1. IENG 486 Statistical Quality & Process Control
11/18/2018
IENG Lecture 07b
Comparison of Spread (Variances) & Operating Characteristic Curves
11/18/2018
IENG 486 Statistical Quality & Process Control
(c) D.H. Jensen & R. C. Wurl

2. IENG 486 Statistical Quality & Process Control
11/18/2018
Assignment:
Preparation:
Print Hypothesis Test Tables from Materials page
Have this available in class …or exam!
Reading:
Chapter 4:
4.1.1 through 4.3.4; (skip 4.3.5); through 4.4.3; (skip rest)
HW 2:
CH 4: # 1, 5, 9, 11, 17, 18, 21, 22* *uses Fig.4.7, p. 126
11/18/2018
IENG 486 Statistical Quality & Process Control
(c) D.H. Jensen & R. C. Wurl

3. Comparison of Variances
The second types of comparison are those that compare the spread of two distributions. To do this:
Compute the ratio of the two variances, and then compare the ratio to one of two known distributions as a check to see if the magnitude of that ratio is sufficiently unlikely for the distribution.
Assumption that the data come from Normal distributions is very important. Assess how normally data are distributed prior to conducting either test.
Definitely Different
Probably Different
Probably NOT Different
Definitely NOT Different
11/18/2018
IENG 486 Statistical Quality & Process Control

4. Situation VII: Variance Test With 0 Known
Used when:
existing comparison process has been operating without much change in variation for a long time
Procedure:
form ratio of a sample variance (t-distribution variable) to a population variance (Normal distribution variable),
v = n - 1 degrees of freedom
11/18/2018
IENG 486 Statistical Quality & Process Control

5. Situation VIII: Variance Test With 0 Unknown
Used when:
worst case variation comparison process for when there is not have enough prior history
Procedure:
form ratio of a sample variances (two 2-distributions),
v1 = n1 – 1 degrees freedom for numerator, and
v2 = n2 – 1 degrees freedom for the denominator
Note:
11/18/2018
IENG 486 Statistical Quality & Process Control

6. Table for Variance Comparisons
Decision on which test to use is based on answering the following:
Do we know the theoretical variance (s2) or should we estimate it by the sample variances (s2) ?
What are we trying to decide (alternate hypothesis)?
11/18/2018
IENG 486 Statistical Quality & Process Control

7. Table for Variance Comparisons
These questions tell us:
What sampling distribution to use
What test statistic(s) to use
What criteria to use
How to construct the confidence interval
Four primary test statistics for variance comparisons
Two sampling distributions
Two confidence intervals
Six alternate hypotheses
Table construction
Note: F1-a, v1, v2 =
Fa, v2, v1
11/18/2018
IENG 486 Statistical Quality & Process Control

8. Operating Characteristic (OC) Curve
IENG 486 Statistical Quality & Process Control
11/18/2018
Operating Characteristic (OC) Curve
Relates the size of the test difference (d) to the Type II Error (b) for a given risk of Type I Error (a)
Ex:
for 2-sided z-statistic
Designing an experiment involves a trade-off in sample size versus power of the test to detect a difference (effect)
The greater the difference in means (d), the smaller the chance of Type II Error for a given sample size and a.
As the sample size increases, the chance of Type II Error decreases for a specified a and given difference in means (d).
11/18/2018
IENG 486 Statistical Quality & Process Control
(c) D.H. Jensen & R. C. Wurl

9. Operating Characteristic Curve
11/18/2018
IENG 486 Statistical Quality & Process Control

10. IENG 486 Statistical Quality & Process Control
O.C. Curve Use
Agree on acceptable a
Estimate anticipated d and s to compute d:
d = | m1 - m2| = |d|
s s
Look for where d intersects with desired b (Probability of keeping H0 when H0 is false) to estimate required sample size (n)
11/18/2018
IENG 486 Statistical Quality & Process Control

11. OC Curve Example (uses Fig 4.7, p.126)
Assume our previous problem had a process std. dev. of 18 (instead of 5), and the same means
(125 population & spec, 134 supplier sample).
Assume the boss wants  = 0.05 of exceeding either the high or low spec. for such a sample.
Probability of what (in English)?
Assume supplier needs  = 0.2
What sample size is needed to fit these constraints?
11/18/2018
IENG 486 Statistical Quality & Process Control

12. Operating Characteristic Curve
β = 0.2
d = 0.5
11/18/2018
IENG 486 Statistical Quality & Process Control