1. Statistical Process Control
Hypothesis Tests

2. Hypothesis Tests
An Hypothesis is a guess about a situation that can be tested, and the test outcome can be either true or false.
The Null Hypothesis has a symbol H0, and is always the default situation that must be proven unlikely beyond a reasonable doubt.
The Alternative Hypothesis is denoted by the symbol HA and can be thought of as the opposite of the Null Hypothesis - it can also be either true or false, but it is always false when H0 is true and vice-versa.

3. Hypothesis Testing Errors
Type I Errors occur when a test statistic leads us to reject the Null Hypothesis when the Null Hypothesis is true in reality.
The chance of making a Type I Error is estimated by the parameter  (or level of significance), which quantifies the reasonable doubt.
Type II Errors occur when a test statistic leads us to fail to reject the Null Hypothesis when the Null Hypothesis is actually false in reality.
The probability of making a Type II Error is estimated by the parameter .

4. Types of Hypothesis Tests
Hypothesis Tests & Rejection Criteria
H0: MA is not better than M0
H0: MA is not different than M0
H0: MA is not better than M0
HA: MA is lower than M0
HA: MA is different than M0
HA: MA is higher than M0 Dm Dm Dm   2  2  θ0 θ0 θA θA θA
-θ0
+θ0 θA
One-Sided Test
Statistic < Rejection Criterion
H0: θA ≥ θ0
HA: θA < θ0
Two-Sided Test
Statistic < -½ Rejection Criterion or
Statistic > +½ Rejection Criterion
H0: -θ0 ≤ θA ≤ +θ0
HA: θA< -θ0 or +θ0< θA
One-Sided Test
Statistic > Rejection Criterion
H0: θA ≤ θ0
HA: θA > θ0

5. Hypothesis Testing Steps
State the null hypothesis (H0) from one of the alternatives:
that the test statistic mA = m0 , mA ≥ m0 , or mA ≤ m0 .
Choose the alternative hypothesis (HA) from the alternatives:
mA ¹ m0 , mA < m0 , or mA > m0 . (Respective to above!)
Choose a significance level of the test (a).
Select the appropriate test statistic and establish a critical region.
(If the decision is to be based on a P-value, it is not necessary to have a critical region)
Compute the value of the test statistic () from the sample data.
Decision: Reject H0 if the test statistic has a value in the critical region (or if the computed P-value is less than or equal to the desired significance level a); otherwise, do not reject H0.

6. Situation I: Means Test, Both σ0 and μ0 Known
Used with:
an existing process with good deal of data showing the variation and location are stable
Procedure:
use the the z-statistic to compare sample mean with population mean 0

7. Sampling Distribution of the Mean from the Normal Distribution
TM 720: Statistical Process Control
11/13/2018 2:35 PM
Sampling Distribution of the Mean from the Normal Distribution
Take a random sample, x1, x2, …, xn, from a normal population with mean μ and standard deviation σ, i.e.,
Compute the sample average x
Then x will be normally distributed with mean μ and standard
deviation:
that is:
(c) D.H. Jensen & R.C. Wurl

8. Ex. Sampling Distribution of x
TM 720: Statistical Process Control
11/13/2018 2:35 PM
Ex. Sampling Distribution of x
When a process is operating properly, the mean density of a liquid is 10 with standard deviation 5. Five observations are taken and the average density is 15.
What is the distribution of the sample average?
r.v. x = density of liquid
(c) D.H. Jensen & R.C. Wurl

9. Ex. Sampling Distribution of x
TM 720: Statistical Process Control
11/13/2018 2:35 PM
Ex. Sampling Distribution of x
What is the probability the sample average is greater than 15?
Would you conclude the process is operating properly?
(c) D.H. Jensen & R.C. Wurl

11. Ex. Sampling Distribution of x
TM 720: Statistical Process Control
11/13/2018 2:35 PM
Ex. Sampling Distribution of x
What is the probability the sample average is greater than 15?
Would you conclude the process is operating properly?
(c) D.H. Jensen & R.C. Wurl

12. Statistical Process Control
Operating Characteristic Curves

13. TM 720: Statistical Process Control
11/13/2018 2:35 PM
Hypothesis Testing
Significance Level of a Hypothesis Test: A hypothesis test with a significance level or size  rejects the null hypothesis H0 if a p-value smaller than is obtained, and accepts the null hypothesis H0 if a p-value larger than  is obtained. In this case, the probability of a Type I error (the probability of rejecting the null hypothesis when it is true) is equal to .
True Situation
Test Conclusion
CORRECT
Type I Error ()
H0 is False
Type II Error ()
H0 is True
(c) D.H. Jensen & R.C. Wurl

14. a error
2.23
a =
1 - a
In words: If the true mean is 10, the probably that we would see a sample mean of 15 or higher is only 1.3%. If we say that the mean has shifted to something larger than 10, there is a 1.3% chance we are wrong.

15. b error
Type II error ( b ) can only be calculated if we want to test for a specific shift. Suppose for example that it is important that detect a shift of m = 12 in the liquid density level.
HO HA 10 c 12

16. b error
Viscosity is supposed to be 10. I need to be able to detect if viscosity has shifted to 12. We agree to use 11 as the critical value for the test.
HO HA 10 c 12

17. b error
Viscosity is supposed to be 10. I need to be able to detect if viscosity has shifted to 12.
HO HA 10 c 12

18. b error
Viscosity is supposed to be 10. I need to be able to detect if viscosity has shifted to 12.
HO HA 10 c 12
In words: We have a 32.6% chance of buying product when the mean has shifted. The supplier has a 32.6% chance of taking back good product. How do we resolve this ?

19. Operating Characteristic Curve
We know that a small change in the mean is hard to detect.
For a given sample, we want to minimize our risk of accepting product where the mean has shifted (b error).
For a given sample, the vendor wants to minimize his/her risk of having to buy back product that is good (a error).
I can calculate b risk for any given sample size and mean difference that is of interest.

20. b error
Type II error ( b ) can only be calculated if we want to test for a specific shift. Suppose for example that it is important that detect a shift of m = 15 in the liquid density level.
HO HA 10 c 12

21. Operating Characteristic Curve

22. OC Curve Example (uses Fig 3-7, p.111)
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)?
Contracting an incapable supplier, based on a bad-luck test outcome
Assume supplier needs  = 0.2
Unfairly being the incapable supplier, based on a bad-luck test outcome
What sample size is needed to fit these constraints?

23. OC Curve Example (uses Fig 3-7, p.111)
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)?
Contracting an incapable supplier, based on a bad-luck test outcome
Assume supplier needs  = 0.2
Unfairly being the incapable supplier, based on a bad-luck test outcome
What sample size is needed to fit these constraints?
d = ( )/18 = 0.5

24. Two-Sided Operating Characteristic Curve,  = 0.05
β =
d = 0.5

25. Two-Sided Operating Characteristic Curve,  = 0.05
β =
d = 0.5

26. Statistical Process Control
Testing for Mean Differences

27. Situation III: Means Test Unknown σ(s) and Known μ0
Used when:
have good control over the center of the distribution, but the variation changed from time to time
Procedure:
use the the t-statistic to compare both sample means
v = n – 1 degrees of freedom

28. TM 720: Statistical Process Control
11/13/2018 2:35 PM
Testing Example
Single Sample, Two-Sided t-Test:
H0: µ = µ0 versus HA: µ ¹ µ0
Test Statistic:
Critical Region: reject H0 if |T| > t/2,n-1
P-Value: 2 • P(X ³ |T|), where the random variable x has a t-distribution with n _ 1 degrees of freedom
(c) D.H. Jensen & R.C. Wurl

29. TM 720: Statistical Process Control
11/13/2018 2:35 PM
Hypothesis Testing
H0: μ = μ0 versus HA: μ  μ0
tn-1 distribution
-|T|
|T|
(c) D.H. Jensen & R.C. Wurl

30. Statistics and Sampling
TM 720: Statistical Process Control
11/13/2018 2:35 PM
Statistics and Sampling
Objective of statistical inference:
Draw conclusions/make decisions about a population based on a sample selected from the population
Random sample – a sample, x1, x2, …, xn , selected so that observations are independently and identically distributed (iid).
Statistic – function of the sample data
Quantities computed from observations in sample and used to make statistical inferences
e.g measures central tendency
(c) D.H. Jensen & R.C. Wurl

31. 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

32. Situation II: Means Test σ(s) Known and μ(s) Unknown
Used when:
the means from two existing processes may differ, but the variation of the two processes is stable, so we can estimate the population variances pretty closely.
Procedure:
use the the z-statistic to compare both sample means

33. Situation IV: Means Test Unknown σ(s) and μ(s), Similar s2
Used when:
logical case for similar variances, but no real "history" with either process distribution (means & variances)
Procedure:
use the the t-statistic to compare using pooled S, v = n1 + n2 – 2 degrees of freedom

34. Situation V: Means Test Unknown σ(s) and μ(s), Dissimilar s2
Used when:
worst case data efficiency - no real "history" with either process distribution (means & variances)
Procedure:
use the the t-statistic to compare,
degrees of freedom given by:

35. Situation VI: Means Test Paired but Unknown σ(s)
Used when:
exact same sample work piece could be run through both processes, eliminating material variation
Procedure:
define variable (d) for the difference in test value pairs
(di = x1i - x2i) observed on ith sample, v = n - 1 dof

36. TM 720: Statistical Process Control
11/13/2018 2:35 PM
Ex. Surface Roughness
Surface roughness is normally distributed with mean 125 and std dev of 5. The specification is 125 ± and we have calculated that 98% of parts are within specs during usual production. This has been the case for a long time.
My supplier of these parts has sent me a large shipment. I take a random sample of 10 parts. The sample average roughness is 134 which is within specifications.
Test the hypothesis that the lot roughness is higher than specifications at  = 0.05.
(c) D.H. Jensen & R.C. Wurl

37. TM 720: Statistical Process Control
11/13/2018 2:35 PM
e.g. Surface Roughness
Check the hypothesis that the sample of size 10, and with an average of 134 comes from a population with mean 125 and standard deviation of 5.
One-Sided Test
H0:  ≤ 0
HA:  > 0
Test Statistic: =
Critical Value:
Z = 1.645
Should I reject H0?
Yes! Since 5.69 > 1.645, it is likely that it exceeds the roughness.
Alpha
One-sided
Two-sided
Level (α) z
0.1
0.05
(c) D.H. Jensen & R.C. Wurl

38. draw the distributions for the surface roughness and sample average
TM 720: Statistical Process Control
11/13/2018 2:35 PM
draw the distributions for the surface roughness and sample average
134
134
(c) D.H. Jensen & R.C. Wurl

39. TM 720: Statistical Process Control
11/13/2018 2:35 PM
e.g. Surface Roughness
Find the probability that the sample of size 10, and with an average of 134 does not come from a population with mean 125 and standard deviation of 5. =
Should I accept this shipment?
(c) D.H. Jensen & R.C. Wurl

40. e.g. Surface Roughness Cont'd
TM 720: Statistical Process Control
11/13/2018 2:35 PM
e.g. Surface Roughness Cont'd
For future shipments, suggest good cutoff values for the sample average
(i.e., accept shipment if average of 10 observations is between what and what)?
We know that encompasses over 99% of the probability mass of the distribution for x
(c) D.H. Jensen & R.C. Wurl

41. Concept Map for Hypothesis Tests

42. 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