1. IENG 486 Statistical Quality & Process Control
4/17/2017
IENG Lecture 06
Hypothesis Testing & Excel Lab
4/17/2017
IENG 486 Statistical Quality & Process Control
(c) , D.H. Jensen & R. C. Wurl

2. IENG 486 Statistical Quality & Process Control
4/17/2017
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: # 1a,b; 5a,c; 9a,c,f; 11a,b,d,g; 17a,b; 18, 21a,c; 22* *uses Fig.4.7, p. 126
4/17/2017
IENG 486 Statistical Quality & Process Control
(c) , D.H. Jensen & R. C. Wurl

3. Relationship with Hypothesis Tests
Assuming that our process is Normally Distributed and centered at the mean, how far apart should our specification limits be to obtain 99. 5% yield?
Proportion defective will be 1 – .995 = .005, and if the process is centered, half of those defectives will occur on the right tail (.0025), and half on the left tail.
To get 1 – = 99.75% yield before the right tail requires the upper specification limit to be set at
 .
4/17/2017
TM 720: Statistical Process Control

4. TM 720: Statistical Process Control
4/17/2017
TM 720: Statistical Process Control

5. Relationship with Hypothesis Tests
Assuming that our process is Normally Distributed and centered at the mean, how far apart should our specification limits be to obtain 99. 5% yield?
Proportion defective will be 1 – .995 = .005, and if the process is centered, half of those defectives will occur on the right tail (.0025), and half on the left tail.
To get 1 – = 99.75% yield before the right tail requires the upper specification limit to be set at + 2.81.
By symmetry, the remaining .25% defective should occur at the left side, with the lower specification limit set at  – 2.81
If we specify our process in this manner and made a lot of parts, we would only produce bad parts .5% of the time.
4/17/2017
IENG 486 Statistical Quality & Process Control

6. IENG 486 Statistical Quality & Process Control
Hypothesis Tests
An Hypothesis is a guess about a situation, that can be tested and can be either true or false.
The Null Hypothesis has a symbol H0, and is always the default situation that must be proven wrong 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.
4/17/2017
IENG 486 Statistical Quality & Process Control

7. 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 .
4/17/2017
IENG 486 Statistical Quality & Process Control

8. IENG 486 Statistical Quality & Process Control
4/17/2017
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 x P(X ³ |t|), where the random variable X has a t-distribution with n _ 1 degrees of freedom
4/17/2017
IENG 486 Statistical Quality & Process Control
(c) , D.H. Jensen & R. C. Wurl

9. IENG 486 Statistical Quality & Process Control
4/17/2017
Hypothesis Testing
H0: m = m 0 versus HA: m  m 0
P-value = P(X£-|t|) + P(X³|t|)
tn-1 distribution
Critical Region: if our test statistic value falls into the region (shown in orange), we reject H0 and accept HA
-|t|
|t|
4/17/2017
IENG 486 Statistical Quality & Process Control
(c) , D.H. Jensen & R. C. Wurl

10. Types of Hypothesis Tests
Hypothesis Tests & Rejection Criteria 0 0   2  2  θ0 θ0 θ θ θ θ0 θ0 θ 0
One-Sided Test
Statistic < Rejection Criterion
H0: θ ≥ θ0
HA: θ < θ0
Two-Sided Test
Statistic < -½ Rejection Criterion or
Statistic > +½ Rejection Criterion
H0: θ = θ0
HA: θ ≠ θ0
One-Sided Test
Statistic > Rejection Criterion
H0: θ ≤ θ0
HA: θ > θ0
4/17/2017
IENG 486 Statistical Quality & Process Control

11. Hypothesis Testing Steps
State the null hypothesis (H0) from one of the alternatives:
that the test statistic q = q0 , q ≥ q0 , or q ≤ q0 .
Choose the alternative hypothesis (HA) from the alternatives:
q ¹ q0 , q < q0 , or q > q0 . (Respectively!)
Choose a significance level of the test (a).
Select the appropriate test statistic and establish a critical region (q0).
(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.
4/17/2017
IENG 486 Statistical Quality & Process Control

12. IENG 486 Statistical Quality & Process Control
4/17/2017
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
4/17/2017
IENG 486 Statistical Quality & Process Control
(c) , D.H. Jensen & R. C. Wurl

13. IENG 486 Statistical Quality & Process Control
4/17/2017
Hypothesis Testing
P-Value: One way to think of the P-value for a particular H0 is: given the observed data set, what is the probability of obtaining this data set or worse when the null hypothesis is true. A “worse” data set is one which is less similar to the distribution for the null hypothesis.
P-Value
0.01
0.10 1
H0 not plausible
Intermediate
area
H0 plausible
4/17/2017
IENG 486 Statistical Quality & Process Control
(c) , D.H. Jensen & R. C. Wurl

14. Statistics and Sampling
IENG 486 Statistical Quality & Process Control
4/17/2017
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
4/17/2017
IENG 486 Statistical Quality & Process Control
(c) , D.H. Jensen & R. C. Wurl

15. Sampling Distribution
IENG 486 Statistical Quality & Process Control
4/17/2017
Sampling Distribution
Sampling Distribution – Probability distribution of a statistic
If we know the distribution of the population from which sample was taken,
we can often determine the distribution of various statistics computed from a sample
4/17/2017
IENG 486 Statistical Quality & Process Control
(c) , D.H. Jensen & R. C. Wurl

16. e.g. Sampling Distribution of the Average from the Normal Distribution
IENG 486 Statistical Quality & Process Control
4/17/2017
e.g. Sampling Distribution of the Average from the Normal Distribution
Take a random sample, x1, x2, …, xn, from a normal population with mean and standard deviation , i.e.,
Compute the sample average
Then will be normally distributed with mean and std deviation
That is
4/17/2017
IENG 486 Statistical Quality & Process Control
(c) , D.H. Jensen & R. C. Wurl

17. Ex. Sampling Distribution of x
IENG 486 Statistical Quality & Process Control
4/17/2017
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 Ans: since the samples come from a normal distribution, and are added together in the process of computing the mean:
4/17/2017
IENG 486 Statistical Quality & Process Control
(c) , D.H. Jensen & R. C. Wurl

18. Ex. Sampling Distribution of x (cont'd)
IENG 486 Statistical Quality & Process Control
4/17/2017
Ex. Sampling Distribution of x (cont'd)
What is the probability the sample average is greater than 15?
Would you conclude the process is operating properly?
4/17/2017
IENG 486 Statistical Quality & Process Control
(c) , D.H. Jensen & R. C. Wurl

19. IENG 486 Statistical Quality & Process Control
4/17/2017
IENG 486 Statistical Quality & Process Control

20. Ex. Sampling Distribution of x (cont'd)
IENG 486 Statistical Quality & Process Control
4/17/2017
Ex. Sampling Distribution of x (cont'd)
What is the probability the sample average is greater than 15?
Would you conclude the process is operating properly?
4/17/2017
IENG 486 Statistical Quality & Process Control
(c) , D.H. Jensen & R. C. Wurl