Transcript Comparison of Location (Means) & CIs

IENG 486 - Lecture 07
Comparison of Location (Means)
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Control
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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:

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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
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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 0 and 0 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 (adjust for any safety factor 0)
z0 
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x  μ0  Δ 0 
 σ0 


 n
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Situation II: Means Test
(s) Known and (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
(adjust for any safety factor 0)
z0 
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x1  x 2  Δ 0 
σ12 σ 22

n1 n2
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Situation III: Means Test
Unknown (s) and Known 0
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
(adjust for any safety factor 0)
x  μ0  Δ 0 
t0 
S
n
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v = n – 1 degrees of freedom
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Situation IV: Means Test Unknown (s) and 0,
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
t0 
x1  x2  Δ 0 
Sp
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1 1

n1 n2
(n1  1)S12  (n2  1)S22
Sp 
n1  n2  2
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Situation V: Means Test
Unknown  and 0, 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  x2  Δ 0 
S12 S 22

n1 n2
S
S 


n n 
2 
 1
2
1
v
2
2
2
2
2
S 
S 




n 
n 
 1  2
n1  1
n2  1
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1
2
2
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Situation VI: Means Test
Paired but Unknown (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  0
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:





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Do we know the population variance (2) or should
we estimate it by the sample variance (s2) ?
Do we know the theoretical mean (), or should we
estimate it by the sample mean (y) ?
Do we know if the samples have equal-variance
(12 = 22) ?
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



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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.
 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.
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ex. cont'd
Draw the distributions for the surface
roughness and sample average
113.35
110
115
136.65
120
125
130
135
140
x
134
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 with average 134
comes from a population with mean 125 and std dev of 5.
 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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