1. Statistical Process Control
Hypothesis Tests II

2. Comparison of Means
The first types of comparison are those that compare the location of two distributions. To do this:
Compare the difference in the mean values for the two distributions, and check to see if the magnitude of their difference is sufficiently large relative to the amount of variation in the distributions
Which type of test statistic we use depends on what is known about the process(es), and how efficient we can be with our collected data
Definitely Different
Probably Different
Probably NOT Different
Definitely NOT Different

3. Test What
Diff 2 means
Pre/Post Paired
True mean
Std Devs Known?
Std Devs Known?
yes no
yes no
n>30
n>30
yes no
yes no

4. 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.
The assumption that the data come from Normal distributions is very important. Assess how normally data are distributed prior to conducting either test.
Probably Different
Probably NOT Different
Definitely NOT Different
Definitely Different

5. Test What
Diff 2 means
Pre/Post Paired
True mean
True variance
Ratio variances
Std Devs Known?
Std Devs Known?
yes no
yes no
n>30
n>30
yes no
yes no

6. 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

7. Example
A machine automatically controls the amount of ribbon on a tape. The machine is judged effective if the variance is less than cm2. A sample of 20 tapes yields a sample variance of s2 = cm2. Should we conclude the machine is effective?
What is null and alternative hypothesis?

8. Example
A machine automatically controls the amount of ribbon on a tape. The machine is judged effective if the variance is less than cm2. A sample of 20 tapes yields a sample variance of s2 = cm2. Should we conclude the machine is effective?

9. Example
We want to reject if S2 >> so it follows we will conduct a right tail test.

10. Example
Criteria = 36.19
21.1 < 36.19
Accept Ho

11. Example
Criteria = 36.19
21.1 < 36.19
Accept Ho
36.19

12. Example P-value Probability that c2 has a value of 21.1 or less
if the true variance is is So
if we say that the variability has increased, we have
an almost 31% chance of being wrong. This is called
The p-value for the test.
21.1

13. Situation VIII: Variance Test With 0 Unknown
Use:
worst case variation comparison process for when there is not enough prior history
Procedure:
form ratio of the sample variances (two 2-distributions),
v1 = n1 – 1 degrees freedom for numerator, and
v2 = n2 – 1 degrees freedom for the denominator
Note:

14. Table for Variance Comparisons
Decision on which test to use is based on answering the following:
Do we know a theoretical variance (s2) or should we estimate it by the sample variances (s2) ?
What are we trying to decide (alternate hypothesis)?

15. 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 = 1/ Fa, v2, v1

16. Grip Strength Example True Corporate Training Example
How could grip strength vary among people in the SPC training room?
Data collection to detect difference in dominant hand mean between the left and right sides of the training room
Expectations?
Direction of comparison?
Significance Level?
Known parameters?
Best test?
Result?

17. Grip Strength Data Results
R-L Side, Equal Variance Dominant Hand Means Comparison:
L = x1 = 129.4, S12 = 2788,
n1 = 34 people
R = x2 = 104.0, S22 = 1225,
n2 = 20 people
Sp = 47.1, v = 52
Two-Sided Test at  = .05
HA: There is a difference
Test: Is | t0 | > t.025, 52?
|1.91| > NO!
Keep the Null Hypothesis:
There is NOT a difference btwn
L & R !

18. Grip Strength Data Results
R-L Side, No Assumptions Dom. Hand Means Comparison:
L = x1 = 129.4, S12 = 2788,
n1 = 34 people
R = x2 = 104.0, S22 = 1225,
n2 = 20 people
v = 51
Two-Sided Test at  = .05
HA: There is a difference
Test: Is | t0 | > t.025, 51?
|2.12| > YES!
Reject the Null Hypothesis:
There IS a difference btwn
L & R!
Why is this wimpy test
significant when the other wasn’t?
ANS: Check the equal
variance assumption!

19. Grip Strength Data Results
Unknown σ0
Variances Comparison:
S12 = 2788
n1 = 34, v1 = 33
S22 = 1225
n2 = 20, v2 = 19
Two-Sided Test at  = .10
HA: There is a difference
Test: Is F0 > F.05, 33, 19?
2.276 > YES!
(Should also check F1– /2, 33, 19)
Reject the Null Hypothesis:
There IS a difference in variance!
At  = .05, this test is just barely not significant
(Should also have checked for Normality with Normal Prob. Plot)

20. Statistical Quality Improvement
Goal: Control and Reduction of Variation
Causes of Variation:
Chance Causes / Common Causes
In Statistical Control
Natural variation / background noise
Assignable Causes / Special Causes
Out of Statistical Control
Things we can do something about - IF we act quickly!
Both can cause defects – because specifications are often set regardless of process capabilities!