INT 506/706: Total Quality Management
Lec #9, Analysis Of Data
Outline
• Confidence Intervals
• t-tests
–1 sample
–2 sample
• ANOVA
2
Hypothesis Testing
Often used to
determine if two
means are equal
Hypothesis Testing
Null Hypothesis (Ho)
H o : 1  2 or H o : 1  2  0
Hypothesis Testing
Alternative Hypothesis (Ha)
H a : 1  2 or H a : 1  2  0
Hypothesis Testing
Uses for
hypothesis
testing
Hypothesis Testing
Assumptions
Confidence Intervals
Estimate +/margin of error
Confidence Intervals
THE TRUE STATE
CONCLUSION DRAWN
Do Not Reject
Ho
Reject
Ho
Ho is TRUE
CORRECT
TYPE I Error
(α risk)
Ho is FALSE
TYPE II Error
(β risk)
You conclude there
is NO difference
when there really is
CORRECT
You
conclude
there is a
difference
when there
really isn’t
Confidence Intervals
Balancing Alpha and Beta Risks
Confidence level = 1 - α
Power = 1 - β
Confidence Intervals
Sample size
Large samples means more
confidence
Less confidence with smaller samples
Confidence Intervals
t-tests
A statistical test that allows
us to make judgments
about the average process
or population
t-tests
Used in 2 situations:
1) Sample to point of
interest (1-sample t-test)
2) Sample to another
sample (2-sample t-test)
t-tests
t-distribution is wider and
flatter than the normal
distribution
1-sample t-tests
Compare a statistical value
(average, standard
deviation, etc) to a value of
interest
1-sample t-tests
X 
t
s/ n
1-sample t-tests
Example
An automobile mfg has a target
length for camshafts of 599.5
mm +/- 2.5 mm. Data from
Supplier 2 are as follows:
Mean=600.23, std. dev. = 1.87
1-sample t-tests
Null Hypothesis – The camshafts from Supplier 2
are the same as the target value
Ho : X  
Alternative Hypothesis – The camshafts from
Supplier 2 are NOT the same as the target value
Ha : X  
1-sample t-tests
X   600.23  599.5
t

 3.90
s/ n
1.87 / 100
1-sample t-tests
2-sample t-tests
Used to test whether or
not the means of two
samples are the same
2-sample t-tests
H o : 1  2 or H o : 1  2  0
“mean of population 1 is the same as the mean of population 2”
H a : 1  2 or H a : 1  2  0
2-sample t-test
Example
The same mfg has data for another supplier and
wants to compare the two:
Supplier 1: mean = 599.55, std. dev. = .62, C.I.
(599.43 – 599.67) – 95%
Supplier 2: mean = 600.23, std. dev. = 1.87, C.I.
(599.86 – 600.60) – 95%
2-sample t-tests
t
( X1  X 2 )   o
2
1
2
2
s
s

n1 n2
2-sample t-tests
ANOVA
Used to analyze the
relationships between several
categorical inputs and one
continuous output
ANOVA
Factors: inputs
Levels: Different sources or
circumstances
ANOVA
Example
Compare on-time delivery performance at three
different facilities (A, B, & C).
Factor of interest: Facilities
Levels: A, B, & C
Response variable: on-time delivery
ANOVA
To tell whether the 3 or more options are
statistically different, ANOVA looks at three
sources of variability
Total: variability among all observations
Between: variation between subgroups means
(factors)
Within: random (chance) variation within each
subgroup (noise, statistical error)
RUN
ANOVA
1
2
3
4
A
58
63
61
62
61
On time deliver
B
62
70
68
69
67.25
Factor means
C
71
66
68
67
68
65.42
Grand Mean
RUN
ANOVA
1
2
3
4
On time deliver
A
B
C
55.007
11.674
31.174
5.840
21.007
0.340
19.507
6.674
6.674
11.674
12.840
2.507
78.03
13.44
26.69
118.17
SS Factor
Factor SS =
4*(Factor mean-Grand mean)^2
SS = (Each value – Grand mean)2
SS Factors
184.92
Total SS
Total SS = ∑ (Each value – Grand mean)2
ANOVA
RUN
(Each mean – Factor mean)2
1
2
3
4
On time deliver
A
B
C
9.000
27.563
9.000
4.000
7.563
4.000
0.000 ∑ 0.563
0.000
1.000
3.063
1.000
14.000
38.750
14.000
66.75
SS Error
184.92
Total SS
ANOVA
Total: variability among all observations
184.92
Between: variation between subgroups means
(factors)
118.17
Within: random (chance) variation within each
subgroup (noise, statistical error)
66.75
ANOVA
Between group variation (factor)
+ Within group variation (error/noise)
Total Variability
118.17
66.75
184.92
ANOVA
ANOVA
ANOVA
Two-way ANOVA
More complex – more factors – more calculations
Example: Photoresist to copper clad, p. 360
ANOVA
ANOVA