Confidence Intervals and Maximum Errors

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Confidence Intervals and Maximum Errors

• By Sidney S. Lewis
• For
• Baltimore Section, ASQ
• February 15, 2005

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• CONFIDENCE INTERVAL
• A confidence interval expresses our belief, or
  confidence, that the interval we construct from
  the data will contain the mean, µ, (for example)
  of the population from which the data were drawn.
  
• Confidence Intervals can be computed on any
  population parameter µ, s, p, c, and even on
  complex parameters, such as Cp and Cpk.

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CONFIDENCE INTERVAL EXAMPLE

• Example of a confidence Interval (C.I.) on the
  population mean µ
• Statement The interval 38.0 - 42.0 contains µ
  with 90 confidence.
• Alternatively, there is a 5 chance that the C.I.
  falls entirely below µ (µ above 42.0), and
  likewise, a 5 chance that the C.I. is entirely
  above µ (µ below 38.0).

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Example

• Pollsters report that 55 of a sample of 1005
  members of the voting population support
  Proposition A.
• The Margin of Error is 3.1
• There is an implied risk of being wrong, usually
  5
• Calcs.

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POPULATION vs. SAMPLES
810
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MAXIMUM ERROR
Logic
s is known to be 2.0 s is known to be 2.0 s is known to be 2.0
A sample of 4 yields X-bar 42.0. A sample of 4 yields X-bar 42.0. A sample of 4 yields X-bar 42.0. A sample of 4 yields X-bar 42.0.

Question? What values of m are probable, Question? What values of m are probable, Question? What values of m are probable, Question? What values of m are probable, Question? What values of m are probable,
with 95 Confidence (a 5). with 95 Confidence (a 5). with 95 Confidence (a 5). with 95 Confidence (a 5).

Solution sXbar s/on 1.0 Solution sXbar s/on 1.0 Solution sXbar s/on 1.0 Solution sXbar s/on 1.0
Za/2 1.96, or about 2 Za/2 1.96, or about 2 Za/2 1.96, or about 2

ME Za/2 sXbar 2.0 ME Za/2 sXbar 2.0 ME Za/2 sXbar 2.0
or from 40.0 to 44.0 or from 40.0 to 44.0 or from 40.0 to 44.0
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MAXIMUM ERROR of the MEAN
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C. I. on the MEAN

• Calculation of the C.I. on the mean µ typically
  uses either the population standard deviation, s,
  if known, or if not, the sample standard
  deviation, s.
• If X-bar is the sample mean, then a 1- a
  confidence interval on µ is
• C.I. X-bar Maximum Error (ME)
• X-bar za/2 s /n if s is known, or
• X-bar ta/2s/n if s is unknown.

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Diameters of 3/4" HR Bars
Diameters of 3/4" HR bars Diameters of 3/4" HR bars Diameters of 3/4" HR bars Diameters of 3/4" HR bars Diameters of 3/4" HR bars X-bar R
0.7466 0.7457 0.7524 0.7495 0.7489 0.7486 0.0067
0.7496 0.7549 0.7542 0.7566 0.7493 0.7529 0.0073
0.7563 0.7436 0.7475 0.7525 0.7492 0.7498 0.0127
0.7491 0.7508 0.7512 0.7482 0.7520 0.7502 0.0038
0.7498 0.7552 0.7508 0.7477 0.7453 0.7497 0.0099
0.7508 0.7480 0.7498 0.7526 0.7532 0.7509 0.0052
0.7498 0.7491 0.7507 0.7514 0.7527 0.7507 0.0037
0.7526 0.7521 0.7496 0.7507 0.7533 0.7516 0.0037
0.7520 0.7470 0.7550 0.7517 0.7404 0.7492 0.0146
0.7463 0.7554 0.7483 0.7507 0.7474 0.7496 0.0091
0.7537 0.7520 0.7501 0.7522 0.7524 0.7521 0.0036
0.7476 0.7535 0.7542 0.7548 0.7515 0.7523 0.0072
0.7516 0.7442 0.7499 0.7509 0.7472 0.7488 0.0074
940
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DATA STATISTICS
X-bar R s
0.7486 0.0067 0.0026
0.7529 0.0073 0.0033
0.7498 0.0127 0.0048
0.7502 0.0038 0.0016
0.7497 0.0099 0.0037
0.7509 0.0052 0.0021
0.7507 0.0037 0.0014
0.7516 0.0037 0.0015
0.7492 0.0146 0.0057
0.7496 0.0091 0.0036
0.7521 0.0036 0.0013

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3/4" HR Bars MEANS and CONFIDENCE INTERVALS
s.00300 z(.90) 1.645 z(.90) 1.645 t(4,.90) 2.132 t(4,.90) 2.132
Sample X-bar s LCI(z) UCI(z) LCI(t) UCI(t)
1 0.7486 0.00263 0.7464 0.7508 0.7461 0.7511
2 0.7529 0.00328 0.7507 0.7551 0.7498 0.7561
3 0.7498 0.00484 0.7476 0.7520 0.7452 0.7544
4 0.7502 0.00157 0.7480 0.7525 0.7488 0.7517
5 0.7497 0.00371 0.7475 0.7519 0.7462 0.7533
6 0.7509 0.00209 0.7487 0.7531 0.7489 0.7529
7 0.7507 0.00142 0.7485 0.7529 0.7494 0.7521
8 0.7516 0.00148 0.7494 0.7539 0.7502 0.7531
9 0.7492 0.00568 0.7470 0.7514 0.7438 0.7546
10 0.7496 0.00360 0.7474 0.7518 0.7462 0.7530
11 0.7521 0.00129 0.7499 0.7543 0.7509 0.7533

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FACTORS AFFECTING THE WIDTH OF A CONFIDENCE
INTERVAL
Factors s or s, n, a
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FACTORS AFFECTING THE WIDTH OF A CONFIDENCE
INTERVAL

• s or s ................... Width increases as s
  or s increases
• Sample size, n ..... Width decreases as n
  increases
• C. I. is proportional to 1/on
• Confidence level 1 - a, or risk a
• Width increases as confidence increases, or
  as risk a decreases.

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CONFIDENCE INTERVALS ON s

• Small samples (nlt30)

Large samples (ngt30)
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CONFIDENCE INTERVALS ON p

• Large samples (npgt5)

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CONFIDENCE INTERVAL ON p, SMALL n
90 C.I. on p for n50, c1

Upper C.I. on p' Upper C.I. on p' Upper C.I. on p' Upper C.I. on p' Lower C.I. on p' Lower C.I. on p' Lower C.I. on p' Lower C.I. on p'
a/2 c P n 1-a/2 c P n
27.9 1 5.0 50 82.7 1 1.5 50
3.38 1 10.0 50 91.1 1 1.0 50
5.32 1 9.0 50 97.4 1 0.5 50
4.25 1 9.5 50 96.4 1 0.6 50
4.87 1 9.2 50 95.2 1 0.7 50
5.00 1 9.14 50 95.0 1 0.72 50

Excel Function Function BINOMDIST(c,n,p,1) BINOMDIST(c,n,p,1) BINOMDIST(c,n,p,1) BINOMDIST(c,n,p,1)
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STATISTICAL CALCULATION EXAMPLE A

• TECHNIQUE 1-SAMPLE TEST OF THE MEAN, SIGMA
  UNKNOWN t-TEST
• SUBJECT Machinability Increased by a New
  Practice?
• GOAL Determine whether the average
  machinability of steel made using a new practice
  in the Melt Shop can increase the machinability,
  with 95 CONFIDENCE (5 a).
• HISTORIC DATA The recent past average
  machinability is 85.0 m0 s 12.2 .
• DATA Machinability data of steel made
  with the new practice are
• 91 99 83 87 98 94 86 92 85 81

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STATISTICAL CALCULATION EXAMPLE A

• Calcs. X-bar 89.6 s 6.196 n 10
• Maximum Error, ME ta/2(df) SX-bar
• 1.833 1.957 3.6 units
• 90 C.I. X-bar ME
• 89.6 3.6 86.0 to 93.2 .
• That means that the machinability should increase
  by at least 1.0 units, and may increase by 8
  units.

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STATISTICAL CALCULATION EXAMPLE B

• TECHNIQUE 1-SAMPLE TEST OF PROPORTIONS Z-TEST
• SUBJECT Cap leakers reduction trial
• GOAL Determine whether the rates of leaking
  caps are lower if a new cap design is used, with
  95 CONFIDENCE (a 5).
• HISTORIC DATA Cap leaker rate 1.2 p.
• DATA A trial using 2000 caps of a new design
  found 18 leaking caps. p 0.90.

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STATISTICAL CALCULATION EXAMPLE B

• FORMULAS
• where p is in percent.
• CALCS p 18/2000 0.90 D p0 p 1.2
  - 0.90 0.30
• Maximum Error, ME Z.05 sp 1.645 0.243
  0.400
• 90 C.I. (2-tail) on the difference (D - d0)
  ME (0.90 1.20) 0.400 0.10 to -0.70
• 90 C.I. on p p ME 0.90 0.40 0.5 to
  1.3

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STATISTICAL CALCULATION EXAMPLE B

• CONCLUSION
• The long term leaker rate of the new caps may be
  0.7
• lower than the old caps, but it may also be 0.1
  higher,
• which if true, says to avoid the new caps.
  Therefore the
• data are insufficient to show, with 95
  confidence, that
• the new caps are definitely better, which
  confirms the
• test of hypothesis.

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STATISTICAL CALCULATION EXAMPLE C

• TECHNIQUE 1-SAMPLE TEST OF A SAMPLE STANDARD
  DEVIATIONCHI-SQUARED TEST
• SUBJECT XYZ Digital Blood Pressure Monitor
  measures of systolic blood pressure
• GOAL To determine if the monitor has become more
  variable than when new.
• HISTORIC DATA Early evaluation of this monitor
  found the standard deviation to be 2.5 units.
• DATA Using the monitor, the systolic blood
  pressure of a patient was measured 7 times over a
  ten minute period. The patient sat quietly
  throughout the testing. The results were 144,
  147, 147, 149, 140, 140, 144, from which s 3.51.

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STATISTICAL CALCULATION EXAMPLE C

• CONFIDENCE INTERVAL a 2-tail, 90 confidence
  interval will be calculated
• The critical values of c2 are

With 90 confidence, the true standard deviation
lies between 1.83 and 5.05 units, which
includes the earliest determined standard
deviation of 2.5.
890.72