04/20/23IENG 486 Statistical Quality & Process

Control 1

IENG 486 - Lecture 07

Comparison of Location (Means)

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

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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

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

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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)

0 00

0

xz

n

μ Δ

σ

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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)

1 2 00 2 2

1 2

1 2

x xz

n n

Δ

σ σ

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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)

v = n – 1 degrees of freedom 0 00

xt

S

n

μ Δ

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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

1 2 00

1 2

1 1p

x xt

Sn n

Δ

2nn

S)1n(S)1n(S

21

222

211

p

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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:

1 2 00 2 2

1 2

1 2

x xt

S S

n n

Δ

1n

n

S

1n

n

S

n

S

n

S

v

2

2

2

22

1

2

1

21

2

2

22

1

21

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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

n

Sd

td

00

1n

dd

S

n

1i

2

i

d

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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 (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 = 2

2) ? 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.

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'dDraw the distributions for the surface

roughness and sample average

125 130 135 140120115110x

. . ~ ( 125, 5)r v x N

125

129.74120.27

x

. . ~ ( 125, 5/ 10 1.58)xr v x N

136.65113.35

134

134

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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 encompasses over 99% of the probability mass of the distribution for

3 x x