Transcript TM 720 Lecture 04: Comparison of Means, CIs, & OC Curves

ENGM 720 - Lecture 04
Comparison of Means,
Confidence Intervals (CIs),
& Operating Characteristic
(OC) Curves
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Assignment:

Reading:
•
•

Chapter 4
•
•
Finish reading through 4.3.4
Begin reading 4.4 through 4.4.3
Chapter 8
•
Begin reading 8 through 8.3
Assignments:
•
•
•
Obtain the Hypothesis Test (Chart &) Tables – Materials Page
Obtain the Exam Tables DRAFT – Materials Page
•
Verify accuracy as you work assignments
Access New Assignment and Previous Assignment Solutions:
•
•
Download Assignment 2 Solutions
Download Assignment 3 Instructions
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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.
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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 .
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Types of Hypothesis Tests
Hypothesis Tests & Rejection Criteria
H0: MA is not different than M0
H0: MA is not better than M0
HA: MA is different than M0
HA: MA is lower than M0

θA θ0
Dm


2
2
θA -θ0
Dm
+θ0 θA
H0: MA is not better than M0
HA: MA is higher than M0

θ0 θA
Dm
One-Sided Test
Statistic < Rejection Criterion
Two-Sided Test
Statistic < -½ Rejection Criterion
or
Statistic > +½ Rejection Criterion
One-Sided Test
Statistic > Rejection Criterion
H0: θA ≥ θ0
HA: θA < θ0
H0: -θ0 ≤ θA ≤ +θ0
HA: θA< -θ0 or +θ0< θA
H0: θA ≤ θ0
HA: θA > θ0
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Hypothesis Testing Steps
1. State the null hypothesis (H0) from one of the alternatives:
that the test statistic MA = M0 , MA ≥ M0 , or MA ≤ M0 .
2. Choose the alternative hypothesis (HA) from the alternatives:
MA  M0 , MA < M0 , or MA > M0 . (Respective to above!)
3. Choose a significance level of the test ().
4. 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)
5. Compute the value of the test statistic () from the sample data.
6. 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.
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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
•
n x  0 )
t=
s
P-Value: 2 • P(X  |t|), where the random variable x
has a t-distribution with n _ 1 degrees of freedom
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Hypothesis Testing
H0: μ = μ0 versus HA: μ  μ0
tn-1 distribution
P-value = P(X-|t|) + P(X|t|)
-|t|
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|t|
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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

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H0 is True
H0 is False
H0 is True
CORRECT
Type II Error ()
H0 is False
Type I Error ()
CORRECT
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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 0.01
H0
not plausible
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1
0.10
Intermediate
area
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H0
plausible
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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. x = 1 n x measures central tendency
i
n i =1
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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,
ex:
•
•
•
When the CLT applies, the distribution is Normal
When sampling for defective units in a large population, use
the Binomial distribution
When working with the sum of squared Normal distributions,
use the 2-distribution
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e.g. 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., x ~ N(μ, σ)

Compute the sample average x

Then x will be normally distributed with mean μ and standard
deviation: σ
n
that is:
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 σ 
x ~ N(μ, σ x ) = N μ,

n

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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 

x ~ N μ = 10, σ =

5

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Ex. Sampling Distribution of x
(cont'd)

What is the probability the sample average is greater
than 15?
x  μ  Δ 15  10
5
z=
0
σ0
n
0
=
5
5
=
2.24
= 2.23
Φ( z ) = Φ(2.23) = ?

Would you conclude the process is operating properly?
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Ex. Sampling Distribution of x
(cont'd)

What is the probability the sample average is greater
than 15?
x     15  10
5
z=
0
0
n
0
=
5
5
=
2.24
= 2.23
 ( z ) =  (2.23) = 0.98713
1  0.98713= 0.01287  or  1.3%

Would you conclude the process is operating properly?
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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
Definitely Different
Probably Different
Probably NOT
Different
Definitely NOT
Different
• 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
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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
x  0
z0 =
 0 


 n
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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
z0 =
x1  x 2
 12
n1
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
 22
n2
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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
x  0
t0 =
S
n
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v = n – 1 degrees of freedom
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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
x1  x 2
t0 =
1 1
Sp

n1 n2
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(n1  1)S12  (n2  1)S 22
Sp =
n1  n2  2
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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:
t0 =
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x1  x 2
S12 S22

n1 n2
2
S
S 



n1 n2 

v=
2
2
2
2
 S1 
 S2 
 
 
 n1    n2 
n1  1 n2  1
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1
2
2
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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
d
t0 =
Sd
n
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 d  d)
n
2
i
Sd =
i =1
n 1
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Table for Means Comparisons

Decision on which test to use is based on
answering (at least some of) the following:
•
•
•
•
•
Do we know the population variance (σ2) or should we estimate
it by the sample variance (s2)?
Do we know the theoretical mean (μ), or should we estimate it
by the sample mean ( y ) ?
Do we know if the samples have equal-variance (σ12 = σ22)?
Have we conducted a paired comparison?
What are we trying to decide (alternate hypothesis)?
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Table for Means 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
Six major test statistics for mean comparisons
•
•
•
Two sampling distributions
Six confidence intervals
Twelve alternate hypotheses
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Ex. Surface Roughness

Surface roughness is normally distributed with mean 125
and std dev of 5. The specification is 125 ± 11.65 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.
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e.g. Surface Roughness Cont'd

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:
y  0
z0 =

n


z0 =
=
Critical Value:
•
Z = 1.645
Should I reject H0?
•
134  125
9
=
= 5.69
5
1.58
10
Alpha One-sided Two-sided
Level (α)
z
z
0.1
1.28155 1.64485
0.05
1.64485 1.95996
Yes! Since 5.69 > 1.645, it is likely that it exceeds the roughness.
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ex. cont'd
draw the distributions for the surface
roughness and sample average
113.35
110
115
134
120
125
130
136.65
135
140
x
r.v. x ~ N (  = 125, = 5)
134
x
125
120.27
129.74
r.v. x ~ N (  = 125, x = 5/ 10 = 1.58)
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e.g. Surface Roughness Cont'd

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.
z0 =
y  0

n
=
134  125
9
z0 =
=
= 5.69
5
1.58
10
P  value =1 (z0 ) = 1 (5.69)  1 1 = 0

Should I accept this shipment?
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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   3 x encompasses over 99%
of the probability mass of the distribution for x
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Operating Characteristic (OC)
Curve

Relates the size of the test difference to Type II
Error () for a given risk of Type I Error ()

Designing a test involves a trade-off in sample
size versus the power of the test to detect a
difference
•
•
The greater the difference in means (d), the smaller the
chance of Type II Error () for a given sample size and .
As the sample size increases, the chance of Type II Error ()
decreases for a specified  and given difference in means (d).
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Operating Characteristic Curve
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O.C. Curve Use



Agree on acceptable 
•
Need to have an OC curve for the correct hypothesis
test and the correct level
Estimate anticipated d and  to compute d:
•
d = | 1 - 2| = |d|


Look for where d intersects with desired 
(Probability of accepting H0) to estimate the
required sample size (n)
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OC Curve Example


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
• Probability of what (in English)?
•

(uses Fig 3-7, p.111)
Unfairly being the incapable supplier, based on a bad-luck test outcome
What sample size is needed to fit these constraints?
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Two-Sided Operating Characteristic Curve,  = 0.05
n = 30
β=
d = 0.5
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Estimation of Process
Parameters


In SPC:
•
the probability distribution is used to model a quality
characteristic (e.g. dimension of a part, viscosity of a
fluid)
Therefore:
•
we are interested in making inferences about the
parameters of the probability distribution
•

(e.g. mean μ and variance σ2)
Since:
•
Values of these parameters are generally not known,
so we need to estimate them from sample data
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Point Estimate


Numerical value, computed from a sample of data, used to
estimate a parameter of a distribution
Example:
• Say we take n = 50 measurements of a quality
characteristic
• Sample mean is point estimate of μ
n
i.e.
x
 i
i
X = =1
•
n
Sample variance is point estimate of σ2
i.e.
n
s =
2
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 X
X)
2
i
i =1
n 1



=
n

i =1


X   nX 2 

=
n 1
2
i
n

i =1
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n


X

i
  i =1 
2
X i  
n

2
n 1
38
Confidence Intervals

A confidence interval for an unknown parameter  is an
interval that contains a set of likely values of the
parameter. It is associated with a confidence level
1- , which measures the probability that the confidence
interval actually contains the unknown parameter.
θ
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Confidence Interval (C.I.)
(Interval Estimate)

A C.I. is an interval that, with some probability, includes the true
value of the parameter

Ex. C.I. of mean μ is
P{L  μ  U } = 1  α
•
•
•

L - lower confidence limit
U - upper confidence limit
(1-) - probability that true value of parameter lies in interval
(we pick )
The interval L  μ  U is called a 100(1-)% C.I. for the mean
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C.I. on the Mean of Normal
Distribution with Variance
Unknown

Suppose x ~ N   , ) , and
We don't know the true mean μ or true variance σ2

A 100(1-)% C.I. for the unknown (true) mean μ is:

x  t ,n1
2
•
•
•
•
S
S
   x  t ,n1
2
n
n
x - sample mean
s - sample standard deviation
n - number of observations in sample
t  ,n 1 - value of t distribution
2
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Ex. C.I. on the Mean of Normal
Distribution with Variance
Unknown


Automatic filler deposits liquid in a container.
WANT: 95% C.I. on the mean amount (ounces) per container
• Collect random sample: x1, x2, …, xn
say n = 10
•
Compute sample average:
X=
•
n
1
n
X
i =1
i
= 1.6
Compute sample variance:
n
S2 =
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  xi  x )
i =1
n 1
n
2
=
2
x

nx
i
2
i =1
n 1
= 0.1
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Ex. C.I. on Mean cont'd

Find the t-distribution value:
•
•
Look in Table (Appendix IV)
Want a 95% C.I. so, 100(1 - )% = 95%   = 0.05
 = degrees of freedom = (n -1) = 9
so …
tα
2
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,n 1
= t .05
2
,10 1
= t.025 ,9 = ?
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Ex. C.I. on Mean cont'd

Find the t-distribution value:
• Look in Table (Appendix IV)
• Want a 95% C.I. so, 100(1 - )% = 95%   = 0.05
 = degrees of freedom = (n -1) = 9
so … t α
2

,n 1
= t .05
2
,101
= t.025 ,9 = 2.262
Substitute into C.I.
x  t ,n1
2
S
S
   x  t ,n1
2
n
n
 0.1 
 0.1 
  μ  1.6  2.262

= 1.6  2.262



 10 
 10 
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- or - = 1.37  μ  1.83
45
Interpretation of a 95% C.I.



Repeat sampling 10,000 (or many, many) times & obtain C.I.s
Each C.I. will have (slightly) different center point and width
On average, 95% of the C.I.s will include the true mean

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C.I.s on Other Parameters
and Quantities

Same procedure, different formulas

For example, C.I. on
• Mean (of any distribution) when variance is known
• Variance of a normal distribution
• Difference in two means (of any distribution) when
variances are known
• Difference in two means from normal distribution when
variances are unknown
• Ratio of variances of two normal distributions
• etc. ...

(See textbook Sections 4.3.1, 4.3.4 to review derivations)
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Questions & Issues
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