04/20/23IENG 486 Statistical Quality & Process
Control 1
IENG 486 - Lecture 06
Hypothesis Testing & Excel Lab
04/20/23 IENG 486 Statistical Quality & Process Control 2
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); 4.3.6 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
04/20/23 TM 720: Statistical Process Control 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 – .0025 = 99.75% yield before the right tail requires the upper specification limit to be set at
+ 2.81.
04/20/23 TM 720: Statistical Process Control 4
04/20/23 IENG 486 Statistical Quality & 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 – .0025 = 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.
04/20/23 IENG 486 Statistical Quality & Process Control 6
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.
04/20/23 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 .
04/20/23 IENG 486 Statistical Quality & Process Control 8
Testing Example
Single Sample, Two-Sided t-Test: H0: µ = µ0 versus HA: µ µ0
Test Statistic:
Critical Region: reject H0 if |t| > t/2,n-1
P-Value: 2 x P(X |t|), where the random variable X has a t-distribution with n _ 1 degrees of freedom
,0
s
xnt
04/20/23 IENG 486 Statistical Quality & Process Control 9
Hypothesis Testing
H0: = 0 versus HA: 0
tn-1 distribution
0-|t| |t|
P-value = P(X-|t|) + P(X|t|)
Critical Region: if our test statistic value falls into the region (shown in orange), we reject H0 and accept HA
04/20/23 IENG 486 Statistical Quality & Process Control 10
2
2
θ0θ
One-Sided TestStatistic < Rejection Criterion
H0: θ ≥ θ0
HA: θ < θ0
Types of Hypothesis Tests
Hypothesis Tests & Rejection Criteria
0 0
Two-Sided TestStatistic < -½ Rejection Criterion
orStatistic > +½ Rejection Criterion
H0: θ = θ0
HA: θ ≠ θ0
One-Sided TestStatistic > Rejection Criterion
H0: θ ≤ θ0
HA: θ > θ0
θθ0θ0θ θθ00
04/20/23 IENG 486 Statistical Quality & Process Control 11
Hypothesis Testing Steps
1. State the null hypothesis (H0) from one of the alternatives:
that the test statistic , ≥, or ≤.
2. Choose the alternative hypothesis (HA) from the alternatives:
, , or . (Respectively!)
Choose a significance level of the test (.
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)
1. Compute the value of the test statistic () from the sample data.
2. 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 ); otherwise, do not reject H0.
04/20/23 IENG 486 Statistical Quality & Process Control 12
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
Tes
t C
on
clu
sio
n
CORRECTType I Error
()H0 is False
Type II Error ()
CORRECTH0 is True
H0 is FalseH0 is True
04/20/23 IENG 486 Statistical Quality & Process Control 13
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.
H0
not plausibleH0
plausibleIntermediate
area
0 0.01 0.10 1
P-Value
04/20/23 IENG 486 Statistical Quality & Process Control 14
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
n
iixn
x1
1
04/20/23 IENG 486 Statistical Quality & Process Control 15
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
04/20/23 IENG 486 Statistical Quality & Process Control 16
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
x
x
n
n
NNxx
,),(~
),(~ Nx
04/20/23 IENG 486 Statistical Quality & Process Control 17
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:
5
5,10~ Nx
04/20/23 IENG 486 Statistical Quality & Process Control 18
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?
?)24.2()(
24.236.2
5
5
51015
0
00
z
n
xz
04/20/23 IENG 486 Statistical Quality & Process Control 19
04/20/23 IENG 486 Statistical Quality & Process Control 20
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?
%3.101255.098745.01
98745.0)24.2()(
24.236.2
5
5
51015
0
00
or
z
n
xz