LSSG Black Belt Training

1
LSSG Black Belt Training
Hypothesis Testing
2
Introduction

• Always about a population parameter
• Attempt to prove (or disprove) some assumption
• Setup
• alternate hypothesis What you wish to prove
• Example Person is guilty of crime
• null hypothesis Assume the opposite of what is
  to be proven. The null is always stated as an
  equality.
• Example Person is innocent

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The test

• Take a sample, compute statistic of interest.
• The evidence gathered against defendant
• How likely is it that if the null were true, you
  would get such a statistic? (the p-value)
• How likely is it that an innocent person would
  be found at the scene of crime, with gun in hand,
  etc.
• If very unlikely, then null must be false, hence
  alternate is proven beyond reasonable doubt.
• If quite likely, then null may be true, so not
  enough evidence to discard it in favor of the
  alternate.

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Types of Errors
5
The Testing Process

• Set up Hypotheses (Null and Alternate )
• Pick a significance level (alpha)
• Compute critical value (based on alpha)
• Compute test statistic (from sample)
• Conclude If test statistic critical value,
  then Alternate Hypothesis proven at alpha level
  of significance.

6
Hypothesis Testing Roadmap
7
Parametric Tests

• Use parametric tests when
• The data are normally distributed
• The variances of populations (if more than one is
  sampled from) are equal
• The data are at least interval scaled

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One sample z - test

• Used when testing to see if sample comes from a
  known population. A sample of 25 measurements
  shows a mean of 16. Test whether this is
  significantly different from a the hypothesized
  mean of 15, assuming the population standard
  deviation is known to be 4.

One-Sample Z Test of mu 15 vs not 15 The
assumed standard deviation 4 N Mean SE
Mean 95 CI Z P 25
17.0000 0.8000 (15.4320, 18.5680) 2.50 0.012
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Z-test for proportions

• 70 of 200 customers surveyed say they prefer the
  taste of Brand X over competitors. Test the
  hypothesis that more than 66 of people in the
  population prefer Brand X.

Test and CI for One Proportion Test of p 0.66
vs p 0.66
95
Lower Sample X N Sample p Bound
Z-Value P-Value 1 140 200 0.700000
0.646701 1.19 0.116
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One sample t-test
BP Reduction 10 12 9 8 7 12 14 13 15 16 18 12 18
19 20 17 15
The data show reductions in Blood Pressure in a
sample of 17 people after a certain treatment. We
wish to test whether the average reduction in BP
was at least 13, a benchmark set by some other
treatment that we wish to match or better.
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One Sample t-test Minitab results

• One-Sample T BP Reduction
• Test of mu 13 vs 13
• 
  95
• 
  Lower
• Variable N Mean StDev SE Mean
  Bound T P
• BP Reduction 17 13.8235 3.9248 0.9519
  12.1616 0.87 0.200

The p-value of 0.20 indicates that the reduction
in BP could not be proven to be greater than 13.
There is a 0.20 probability that it is not
greater than 13.
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Two Sample t-test
You realize that though the overall reduction is
not proven to be more than 13, there seems to be
a difference between how men and women react to
the treatment. You separate the 17 observations
by gender, and wish to test whether there is in
fact a significant difference between genders.

• M F
• 10 15
• 12 16
• 9 18
• 8 12
• 7 18
• 12 19
• 14 20
• 13 17
• 15

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Two Sample t-test
The test for equal variances shows that they are
not different for the 2 samples. Thus a 2-sample
t test may be conducted. The results are shown
below. The p-value indicates there is a
significant difference between the genders in
their reaction to the treatment.

• Two-sample T for BP Reduction M vs BP Reduction F
• N Mean StDev SE Mean
• BP Red M 8 10.63 2.50 0.89
• BP Red F 9 16.67 2.45 0.82
• Difference mu (BP Red M) - mu (BP Red F)
• Estimate for difference -6.04167
• 95 CI for difference (-8.60489, -3.47844)
• T-Test of difference 0 (vs not ) T-Value
  -5.02 P-Value 0.000
• DF 15
• Both use Pooled StDev 2.4749

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Assignment

• From the book Doing Data Analysis with Minitab
  14 by Robert Carver
• Pages 138 142 Choose any 3 of the datasets
  mentioned on those pages and answer the related
  questions. 1-sample means
• Pages 148 -151 Choose any 3 of the datasets
  mentioned on those pages and answer the related
  questions. 1-sample proportions
• Pages 165 -168 Choose any 3 of the datasets
  mentioned on those pages and answer the related
  questions. 2-sample tests

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Basics of ANOVA

• Analysis of Variance, or ANOVA is a technique
  used to test the hypothesis that there is a
  difference between the means of two or more
  populations. It is used in Regression, as well as
  to analyze a factorial experiment design, and in
  Gauge RR studies.
• The basic premise of ANOVA is that differences
  in the means of 2 or more groups can be seen by
  partitioning the Sum of Squares. Sum of Squares
  (SS) is simply the sum of the squared deviations
  of the observations from their means. Consider
  the following example with two groups. The
  measurements show the thumb lengths in
  centimeters of two types of primates.
• Total variation (SS) is 28, of which only 4
  (22) is within the two groups. Thus 24 of the 28
  is due to the differences between the groups.
  This partitioning of SS into between and
  within is used to test the hypothesis that the
  groups are in fact different from each other.
• See www.statsoft.com for more details.

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Results of ANOVA
The results of running an ANOVA on the sample
data from the previous slide are shown here. The
hypothesis test computes the F-value as
the ratio of MS Between to MS Within. The
greater the value of F, the greater
the likelihood that there is in fact a difference
between the groups. looking it up in an
F-distribution table shows a p-value of
0.008, indicating a 99.2 confidence that the
difference is real (exists in the Population,
not just in the sample).

• One-way ANOVA Type A, Type B
• Source DF SS MS F P
• Factor 1 24.00 24.00 24.00 0.008
• Error 4 4.00 1.00
• Total 5 28.00
• ___________________________________
• S 1 R-Sq 85.71 R-Sq(adj) 82.14

Minitab Stat/ANOVA/One-Way (unstacked)
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Two-Way ANOVA
Is the strength of steel produced different for
different temperatures to which it is heated and
the speed with which it is cooled? Here 2 factors
(speed and temp) are varied at 2 levels each, and
strengths of 3 parts produced at each combination
are measured as the response variable.
Strength Temp Speed 20.0 Low Slow 22.0 Low Slow 21
.5 Low Slow 23.0 Low Fast 24.0 Low Fast 22.0 Low F
ast 25.0 High Slow 24.0 High Slow 24.5 High Slow 1
7.0 High Fast 18.0 High Fast 17.5 High Fast

• Two-way ANOVA Strength versus Temp, Speed
• Source DF SS MS F
  P
• Temp 1 3.5208 3.5208 5.45 0.048
• Speed 1 20.0208 20.0208 31.00 0.001
• Interaction 1 58.5208 58.5208 90.61 0.000
• Error 8 5.1667 0.6458
• Total 11 87.2292
• S 0.8036 R-Sq 94.08 R-Sq(adj) 91.86

The results show significant main effects as well
as an interaction effect.
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Two-Way ANOVA
The box plots give an indication of the
interaction effect. The effect of speed on the
response is different for different levels of
temperature. Thus, there is an interaction effect
between temperature and speed.
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Assignment

• From the book Doing Data Analysis with Minitab
  14 by Robert Carver
• Pages 192 194 Choose any 3 of the datasets
  mentioned on those pages and answer the related
  questions. 1-way ANOVA
• Pages 204 206 Choose any 3 of the datasets
  mentioned on those pages and answer the related
  questions. 2-way ANOVA

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DOE Overview

• A design of experiment involves controlling
  specific inputs (factors) at various levels
  (typically 2 levels, like High and Low
  settings) to observe the change in output as a
  result, and analyzing the data to determine the
  significance and relative importance of factors.
• The simplest case would be to vary a single
  factor, say temperature, while baking cookies.
  Keeping all else constant, we can set temperature
  at 350 degrees and 400 degrees, and make several
  batches at those two temperatures, and measure
  the output desired in this case it could be a
  rating by experts of crispiness of the cookies on
  a scale of 0-10.

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Full Factorial Designs

• A 2F Factorial design implies that there are
  factors at 2 levels each. The case described on
  the previous slide with only one factor is the
  simplest. Having two factors at 2 levels would
  give us four combinations. Three factors yield 8
  combinations, 4 would yield 16, and so forth.
• The following table shows the full factorial
  (all 8 combinations) design for 3 factors
• temperature,
• baking time, and
• amount of butter,
• each at two levels HIGH and LOW.

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Fractional Factorials

• The previous example would require 8 different
  setups to bake the cookies. For each setup, one
  could bake several batches, say 4 batches, to get
  a measure of the internal variation. In practice,
  as the number of factors tested grows, it is
  difficult to even create all the setups needed,
  much less have replications within a setup.
• An alternative is to use fractional factorial
  designs, typically a ½ or ¼. As the name
  suggests, a ½ factorial design with 3 factors
  would only require 4 of the 8 combinations to be
  tested. This entails some loss of resolution,
  usually a confounding of interaction terms, which
  may be of no interest to the experimenter, and
  can be sacrificed.

Minitab Stat/DOE/Create Factorial Design/Display
Factorial Designs
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Running the Experiment Outcome Values

• Once the settings to be used are determined, we
  can run the experiment and measure the values of
  the outcome variable. This table shows the values
  of the outcome variable Crisp, showing the
  crispiness index for the cookies, for each of the
  8 settings of the full factorial experiment.

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Analysis of Data

• Analyzing the data in Minitab for the main
  effects and ignoring interaction
• terms, we get the following output

Factorial Fit Crispiness versus Temp, Time,
Butter Estimated Effects and Coefficients for
Crispiness (coded units) Term Effect
Coef SE Coef T P Constant
7.2500 0.3750 19.33 0.000 Temp 3.0000
1.5000 0.3750 4.00 0.016 Time
0.5000 0.2500 0.3750 0.67 0.541 Butter
1.5000 0.7500 0.3750 2.00 0.116 S
1.06066 R-Sq 83.64 R-Sq(adj) 71.36
Analysis of Variance for Crispiness (coded
units) Source DF Seq SS Adj SS
Adj MS F P Main Effects 3 23.000
23.000 7.667 6.81 0.047 Residual Error
4 4.500 4.500 1.125 Total
7 27.500 Estimated Coefficients for Crispiness
using data in uncoded units Term
Coef Constant 7.25000 Temp
1.50000 Time 0.250000 Butter 0.750000
Note that only the temperature is significant
(p-value lower than 0.05). The effect of
temperature is 3.00, which means that if temp.
is set at HIGH, crispiness will increase by 3.00
units on average, compared to the LOW setting.
Minitab Stat/DOE/Create Factorial Design/Analyze
Factorial Design
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Assignment

• From the book Doing Data Analysis with Minitab
  14 by Robert Carver
• Pages 309 310 Answer any 4 of the 7 questions
  on those pages. DOE

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Hypothesis Testing Roadmap
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Non-Parametric Tests

• Use non-parametric tests
• When data are obviously non-normal
• When the sample is too small for the central
  limit theorem to lead to normality of averages
• When the distribution is not known
• When the data are nominal or ordinal scaled
• Remember that even non-parametric tests have
  some assumptions about the data.

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The sign test

• The story
• A patient sign-in process at a hospital is being
  evaluated, and the time lapse between arrival and
  seeing a physician is recorded for a random
  sample of patients. You believe that currently
  the median time is over 20 minutes, and wish to
  test the hypothesis.

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The sign test data
Data for the test are as follows

• Process
• Time
• 5
• 7
• 15
• 30
• 32
• 35
• 62
• 75
• 80
• 85
• 95
• 100

The histogram of the data shows that it is
non-normal, and the sample size is too small for
the central limit theorem to apply. The data are
at least ordinal in nature (here they are ratio
scaled), satisfying the assumption of the sign
test.
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Sign test - analysis

• Since the hypothesis is that the median is
  greater than 20, the test compares each value to
  20. Those that are smaller get a negative sign,
  those that are larger than 20 get a positive one.
  The sign test then computes the probability that
  the number of negatives and positives observed
  would come about through random chance if the
  null were true (that the median time is 20
  minutes).

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Sign Test in Minitab - Results

• Sign Test for Median Process Time
• Sign test of median 20.00 versus 20.00
• N Below Equal Above P
  Median
• Process Time 12 3 0 9
  0.0730 48.50

In this data, there are 9 observations above 20,
and 3 of them below. This can be shown to have a
.0730 probability of occurring, even if the
median time for the population is in fact not
greater than 20. Thus, there is insufficient
evidence to prove the hypothesis (to reject the
null) at the 5 level, but enough if you are
willing to take a 7.3 risk.
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The sign test other applications

• The sign test can also be used for testing the
  value of the median difference between paired
  samples, as illustrated in the following link.
  The difference between values in a paired sample
  can be treated as a single sample, so any
  1-sample hypothesis test can be applied. In such
  a case, the assumption is that the pairs are
  independent of each other.
• The equivalent parametric tests for the sign
  test are the 1-sample z test and the 1-sample
  t-test.
• http//davidmlane.com/hyperstat/B135165.html

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Wilcoxon Signed-Rank Test

• A test is conducted where customers are asked to
  rate two products based on various criteria, and
  come up with a score between 0 and 100 for each.
  The testers goal is to check whether product A,
  the new version, is perceived to be superior to
  product B. The null hypothesis would be that they
  are equal to each other.

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Wilcoxon Signed-Rank Test

• The data

The measures are rankings by people, so the data
are not necessarily interval scaled, and
certainly not ratio scaled. Thus a paired sample
t-test is not appropriate. A non-parametric
equivalent of that is the Wicoxon Signed-Rank
Test. This is similar to the sign test, but more
sensitive.
Prod A Prod B Diff 55 50 5 60 62
-2 77 70 7 82 78 4 99 90 9 92 95 -3 86 90
-4 84 80 4 90 86 4 72 71 1
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Wilcoxon Signed-Rank Test

• Unlike the sign test, which only looks at whether
  something is larger or smaller, this tests uses
  the magnitudes of the differences, rank orders
  them, and then applies the sign of the difference
  and computes the sum of those ranks. This
  statistic (called W) has a sampling distribution
  that is approximately normal.
• For details on the technique, see the link below.
• Assumptions are
• Data are at least ordinal in nature
• The pairs are independent of each other
• Dependent variable is continuous in nature
• http//faculty.vassar.edu/lowry/ch12a.html

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Wilcoxon test in Minitab - Results

• Wilcoxon Signed Rank Test Diff
• Test of median 0.000000 versus median
  0.000000
• N
• for Wilcoxon
  Estimated
• N Test Statistic P Median
• Diff 10 10 44.5 0.046 2.500

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Mann-Whitney Test Two Samples

• The Story
• Customers were asked to rate a service in the
  past, and 10 people did so. After some
  improvements were made, data were collected
  again, with a new random set of customers. Twelve
  people responded this time.
• There is no pairing or matching of data, since
  the samples of customers for the old and the new
  processes are different.
• http//faculty.vassar.edu/lowry/ch11a.html

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Mann-Whitney Test Two Samples

• Old New
• 60 85
• 70 85
• 85 90
• 78 94
• 90 90
• 68 70
• 35 75
• 80 90
• 80 90
• 75 100
• 95
• 90

Note that the assumptions of a 2-sample t-test
are violated because the data are not interval
scaled, and may not be normally distributed. The
Mann-Whitney Test is the non-parametric
alternative to the 2-sample t-test.
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Mann-Whitney Test Two Samples

• The Mann-Whitney test rank orders all the data,
  with both columns combined into one. The ranks
  are then separated by group so the raw data is
  now converted to ranks. The sum of the ranks for
  each column is computed.
• The sums of ranks are expected to be in
  proportion to the sample sizes, if there is no
  difference between the groups. Based on this
  premise, the actual sum is compared to the
  expected sum, and the statistic is tested for
  significance.
• See details with another example on this link
  from Vassar Univ.
• http//faculty.vassar.edu/lowry/ch11a.html

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Mann-Whitney Test in Minitab - Results

• Mann-Whitney Test and CI Old, New
• N Median
• Old 10 76.50
• New 12 90.00
• Point estimate for ETA1-ETA2 is -14.00
• 95.6 Percent CI for ETA1-ETA2 is (-22.00,-5.00)
• W 72.5
• Test of ETA1 ETA2 vs ETA1 at 0.0028
• The test is significant at 0.0025 (adjusted for
  ties)

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Kruskal-Wallis Test 3 or more samples

• Here the data would be similar to the
  Mann-Whitney test, except for having more than 2
  samples. For parametric data, one would conduct
  an ANOVA to test for differences between 3 or
  more populations. The Kruskal-Wallis test is thus
  a non-parametric equivalent of ANOVA.

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Kruskal-Wallis Test Data
Rating Factor 7 Adults 5 Adults 6 Adults 4 Adults
2 Adults 6 Adults 5 Adults 9 Teens 9 Teens 8 Teens
5 Teens 9 Teens 10 Teens 7 Teens 8 Teens 3 Childr
en 4 Children 3 Children 5 Children 10 Children 2
Children

• Adults Teens Children
• 7 9 3
• 5 9 4
• 6 8 3
• 4 5 5
• 2 9 10
• 6 10 2
• 5 7
• 8

The data show ratings of some product by three
different groups. The same data are shown stacked
on the right to perform the test in Minitab.
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Kruskal-Wallis Test

• The Kruskal-Wallis test proceeds very similarly
  to the Mann-Whitney test. The data are all ranked
  from low to high values, and the ranks then
  separated by group. For each group, the ranks are
  summed and averaged.
• Each group average is compared to the overall
  average, and the deviation measured, weighted by
  the number of observations in each group. If the
  groups were identical, the deviations from the
  grand mean would be a small number (not 0, as
  one might intuitively think) that can be
  computed.
• The actual difference is compared to the expected
  one (H statistic computed) to complete the test.
  See the link below for details of the
  computation, if interested.
• http//faculty.vassar.edu/lowry/ch14a.html

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Kruskal-Wallis Test Minitab Results

• Kruskal-Wallis Test Rating versus Factor
• Kruskal-Wallis Test on Rating
• Factor N Median Ave Rank Z
• Adults 7 5.000 8.6 -1.23
• Children 6 3.500 7.2 -1.79
• Teens 8 8.500 15.9 2.86
• Overall 21 11.0
• H 8.37 DF 2 P 0.015
• H 8.48 DF 2 P 0.014 (adjusted for ties)

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Moods Median Test

• Mood median test for Rating
• Chi-Square 10.52 DF 2 P 0.005
• Individual
  95.0 CIs
• Factor N Median Q3-Q1
  ----------------------------------
• Adults 6 1 5.00 2.00
  (------------)
• Children 5 1 3.50 3.50
  (---------------------------)
• Teens 1 7 8.50 1.75
  (---------)
• 
  ----------------------------------
• 
  4.0 6.0 8.0
• Overall median 6.00

The Moods median test is an alternative to
Kruskal-Wallis. It is generally more robust
against violations of assumptions, but less
powerful.
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Friedmans Test

• Friedmans Test is the non-parametric equivalent
  to a randomized block design in an ANOVA. In
  other words, there are 3 or more groups, but each
  row of values across the groups are matched.
• The story
• A persons performance is rated in a normal
  state, rated again after introducing noise in the
  environment, and finally with the introduction of
  classical music in the background. This is done
  for a sample of 7 employees.

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Friedmans Test Data

• Normal Noise Music
• 7 5 8
• 8 4 8
• 6 6 8
• 9 5 8
• 5 5 7
• 7 4 9
• 8 4 9

Perform Group Block 7 Normal 1 8 Normal 2 6 Normal
3 9 Normal 4 5 Normal 5 7 Normal 6 8 Normal 7 5 N
oise 1 4 Noise 2 6 Noise 3 5 Noise 4 5 Noise 5 4 N
oise 6 4 Noise 7 8 Music 1 8 Music 2 8 Music 3 8 M
usic 4 7 Music 5 9 Music 6 9 Music 7
The data show the ratings of performance by
person in each of 3 conditions. The same data are
stacked in the table to the right, for doing the
test in Minitab. Each person represents a block
of data, since the 3 numbers for that person are
related.
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Friedmans Test - Analysis

• Friedmans test also ranks the ratings, but this
  time the ranking is done internally within each
  row the three scores for each person are ranked
  1, 2, and 3. These ranks are then summed and
  averaged.
• If the groups are identical, then one would
  expect no difference in the sum or mean of
  rankings for each group. In other words, if the
  conditions did not affect the performance rating,
  the rankings would either be the same, or vary
  randomly across people to yield equal sums.
• The sums are compared to this expectation to test
  the hypothesis. See the following link for more
  details.
• http//faculty.vassar.edu/lowry/ch15a.html

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Friedmans Test in Minitab Results.

• Friedman Test Perform versus Group blocked by
  Block
• S 9.50 DF 2 P 0.009
• S 10.64 DF 2 P 0.005 (adjusted for ties)
• Sum
• Est of
• Group N Median Ranks
• Music 7 8.000 19.5
• Noise 7 4.667 8.0
• Normal 7 7.333 14.5
• Grand median 6.667

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Assignment

• From the book Doing Data Analysis with Minitab
  14 by Robert Carver
• Pages 293 294 Choose any 3 datasets on those
  pages and answer the related questions.
  Non-parametric tests