int 506/706: total quality management lec #9, analysis of data
TRANSCRIPT
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)
0:or : 2121 oo HH
Hypothesis Testing
Alternative Hypothesis (Ha)
0:or : 2121 aa HH
Hypothesis Testing
Uses for hypothesis
testing
Hypothesis Testing
Assumptions
Confidence Intervals
Estimate +/- margin of error
Confidence Intervals
CONCLUSION DRAWN
Do Not RejectHo
RejectHo
THE TRU
E STATE
Ho is TRUE CORRECTTYPE I Error
(α risk)
Ho is FALSETYPE II Error
(β risk) CORRECT
You conclude there is a difference when there really isn’t
You conclude there is NO difference
when there really is
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
ns
Xt
/
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
Alternative Hypothesis – The camshafts from Supplier 2 are NOT the same as the target value
: XHo
: XH a
1-sample t-tests
90.3100/87.1
5.59923.600
/
ns
Xt
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
0:or : 2121 oo HH
0:or : 2121 aa HH
“mean of population 1 is the same as the mean of population 2”
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
2
22
1
21
21 )(
ns
ns
XXt o
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)
ANOVA
On time deliverA B C
1 58 62 712 63 70 663 61 68 684 62 69 67
61 67.25 68 65.42Grand Mean
RUN
Factor means
ANOVA
On time deliverA B C
1 55.007 11.674 31.1742 5.840 21.007 0.3403 19.507 6.674 6.6744 11.674 12.840 2.507
78.03 13.44 26.69 SS Factors184.92
SS Factor Total SS118.17
RUN
Factor SS = 4*(Factor mean-Grand mean)^2
SS = (Each value – Grand mean)2
Total SS = ∑ (Each value – Grand mean)2
ANOVA
On time deliverA B C
1 9.000 27.563 9.0002 4.000 7.563 4.0003 0.000 0.563 0.0004 1.000 3.063 1.000
14.000 38.750 14.000184.92
SS Error Total SS
RUN
66.75
(Each mean – Factor mean)2
∑
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) 118.17 + Within group variation (error/noise)
66.75
Total Variability 184.92
ANOVA
ANOVA
ANOVA
Two-way ANOVA
More complex – more factors – more calculations
Example: Photoresist to copper clad, p. 360
ANOVA
ANOVA