not always practical or possible to measure/query every
item/person in the population, you take a random sample.
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Terms and Labels: Population vs. Sample
Population = Sample =
Term
Parameter Statistic
Count of items N n
m
x
Mean
~ ~
Median m x
Standard Dev. s S
m x
Estimators =
s s
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Population Parameters vs. Sample Statistics
Random Samples
of Size, n = 4
Population
x1 , s1
x2 , s2
x3 , s3
m, s x4 , s4
Population Parameters; Sample Statistics; Mean, x-bar,
Mean, m (mu), and Standard Deviation, s (sigma) and Standard Deviation, s
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Central Limit Theorem
If:
x1, x2, …, xn are independent measurements (i.e., a random sample of size n)
from a population, where the mean of x is m, when
the standard deviation of x is given as s,
Then:
The distribution of x X X 1 X 2 x3 X n
n
has mean and standard
deviation given by: s Standard Error
mX m and sX of the Mean
n
In addition, when n is sufficiently large, then the distribution of x- bar is
approximately normal (“bell-shaped curve”). More on sample sizes later...
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Variability of Means
Sample statistics estimate population parameters by inference:
For a given sample ( x, s, n ), we can estimate population
parameters of m s by inference.
As the sample size increases we are more confident that our
sample statistic is a more valid estimator of the population
parameter.
n=5
sx
n=3
sx sx
n
n=1
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11. UNCLASSIFIED / FOUO
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National Guard
Black Belt Training
Confidence Intervals
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What Is a Confidence Interval?
We know that when we take the average of a sample,
it is probably not exactly the same as the average of
the population.
Confidence intervals help us determine the likely
range of the population parameter.
For example, if my 95% confidence interval is 5 +/-
2, then I have 95% confidence that the mean of the
population is between 3 and 7.
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What Is a Confidence Interval? (Cont.)
Usually, confidence intervals have an additive
uncertainty:
Estimate ± Margin of Error
Sample Statistic ± [ ___ X ___ ]
Example: Confidence Measure of
x, s Factor Variability
Note: Detailed formulas may be found in the appendix.
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Why Do We Need Confidence Intervals?
Sample statistics, such as Mean and Standard
Deviation, are only estimates of the population’s
parameters.
Because there is variability in these estimates from
sample to sample, we can quantify our uncertainty
using statistically-based confidence intervals.
Confidence intervals provide a range of plausible
values for the population parameters (m and s).
Any sample statistic will vary from one sample to
another and, therefore, from the true population or
process parameter value.
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Exercise
Let’s look at a population that has a normal distribution with:
known mean value = 65
standard deviation = 4
(This has been generated in dataset Confidence.mtw)
Each member in the class will randomly sample 25 data points from
this population. (In Minitab, use Calc>Random Data>Sample from
Columns.)
Sample 25 rows of data from C1 and store the results in C2.
Use graphical descriptive statistics to calculate the 95% confidence
interval for the mean and sigma based on your sample of 25 data
points. Do they include the mean, 65, and the sigma, 4?
Based on a class size of 25, we would expect 1 confidence interval to
not contain 65 for the mean, and 1 that does not include 4 for sigma.
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 15
16. Confidence Interval for the Mean (m) with
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Population Standard Deviation (s) Known
Example
A random sample of size, n = 36, is taken and the distribution of
x
is normal. We are given that the population standard deviation
(s) is 18.0. The value of x-bar is an estimator of the population
mean (m),
and the standard error of x-bar is:
s x bar s / n 18.0 / 36 3.0
From the properties of the standardized normal distribution,
there is a 95% chance that m is within the range of ( x-bar +
and - 1.96 times the Standard Error of x-bar).
This is known as the Standard Error of the Mean
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What Values of x-bar Can I Expect?
Distribution of x-bar
.95
95% of all x-bars will
fall into the shaded region,
defined by m ± 1.96(3.0)
.025 .025
Standard Error
m1 - 1.96(3.0) m1 m1+ 1.96(3.0) of the Mean
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18. UNCLASSIFIED / FOUO
But I Don’t Know m, I Only Know x-bar!
We can turn it around.
x-bar lying in the interval m ±
1.96(3.0) is the same thing as
m lying in the interval x-bar ±
(----------- x-barsample A -----------)
1.96(3.0).
(---------- x-barsample B-----------)
Because there is a 95% chance (---------- x-barsample C-----------)
that x-bar lies in the interval
m ± 1.96(3.0), there is a 95%
chance that the interval x-bar m1 - 1.96(3.0) m1 m1+ 1.96(3.0)
± 1.96(3.0) encloses m.
The interval we construct using Observed sample mean, x-barsample C
the observed sample mean is
called a 95% confidence
interval for m.
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 18
19. Confidence Interval for the Mean (m) with
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Population Standard Deviation (s) Known
Another Example
An airline needs an estimate of the average number of
passengers on a newly scheduled flight. Its experience is that
data for the first month of flights is unreliable, but thereafter the
passenger loading settles down.
Therefore, the mean passenger load is calculated for the first 20
weekdays of the second month after initiation of this particular
new flight. If the sample mean (x-bar) is 112.0 and the
population standard deviation (s) is assumed to be 25, find a
95% confidence interval for the true, long-run average number
of passengers on this flight.
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20. Confidence Interval for the Mean (m)
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with Standard Deviation (s) Known
Solution
We assume that the hypothetical population of daily passenger loads for
weekdays is not badly skewed. Therefore, the sampling distribution of
x-bar is approximately normal and the confidence interval results are
approximately correct, even for a sample size of only 20 weekdays.
x -bar 112.0
s 25
s
s x -bar 5.59
20
For a 95% confidence interval, we use z.025= 1.96 in the formula to
obtain 112 1.965.59 or 101.04 to 122.96
We are 95% confident that the long-run mean, m , lies in this interval.
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 20
21. Confidence Interval for the Mean (m) with
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Population Standard Deviation (s) Unknown
A very important point to remember is that for this
example we assumed that we knew the population
standard deviation, and many times that is not the case.
Often, we have to estimate both the mean and the
standard deviation from the sample.
When s is not known, we use the t-distribution rather
than the normal (z) distribution. The t-distribution will
be explained next.
In many cases, the true population s is not known, so
we must use our sample standard deviation (s) as an
estimate for the population standard deviation (s
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 21
22. Confidence Interval for the Mean (m) with
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Standard Deviation (s) Unknown (Cont.)
Since there is less certainty (not knowing m or s ),
the t-distribution essentially “relaxes” or
“expands” our confidence intervals to allow for this
additional uncertainty.
In other words, for a 95% confidence interval, you
would multiply the standard error by a number
greater than 1.96, depending on the sample size.
1.96 comes from the normal distribution, but the
number we will use in this case will come from the
t-distribution.
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23. UNCLASSIFIED / FOUO
What Is This t-Distribution?
The t-distribution is actually a family of distributions.
They are similar in shape to the normal distribution
(symmetric and bell-shaped), although wider, and flatter in
the tails.
How wide and flat the specific t-distribution is depends on
the sample size. The smaller the sample size, the wider
and flatter the distribution tails.
As sample size increases, the t-distribution approaches the
exact shape of the normal distribution.
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24. UNCLASSIFIED / FOUO
An Example of a t-Distribution
0.4
t-distribution
0.3 (n = 5)
frequency
0.2
Area =
0.025
0.1
0.0
-3 -2 -1 0 1 2 2.78 3
t
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Some Selected t-Values
Here are values from the t-distribution for various
sample sizes (for 95% confidence intervals):
Sample Size t-value (.025)*
2 12.71
3 4.30
5 2.78
10 2.26
20 2.09
30 2.05
100 1.98
1000 1.96
* For a 95% CI, = .05. Therefore, for a two tail distribution: /2= .05/2= .025
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 25
26. Confidence Interval for the Mean (m) with
UNCLASSIFIED / FOUO
Population Standard Deviation (s) Unknown
Example
The customer expectation when phoning an order-out pizza shop is that
the average amount of time from completion of dialing until they hear
the message indicating the time in queue is equal to 55.0 seconds (less
than a minute was the response from customers surveyed, so the
standard was established at 10% less than a minute). You decide to
randomly sample at 20 times from 11:30am until 9:30pm on 2 days to
determine what the actual average is. In your sample of 20 calls, you
find that the sample mean, x-bar, is equal to 54.86 seconds and the
sample standard deviation, s, is equal to 1.008 seconds.
The actual data was as follows:
54.1, 53.3, 56.1, 55.7, 54.0, 54.1, 54.5, 57.1, 55.2, 53.8,
54.1, 54.1, 56.1, 55.0, 55.9, 56.0 ,54.9, 54.3, 53.9, 55.0
What is a 95% confidence interval for the true mean call completion
time?
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27. UNCLASSIFIED / FOUO
95% Confidence Interval
for Mean Call Completion Time
x = 54.860
We’re 95% confident that the actual mean
s = 1.008 call completion time is somewhere between
54.389 seconds and 55.331 seconds,
n = 20 based on our sample of 20 calls.
t.025,19 = 2.09 our sample of 20 calls
s
Luckily, we don’t
x t α/2, n1
have to worry about
n
the details of how to
calculate the t-value. 1.008
54.860 2.09
Minitab takes care of 20
that for us.
54.389, 55.331
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Now Let Minitab Calculate
the Confidence Interval
1. Open the Minitab file
PizzaCall.mtw
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29. UNCLASSIFIED / FOUO
Now Let Minitab Calculate
the Confidence Interval (Cont.)
2. Select Stat>
Basic Statistics>
Graphical Summary
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30. UNCLASSIFIED / FOUO
Now Let Minitab Calculate
the Confidence Interval (Cont.)
3. Double click on C-1 to place it
in the Variables box
4. Click on OK
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31. UNCLASSIFIED / FOUO
Now Let Minitab Calculate
the Confidence Interval (Cont.)
Summary for C1
A nderson-Darling Normality Test
We’re 95% confident that
A -Squared 0.60 the actual mean is
P-V alue 0.105
between
Mean
StDev
54.860
1.008
54.388 and 55.332
V ariance 1.016
Skew ness 0.560026 We’re also taking a 5%
Kurtosis -0.509797
N 20 chance that we’re wrong.
Minimum 53.300
1st Q uartile 54.100
Median 54.700
54 55 56 57
3rd Q uartile
Maximum
55.850
57.100
95% Confidence Interval
95% C onfidence Interv al for Mean for Mean (m:
54.388 55.332
54.388 55.332
95% C onfidence Interv al for Median
54.100 55.582
95% Confidence Intervals 95% C onfidence Interv al for StDev 95% Confidence Interval
Mean
0.767 1.472
for Standard Deviation (s:
0.767 1.472
Median
54.00 54.25 54.50 54.75 55.00 55.25 55.50
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32. UNCLASSIFIED / FOUO
Other Types of Confidence Intervals
There are other types of confidence intervals that are
based on the same principles we have learned:
Standard Deviation
Proportions
Median
Others
We will discuss some of these later.
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 32
33. UNCLASSIFIED / FOUO
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National Guard
Black Belt Training
Hypothesis Testing
UNCLASSIFIED / FOUO
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34. UNCLASSIFIED / FOUO
Extending the Concept
of Confidence Intervals
Extending the concept of confidence intervals allows
us to set-up and interpret statistical tests.
We refer to these tests as Hypothesis Tests.
One way to describe a hypothesis test:
Determining whether or not a particular value of
interest is contained within a confidence interval.
Hypothesis testing also gives us the ability to
calculate the probability that our conclusion is wrong.
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The New Car
You buy a one-year old car from the Lemon Lot in
order to save money on gas. The previous owner still
had the original features sticker and you were pleased
to note that the EPA mileage estimate indicated that
the car should get 31 miles per gallon overall.
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 35
36. UNCLASSIFIED / FOUO
The New Car (Cont.)
As soon as you buy the car, you fill up the tank so
that you’ll be ready to take the family for a drive and
to go to work the next day. A few days later, you fill
up again and calculate your gas mileage for that tank.
After you push the “=“ key on your calculator, the
number 27.1 appears.
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37. UNCLASSIFIED / FOUO
The New Car (Cont.)
Should you send the car to a mechanic to check for
problems?
Do you conclude that the EPA estimate is simply wrong?
Do you leave cruel messages on the seller’s answering
machine?
What ARE your conclusions?
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 37
38. UNCLASSIFIED / FOUO
Continuing the Car Situation
At what value of gas consumption should you become
alarmed that you are experiencing anything more
than just random variation?
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 38
39. UNCLASSIFIED / FOUO
The Car Situation (Cont.)
What if we knew this?
Distribution of gas
consumption for this
car
12.8 %
s = 3.46
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 39
40. UNCLASSIFIED / FOUO
Hypothesis Testing
Hypothesis Testing:
Allows us to determine statistically whether or not a
value is cause for alarm (or is simply due to random
variation)
Tells us whether or not two sets of data are different
Tells us whether or not a statistical parameter
(mean, standard deviation, etc.) is statistically
different from a test value of interest
Allows us to assess the “strength” of our conclusion
(our probability of being correct or wrong)
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41. UNCLASSIFIED / FOUO
Hypothesis Testing (Cont.)
Hypothesis Testing Enables Us to:
Handle uncertainty using a commonly accepted
approach
Be more objective (2 persons will use the same
techniques and come to similar conclusions almost all
of the time)
Disprove or “fail to disprove” assumptions
Control our risk of making wrong decisions or coming
to wrong conclusions
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42. UNCLASSIFIED / FOUO
Hypothesis Testing (Cont.)
Some Possible Samples
Sample A
True
Sample B
Population
Distribution
Sample C
Sample D
m
Population
Mean
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43. UNCLASSIFIED / FOUO
Sample Size Concerns
If we sample only one item, how close do we expect
to get to the true population mean?
How well do you think this one item represents the
true mean?
How much ability do we have to draw conclusions
about the mean?
What if we sample 900 items? Now, how close
would we expect to get to the true population
mean?
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44. UNCLASSIFIED / FOUO
Sample Size (Cont.)
The larger our sample, the
closer x-bar is likely to be to
Population the true population mean.
Likely value of x-bar
with a small sample
size
m
Likely value of x-bar
x with a large sample
size
x
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 44
45. UNCLASSIFIED / FOUO
Standard Deviation
What effect would a lot of variation in the population
have on our estimate of the population mean from a
sample?
How would this affect our ability to draw conclusions
about the mean?
What if there is very little variation in the population?
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46. UNCLASSIFIED / FOUO
Standard Deviation (Cont.)
Population with a lot
of variation
m Likely value of x-bar
x with sample size, n
Population with less
variation
m Likely value of x-bar
with sample size, n
x
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47. UNCLASSIFIED / FOUO
Statistical Inferences and Confidence
How much confidence do we have in our estimates?
How close do you think the true mean, m, is to our
estimate of the mean, x-bar?
How certain do we want/need to be about conclusions
we make from our estimates?
If we want to be more confident about our sample
estimate (i.e., we want a lower risk of being wrong),
then we must relax our statement of how close we
are to the true value.
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 47
48. UNCLASSIFIED / FOUO
Statistical Inferences and Confidence
(Cont.)
Population
If we want to have
high confidence in our
conclusions, we must
relax the range in
which we say the true
m mean lies
As we tighten our
estimate of the mean,
our risk of being wrong
x
increases. Thus, our
confidence decreases. x
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 48
49. UNCLASSIFIED / FOUO
Three Factors Drive Sample Sizes
Three concepts affect the conclusions drawn from a single
sample data set of (n) items:
Variation in the underlying population (sigma)
Risk of drawing the wrong conclusions (alpha, beta)
How small a Difference is significant (delta)
Risk
(n)
Variation Difference
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 49
50. UNCLASSIFIED / FOUO
Three Factors: Variation, Risk, Difference
These 3 factors work together. Each affects the others.
Variation: When there’s greater variation, a larger sample
is needed to have the same level of confidence that the
test will be valid. More variation reduces our confidence
interval.
Risk: If we want to be more confident that we are not
going to make a decision error or miss a significant event,
we must increase the sample size.
Difference: If we want to be confident that we can identify
a smaller difference between two test samples, the sample
size must increase.
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51. UNCLASSIFIED / FOUO
Three Factors (Cont.)
Larger samples improve our confidence interval.
Lower confidence levels allow smaller samples.
All of these translate into a specific confidence
interval for a given parameter, set of data, confidence
level and sample size.
They also translate into what types of conclusions
result from hypothesis tests.
Testing for larger differences between the samples,
reduces the size of the sample. This is known as delta
(D).
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 51
52. UNCLASSIFIED / FOUO
An Example
A unit has several quick response forces, QRF. Some forces have over
700 members, with at least 300 on the site at any time.
By regulation, all forces must have a quick response plan, the critical
first phase of which is required to be completed in 10 minutes (600
seconds) or less.
There are two teams that are vying for “most responsive.” They have
taken somewhat different approaches to implementing their quick
response plans and management wants to know which approach is
better: Team 1 or Team 2
Each one has 100 data points for actual responses and drills (Minitab
file Response.mtw)
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55. UNCLASSIFIED / FOUO
Descriptive Statistics – Team 1
Summary for Team 1
A nderson-D arling N ormality Test
A -S quared 0.84
P -V alue 0.029
M ean 599.55
S tD ev 0.62
V ariance 0.38
S kew ness -0.082566
Kurtosis 0.745102
N 100
M inimum 597.80
1st Q uartile 599.20
M edian 599.60
3rd Q uartile 600.00
597.75 598.50 599.25 600.00 600.75 M aximum 601.20
95% C onfidence Interv al for M ean
599.43 599.67
95% C onfidence Interv al for M edian
599.40 599.60
95% C onfidence Interv al for S tD ev
9 5 % C onfidence Inter vals
0.54 0.72
Mean
Median
599.40 599.45 599.50 599.55 599.60 599.65 599.70
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 55
56. UNCLASSIFIED / FOUO
Descriptive Statistics – Team 2
Summary for Team 2
A nderson-D arling N ormality Test
A -S quared 0.29
P -V alue 0.615
M ean 600.23
S tD ev 1.87
V ariance 3.51
S kew ness 0.051853
Kurtosis -0.518286
N 100
M inimum 596.20
1st Q uartile 599.00
M edian 600.20
3rd Q uartile 601.60
597.0 598.5 600.0 601.5 603.0 M aximum 604.20
95% C onfidence Interv al for M ean
599.86 600.60
95% C onfidence Interv al for M edian
599.80 600.60
95% C onfidence Interv al for S tD ev
9 5 % C onfidence Inter vals
1.65 2.18
Mean
Median
599.8 600.0 600.2 600.4 600.6
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 56
57. UNCLASSIFIED / FOUO
Example
The average cycle time for Team 1 is 599.55
seconds.
The average cycle time for Team 2 is 600.23
seconds.
The target cycle time for Phase 1 response is 600
seconds.
Is the difference between the two average cycle times
statistically significant?
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 57
58. UNCLASSIFIED / FOUO
Example (Cont.)
The unit wants to determine if the true averages of
the two teams are really different.
The unit thinks that the 600.23 average of team 2 is
little too high, so there is a need to determine if the
data indicates that the true average is really not equal
to the target of 600 seconds.
The unit will use hypothesis testing to answer these
questions.
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 58
59. UNCLASSIFIED / FOUO
Example
The first hypothesis test to be performed is to determine
whether there is a statistically significant difference between
the means of the two teams. This is called a 2-Sample t
Test.
The real question is whether or not the means are different
enough to indicate that the approaches taken by the two
teams really are centered differently, or are they close
enough that the difference could simply be a result of random
variation?
After that, hypothesis testing can tell us if there is evidence
indicating whether or not each team’s average is different
from the target of 600 seconds.
First, we need to introduce some terminology.
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 59
60. UNCLASSIFIED / FOUO
The Null Hypothesis for a 2-Sample t Test
The 2-Sample t Test is used to test whether or not
the means of two populations are the same.
The null hypothesis is a statement that the
population means for the two samples are equal.
Ho: μ1 = μ2
We assume the null hypothesis is true unless we have
enough evidence to prove otherwise. We say – we
“fail to reject the null”.
If we can prove otherwise, then we “reject the null”
hypothesis and accept the Alternative Hypothesis
HA: μ1 ≠ μ2
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 60
61. UNCLASSIFIED / FOUO
Null Hypothesis for 2-Sample t Test (Cont.)
This is analogous to our judicial system principle of
“innocent until proven guilty”
The symbol used for the null hypothesis is Ho:
H 0 : m1 m2 OR H 0 : m1 m2 0
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 61
62. UNCLASSIFIED / FOUO
The Alternative Hypothesis
for a 2-Sample t Test
The alternative hypothesis is a statement that
represents reality if there is enough evidence to reject
Ho.
If we reject the null hypothesis then we accept the
alternative hypothesis.
This is analogous to being found “guilty” in a court of
law.
The symbol used for the alternative hypothesis is Ha:
H a : m1 m2 OR H a : m1 m2 0
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 62
63. UNCLASSIFIED / FOUO
Our Emergency Response Team Example
In our example, the first hypothesis test will take this
form:
H o : m1 m 2
H a : m1 m 2 Reminder:
We are conducting a 2-
Sample t test to determine if
We can rewrite it in this form: the average cycle time of the
Phase 1 response from our two
H o : m1 m 2 0 teams are different.
H a : m1 m 2 0
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 63
64. UNCLASSIFIED / FOUO
Our Emergency Response Team Example
(Cont.)
If we wanted to specifically test only whether or not
there was enough evidence to indicate that team 2’s
average was greater than team 1’s, it would take this
form:
H o : m1 m 2 0
H a : m1 m 2 0 This is still a 2-Sample t-Test
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 64
65. UNCLASSIFIED / FOUO
Our Emergency Response Team Example
(Cont.)
The second hypothesis test will be a 1-Sample t. It
will take this form for each team:
H o : m1 600
H a : m1 600
When you are testing whether or not a
population mean is equal to a given or
Target value, you use a 1-Sample t
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 65
66. UNCLASSIFIED / FOUO
Hypothesis Test in Minitab
We will use Minitab to conduct our hypothesis tests.
Open the Minitab file Response.mtw
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 66
67. UNCLASSIFIED / FOUO
Hypothesis Test in Minitab:
2-Sample t-Test
Select Stat>
Basic
Statistics>
2-Sample t
to compare
Team 1 to
Team 2
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 67
68. UNCLASSIFIED / FOUO
Hypothesis Test in Minitab (Cont.)
Team 1 and Team 2
are in different
columns, so select
Samples in
different columns
Double click on C1-Supp1
Then double click on
C2-Supp2 to place them
In First and Second boxes
Select Graphs to get the
Graphs dialog box
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 68
69. UNCLASSIFIED / FOUO
Hypothesis Test in Minitab (Cont.)
In the Graphs dialog
box, check both
Boxplots of data
and Dotplots of
data
Click OK here, and
then click on OK in
the previous dialog
box
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 69
70. UNCLASSIFIED / FOUO
Hypothesis Test in Minitab (Cont.)
Boxplot of Team 1, Team 2
605
604
603
602
601
Data
600
599
598
597
596
Team 1 Team 2
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 70
71. UNCLASSIFIED / FOUO
Hypothesis Test in Minitab (Cont.)
Individual Value Plot of Team 1, Team 2
605
604
603
602
601
Data
600
599
598
597
596
Team 1 Team 2
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 71
72. UNCLASSIFIED / FOUO
Hypothesis Test in Minitab (Cont.)
This descriptive output shows up
in your Session Window
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 72
73. UNCLASSIFIED / FOUO
Hypothesis Test in Minitab (Cont.)
The null hypothesis states that the difference
between the two means is zero
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 73
74. UNCLASSIFIED / FOUO
Hypothesis Test in Minitab (Cont.)
We will cover
p-values in more
detail a little later
The p-value here is less than 0.05, so
we can reject the null hypothesis
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 74
75. UNCLASSIFIED / FOUO
Assumptions
The Hypothesis Tests we have discussed make certain
assumptions:
Independence between and within samples
Random samples
Normally distributed data
Unknown Variance
In our example, we did not assume equal variances.
This is the safe choice. However, if we had reason to
believe equal variances, then we could have checked
the “Assume equal variances” box in the dialogue box.
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 75
76. UNCLASSIFIED / FOUO
The Risks of Being Wrong
Error Matrix
Conclusion Drawn
Accept Ho Reject Ho
Type I
Ho True Correct Error
The -Risk)
True
State Type II Error
Correct
Ho False -Risk)
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 76
77. UNCLASSIFIED / FOUO
Type I and Type II Errors
Type I Error
I’ve discovered
Alpha Risk something that really
Producer Risk
isn’t here!
The risk of rejecting the null, and taking action, when
no action was necessary
Type II Error
I’ve missed a
Beta Risk
significant effect!
Consumer Risk
The risk of failing to reject the null when you should
have rejected it.
No action is taken when there should have been action.
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 77
78. UNCLASSIFIED / FOUO
Type I and Type II Errors (Cont.)
The Type I Error is determined up front.
It is the alpha value you choose.
The confidence level is one minus the alpha level.
The Type II Error is determined from the circumstances of the
situation.
If alpha is made very small, then beta increases (all else being
equal).
Requiring overwhelming evidence to reject the null increases the
chances of a type II error.
To minimize beta, while holding alpha constant, requires increased
sample sizes.
One minus beta is the probability of rejecting the null hypothesis
when it is false. This is referred to as the Power of the test.
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 78
79. UNCLASSIFIED / FOUO
Type I and Type II Errors (Cont.)
What type of error occurs when an innocent man is
convicted?
What about when a guilty man is set free?
Does the American justice system place more
emphasis on the alpha or beta risk?
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 79
80. UNCLASSIFIED / FOUO
Exercise
Draw the Type I & II error matrix for airport security.
Do you think the security system at most airports
places more emphasis on the alpha or beta risk?
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 80
81. UNCLASSIFIED / FOUO
The p-Value
If we reject the null hypothesis, the p-value is the
probability of being wrong.
In other words, if we reject the null hypothesis, the p-
value is the probability of making a Type I error.
It is the critical alpha value at which the null
hypothesis is rejected.
If we don’t want alpha to be more than 0.05, then we
simply reject the null hypothesis when the p-value is
0.05 or less.
As we will learn later, it isn’t always this simple.
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 81
82. UNCLASSIFIED / FOUO
UNCLASSIFIED / FOUO
National Guard
Black Belt Training
Power, Delta and Sample
Size
UNCLASSIFIED / FOUO
UNCLASSIFIED / FOUO
83. UNCLASSIFIED / FOUO
Beta, Power, and Sample Size
If two populations truly have different means, but only by a
very small amount, then you are more likely to conclude they
are the same. This means that the beta risk is greater.
Beta only comes into play if the null hypothesis truly is false.
The “more” false it is, the greater your chances of detecting it,
and the lower your beta risk.
The power of a hypothesis test is its ability to detect an effect
of a given magnitude.
Power 1
Minitab will calculate beta for us for a given sample size, but
first let’s show it graphically….
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 83
84. UNCLASSIFIED / FOUO
Beta and Alpha
95% Confidence Limit
(alpha = .05) for mean, m1
(critical value)
Alpha Risk
m1
Here is our first population and its corresponding alpha risk.
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 84
85. UNCLASSIFIED / FOUO
Beta and Alpha (Cont.)
95% Confidence Limit (alpha
= .05) for mean, m1 (critical
value)
m1 m2
D
We want to compare these two populations. Do you think that
we will easily be able to determine if they are different?
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 85
86. UNCLASSIFIED / FOUO
Beta and Alpha (Cont.)
95% Confidence Limit (alpha
= .05) for mean, m1 (critical
value)
Beta Risk
m1 m2
D
If our sample from population 2 is in this grey area, we will
not be able to see the difference. This is called Beta Risk.
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 86
87. UNCLASSIFIED / FOUO
Beta and Delta
If we are trying to see a larger change, we have less Beta
Risk.
95% Confidence Limit
(alpha = .025) for
Beta Risk mean, m1 (critical value)
m1 m2
D
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 87
88. UNCLASSIFIED / FOUO
Beta and Sigma
Now we’re back to our original graphic. What do you think
happens to Beta Risk if the standard deviations of the
populations decrease?
95% Confidence Limit
(alpha = .05) for mean,
m1 (critical value)
Beta Risk
m1 m2
D
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 88
89. UNCLASSIFIED / FOUO
Beta and Sigma (Cont.)
If the standard deviation decreases, Beta Risk decreases.
Reducing variability has the same effect on Beta Risk as
increasing sample size.
Beta Risk
95% Confidence Limit
(alpha = .05) for mean, m1
(critical value)
m1 m2
D
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 89
90. UNCLASSIFIED / FOUO
How Can Power Be Increased?
Power is related to risk, variation, sample size, and
the size of change that we want to detect.
If we want to detect a smaller delta (effect), we
typically must increase our sample size.
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 90
91. UNCLASSIFIED / FOUO
Example:
Power
Let’s use Minitab to determine the beta risk of the hypothesis
test we performed on the two teams.
First, we’ll have to make some assumptions.
We don’t know the TRUE difference in the means, so we’ll assume that
it’s 0.682, the differences in the sample averages.
A variance hypothesis test shows that the variances are not equal.
We will average the variances from Minitab to determine the combined
variance using the following formula:
2 2
s1 s 2
s
2
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 91
92. UNCLASSIFIED / FOUO
Example:
Power (Cont.)
Select; Stat>
Power and Sample Size>
2-Sample t...
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 92
93. UNCLASSIFIED / FOUO
Example:
Power (Cont.)
To calculate Power, we need three things;
1. Sample Size
2. The Difference between the two Means
3. The Average Standard Deviation of the two samples
We can get all this information from our 2-Sample t-Test conducted earlier:
1. Sample Size = 100
2. Difference Between Means = 0.682
(600.230 – 599.548 = 0.682)
3. Average Standard Deviation ??
(See Next Slide)
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 93
94. UNCLASSIFIED / FOUO
Example:
Power (Cont.)
To Calculate Average Standard Deviation
Remember that Standard Deviations are
the Square Roots of the Variance. Since
square roots are not additive (we cannot
add them and divide by two) we have to
convert them back to Variances which are
additive.
StDev Squared = Variance
Team 1 0.619 squared = 0.3832
Team 2 1.870 squared = 3.4969
Sum = 3.8801
Divide by 2 to get Average = 1.9401
And Square Root of Average = 1.3929
So the Average Standard Deviation for the two samples is 1.3929
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 94
95. UNCLASSIFIED / FOUO
Example:
Power (Cont.)
1. Type in Sample Size of 100 here
2. Type in Difference Between
Means of 0.682 here
3. Type in Average
Standard Deviation
of 1.393 here
4. Click on OK
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 95
96. UNCLASSIFIED / FOUO
Example:
Power (Cont.)
The Power = 0.9312
And since Beta = (1 –Power)
Beta = 0.0688.
If the TRUE difference between the two support orgs. was
0.682, we would have a 6.88% chance of not observing this
and therefore concluding that they are the same.
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 96
97. UNCLASSIFIED / FOUO
Example:
Power (Cont.)
In practice, we evaluate the power of a test to
determine its ability to detect a difference of a given
magnitude that we deem important, or practically
significant.
For example, we could calculate the power of a
hypothesis test to see if we could measure a one
minute difference in responsiveness between the two
teams.
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 97
98. UNCLASSIFIED / FOUO
Example:
Power (Cont.)
Let’s say that if the two support organizations’ cycle
times differ by as little as 0.4 seconds, then we need
to analyze the reasons for the differences.
What is the power of our test to detect this
difference?
What is the probability of making a type II error
(concluding that there is no difference when one
exists)?
Use Minitab to individually answer these questions.
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 98
99. UNCLASSIFIED / FOUO
Exercise:
Sample Size
Now that we understand the relationship between
Beta, Power, Delta, and Sample Size, we can use this
information to calculate the sample size necessary to
give us the information we want.
We simply use the same function in Minitab to solve
for sample size rather than power.
This is a very useful and common extension of
Hypothesis Testing.
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 99
100. UNCLASSIFIED / FOUO
Exercise:
Sample Size (Cont.)
Here we enter the
Difference (delta) we wish
to detect, and the minimum
Power value that we are
willing to live with.
We leave Sample sizes
blank.
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 100
101. UNCLASSIFIED / FOUO
Exercise:
Sample Size
Let’s extend our response team cycle time example
Determine what sample size we would need to detect
a difference of 0.4 seconds at a power of 0.90.
What about at a power of 0.95?
What about at a power of 0.95 and an alpha of
0.025?
Hint: Click the Options button in the Power and
Sample Size dialogue box.
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 101
102. UNCLASSIFIED / FOUO
Other Power and Sample Size Scenarios
We can perform these
calculations not only for
the difference between
two means, but for other
tests as well.
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 102
103. UNCLASSIFIED / FOUO
1-Sample t-test in Minitab
Now, we will return to Minitab to test the following hypothesis
about our two support organizations cycle times:
H o : m1 600
H a : m1 600
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 103
104. UNCLASSIFIED / FOUO
Back to the Support Organization Example:
One Sample t-Test
1-Sample t-test in Minitab
Choose Stat>Basic
Statistics>1-Sample t
to test the mean of
each response team against
a standard or spec
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 104
105. UNCLASSIFIED / FOUO
1-Sample t-test in Minitab
Double click on C1 Team 1
and C2 Team 2 to place
them in the dialog box
here.
Type in the Hypothesized
mean, or standard we are
comparing to. Here it is
600.
Click the Graphs
button to get to the
Graphs dialog box.
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 105
106. UNCLASSIFIED / FOUO
1-Sample t-test in Minitab (Cont.)
Select Histogram
of data and
Boxplot of data
Click OK here and on
the previous Screen
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 106
107. UNCLASSIFIED / FOUO
1-Sample t-test in Minitab
Histogram of Team 1
(with Ho and 95% t-confidence interval for the mean)
35
30 This shows the Target
we are testing, along with
25
the Average and the
20 Confidence Interval
Frequency
from the data.
15
10
5
0 _
X
-5 Ho
598.0 598.5 599.0 599.5 600.0 600.5 601.0
Team 1
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 107
108. UNCLASSIFIED / FOUO
1-Sample t-test in Minitab (Cont.) - adj
Boxplot of Team 1
(with Ho and 95% t-confidence interval for the mean)
_
X
Ho
598.0 598.5 599.0 599.5 600.0 600.5 601.0 601.5
Team 1
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 108
109. UNCLASSIFIED / FOUO
1-Sample t-test in Minitab (Cont.)
Histogram of Team 2
(with Ho and 95% t-confidence interval for the mean)
15.0
12.5
10.0
Frequency
7.5
5.0
2.5
0.0 _
X
Ho
597.0 598.5 600.0 601.5 603.0
Team 2
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 109
110. UNCLASSIFIED / FOUO
1-Sample t-test in Minitab (Cont.)
Boxplot of Team 2
(with Ho and 95% t-confidence interval for the mean)
_
X
Ho
596 597 598 599 600 601 602 603 604 605
Team 2
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 110
111. UNCLASSIFIED / FOUO
1-Sample t-test in Minitab (Cont.)
Here is the descriptive output for the 1-Sample t-Test
found in Session Window
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 111
112. UNCLASSIFIED / FOUO
2-Sided and 1-Sided Hypothesis Tests
We have concentrated on 2-sided hypothesis tests.
2-Sided tests determine whether or not two items are equal
or whether a parameter is equal to some value.
Whether an item is less than or greater than another item or
a value is not sought up front. A 2-sided test is a less specific
test.
The alternative hypothesis is “Not Equal”.
Everything we have learned also applies to 1-sided tests.
1-Sided tests determine whether or not an item is less than
(<) or greater than (>) another item or value.
The alternative hypothesis is either (<) or (>).
This makes for a more powerful test (lower beta at a given
alpha and sample size).
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 112
113. UNCLASSIFIED / FOUO
More Detailed Information
Remember to use the Stat Guide button to learn more
about the results and to help you interpret them.
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 113
114. UNCLASSIFIED / FOUO
Hypothesis Test Summary Template
Hypothesis Test Factor (x)
(ANOVA, 1 or 2 sample t - test, Chi Squared, p Value Observations/Conclusion
Regression, Test of Equal Variance, etc) Tested
Significant factor - 1 hour driving time from DC
Example: ANOVA Location 0.030 to Baltimore office causes ticket cycle time to
generally be longer for the Baltimore site
Significant factor - on average, calls requiring
Example: ANOVA Part vs. No Part 0.004 parts have double the cycle time (22 vs 43
hours)
Significant factor - Department 4 has digitized
Example: Chi Squared Department 0.000 addition of customer info to ticket and less
human intervention, resulting in fewer errors
South region accounted for 59% of the defects
Example: Pareto Region n/a due to their manual process and distance from
the parts warehouse
- Example - Optional BB Deliverable
Describe any other observations about the root cause (x) data
UNCLASSIFIED / FOUO
115. UNCLASSIFIED / FOUO
One-Way ANOVA Template Boxplots of Net Hour by Part/No
(means are indicated by solid circles)
After further investigation, possible 150
Boxplot: Part/ No Part Impact on Ticket Cycle Time
reasons proposed by the team are
OEM backorders, lack of technician
- Example -
Net Hours Call Open
certifications and the distance from
the OEM to the client site. It is also 100
caused by the need for technicians to
make a second visit to the end user
to complete the part replacement. 50
Next step will be for the team to
confirm these suspected root causes.
0
Part/No Part
Part
No Part
Analysis of Variance for Net Hour Because the p-value <=
Source DF SS MS F P
Part/No 1 7421 7421 8.65 0.004 0.05, we can be confident
Error 69 59194 858 that calls requiring parts
Total 70 66615 do have an impact on the
Individual 95% CI's For Mean
Level N Mean StDev --+---------+---------+---------+----
ticket cycle time.
No Part 27 21.99 19.95 (--------*---------)
Part 44 43.05 33.70 (------*------)
--+---------+---------+---------+----
Pooled StDev = 29.29 12 24 36 48
Optional BB Deliverable UNCLASSIFIED / FOUO
116. UNCLASSIFIED / FOUO
Linear Regression Template
95% confident that 94.1% of the variation in “Wait Time” is from the “Qty of Deliveries”
Fitted Line Plot
Wait Time = 32.05 + 0.5825 Deliveries
55
S 1.11885
R-Sq 94.1%
R-Sq(adj) 93.9%
50
Wait Time
45
40
- Example -
35
10 15 20 25 30 35
Deliveries
Optional BB Deliverable UNCLASSIFIED / FOUO
117. UNCLASSIFIED / FOUO
Takeaways
Since it is not always practical or possible to measure
every item in the population, you take a random
sample.
A basic understanding of the terms: Population,
Sample, Population Parameter, Sample Statistic,
Sample Mean, and Sample Standard Deviation
How to calculate a confidence interval with the
population standard deviation known
How to calculate a confidence interval with the
population standard deviation unknown
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 117
118. UNCLASSIFIED / FOUO
Takeaways (Cont.)
How Hypothesis tests help us handle uncertainty
The role of sample size, variation, and confidence
level
The null and alternative hypotheses
Type I and Type II errors
Hypothesis tests in Minitab
Stat Guide
p-value
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 118
119. UNCLASSIFIED / FOUO
Takeaways (Cont.)
How to conduct a 1-way and 2-way t-test
How to conduct a Variance test (see Appendix)
How to conduct a Paired t-test (see Appendix)
Understanding of 1-way and 2-way test of proportions
(see Appendix)
Understanding the relationship between Power and
sample size and detectable difference (delta)
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 119
120. UNCLASSIFIED / FOUO
What other comments or questions
do you have?
UNCLASSIFIED / FOUO
121. UNCLASSIFIED / FOUO
References
Hildebrand and Ott, Statistical Thinking for Managers,
4th Edition
Kiemele, Schmidt, and Berdine, Basic Statistics,
4th Edition
Hypothesis Testing - Basic UNCLASSIFIED / FOUO 121
122. UNCLASSIFIED / FOUO
UNCLASSIFIED / FOUO
National Guard
Black Belt Training
APPENDIX
UNCLASSIFIED / FOUO
UNCLASSIFIED / FOUO
123. UNCLASSIFIED / FOUO
Hypothesis Testing - Steps
Step 1: Define the problem objective
Step 2: Determine what data to collect (continuous or attribute)
Step 3: Based on data type, determine the appropriate hypothesis test to use
Step 4: Specify the null (H0) hypothesis and the alternative (H1) hypothesis
Step 5: Select a significance level (degree of risk acceptable), usually 0.05
Step 6: Execute Data Collection plan from step 2
Step 7: From the sample, conduct the hypothesis test using a statistical tool
Step 8: Identify the p-value
Step 9: Compare the p-value to the significance level - if the p-value is less than or
equal to your acceptable risk (your alpha), then the null hypothesis is rejected
Step 10: Translate the decision to the situation
UNCLASSIFIED / FOUO
124. UNCLASSIFIED / FOUO
Decision Tree Matrix
Data Type
Hypothesis to be Tested (Step 3) Tree
(Step 2)
Variable Testing equality of population MEAN (average) to a specific value 1
Variable Testing equality of population MEANS (averages) from two populations 2
Variable Testing equality of population MEANS (averages) from more than two populations 3
Testing equality of population VARIANCES (standard deviation) from more than two
Variable
populations 4
Attribute - Binomial
"Go/No-Go" Testing equality of population PROPORTIONS (binomial data; e.g., pass/fail, go/no
"Pass/Fail" or go, is/is not, etc.) from one or more populations 5
"Defective" Data
Attribute - Poisson
Testing equality of population PROPORTIONS (Poisson data; i.e., frequency of
"Count" or
occurence in time or space) from two or more populations 6
"Defects" data
Testing for ASSOCIATION (not necessarily causal)
Attribute (Contingency
Table Data)
Note: For use with attribute data only. For variable data, use correlation 7
or regression. No decision tree required.
UNCLASSIFIED / FOUO