Confidence, Prediction, and Tolerance Intervals

1
Confidence, Prediction, and Tolerance Intervals

• Engineering Experimental Design
• Valerie L. Young

2
What is a Confidence Interval for the Mean?

• A way of expressing the uncertainty in x as an
  estimate of ?
• x sample mean
• ? population mean
• 95 CI says that
• about 95 of the time, if you estimate an
  interval for ? this way, the true value of ? will
  be inside the interval
• if you collected data this way many times, 95
  times out of 100, the mean of the dataset would
  be in this range

3
What is a Prediction Interval?

• A way of expressing the uncertainty in x as an
  estimate of what the next measured value will be
• x sample mean
• 95 PI means that about 95 of the time, the
  next measurement you make will be inside this
  interval

4
What is a Tolerance Interval?

• A way of determining a range that (with a certain
  confidence level) will contain a certain
  percentage of the population
• An 80 TI with 95 confidence says that about
  95 of the time, 80 of the measurements you
  make will be inside this interval

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In this course . . .

• Confidence intervals are the most important of
  the three
• Prediction intervals will be revisited when we
  learn to place uncertainties on values predicted
  using a model determined by regression
• Tolerance intervals will not be tested on an exam

6
Why the Distinction Between Confidence Interval
and Prediction Interval?

• It is much easier to predict what will happen on
  average, in the long run, than it is to predict
  what will happen in any particular measurement.
• Based on the heights of students in this class,
  what is the average height of an OU student?
• Based on the heights of students in this class,
  how tall will the next person through the door
  be?
• A prediction interval must be wider than a
  confidence interval to allow for this additional
  uncertainty.

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When Do I Do What?
8
When Do I Do What?
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Why the Distinction Between Large and Small
Samples for Confidence Intervals?

• For large samples, the distribution of sample
  means is always normal, regardless of what the
  original population distribution looks like. So,
  you can always use the standard normal (z)
  distribution.
• For small samples, you only get a normal
  distribution of sample means if the population
  distribution is normal, and you have to correct s
  as an estimate of ? by taking into account n.
  (As n increases, s decreases.)

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Normality and Prediction Intervals

• The formulas given here for prediction intervals
  are only valid for normally-distributed data.
• Should use a normal probability plot to check
• Will assume normal distribution when I ask for
  prediction interval in this class
• Nelson, Coffin Copeland discuss how to handle
  non-normal data, should you need to in the future.

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Example 1

• You measure the zinc concentration in the livers
  of 56 fish and find that the mean of these 56
  values is 9.15 ?g Zn / g liver and the standard
  deviation is 1.27 ?g Zn / g liver. What is the
  concentration of zinc in fish liver?

Based on a problem in Devore Farnum, Applied
Statistics for Engineers Scientists, 1999
12
Example 1

• You measure the zinc concentration in the livers
  of 56 fish and find that the mean of these 56
  values is 9.15 ?g Zn / g liver and the standard
  deviation is 1.27 ?g Zn / g liver. What is the
  concentration of zinc in fish liver?
• n 56, so this is a large sample and we can use
  z
• For 95 confidence, we want 5 of the area in
  the 2 tails, or 2.5 of the area in each
• 1 0.025 0.975
• z(area 0.975) 1.96
• CI 1.96 ? 1.27 / sqrt(56) 0.3326
• The mean zinc concentration is 9.2 0.3 ?g Zn /
  g liver (95 confidence interval)
• Note if you use t-critical(?55) then you get
  t-critical2.000 and CI 0.3394. No important
  difference.

Based on a problem in Devore Farnum, Applied
Statistics for Engineers Scientists, 1999
13
Example 2

• You measure the zinc concentration in the livers
  of 56 fish and find that the mean of these 56
  values is 9.15 ?g Zn / g liver and the standard
  deviation is 1.27 ?g Zn / g liver. What
  concentration of zinc do you expect to find in
  the next fish liver you eat?

Based on a problem in Devore Farnum, Applied
Statistics for Engineers Scientists, 1999
14
Example 2

• You measure the zinc concentration in the livers
  of 56 fish and find that the mean of these 56
  values is 9.15 ?g Zn / g liver and the standard
  deviation is 1.27 ?g Zn / g liver. What
  concentration of zinc do you expect to find in
  the next fish liver you eat?
• n 56, so we need t-critical for ?60 (closest
  we can get to 55) and ? 0.025 (half of 0.05).
• t-critical 2.000 (Table B2 in text)
• PI 2.000 ? 1.27 ? sqrt(1(1/56)) 2.562
• Concentration of zinc in next fish will be 9.2
  2.5 ?g Zn / g liver (95 prediction interval)

Based on a problem in Devore Farnum, Applied
Statistics for Engineers Scientists, 1999
15
Example 3

• You measure the zinc concentration in the livers
  of 56 fish and find that the mean of these 56
  values is 9.15 ?g Zn / g liver and the standard
  deviation is 1.27 ?g Zn / g liver. What range
  of zinc concentrations will describe the livers
  of 90 of this type of fish?

Based on a problem in Devore Farnum, Applied
Statistics for Engineers Scientists, 1999
16
Example 3

• You measure the zinc concentration in the livers
  of 56 fish and find that the mean of these 56
  values is 9.15 ?g Zn / g liver and the standard
  deviation is 1.27 ?g Zn / g liver. What range
  of zinc concentrations will describe the livers
  of 90 of this type of fish?
• n 56 and p 0.90, so r 1.6585 and u 1.1787
  (Table B.12 in text use the value for n60)
• TI 1.6585 ? 1.1787 ? 1.27 2.482
• 90 of these fish are likely to have 9.5 2.5
  ?g Zn / g liver in their livers (95 confidence
  level)

Based on a problem in Devore Farnum, Applied
Statistics for Engineers Scientists, 1999
17
Example 4

• Measurements of stabilized viscosity were made on
  five asphalt specimens, resulting in values of
  2781, 2900, 3013, 2856, and 2888 cP. What is the
  viscosity of the asphalt?

Based on a problem in Devore Farnum, Applied
Statistics for Engineers Scientists, 1999
18
Example 4

• Measurements of stabilized viscosity were made on
  five asphalt specimens, resulting in values of
  2781, 2900, 3013, 2856, and 2888 cP. What is the
  viscosity of the asphalt?
• Mean 2887.60 cP, sample std dev 84.03 cP
• n5, so use t-critical for ?4 and ?0.025
• t-critical 2.776 (Table B2 in text)
• CI 2.776 ? 84.03 / sqrt(5) 104
• The mean viscosity of the asphalt is 2890 100
  cP (95 confidence interval)

Based on a problem in Devore Farnum, Applied
Statistics for Engineers Scientists, 1999
19
Example 5

• Measurements of stabilized viscosity were made on
  five asphalt specimens, resulting in values of
  2781, 2900, 3013, 2856, and 2888 cP. If you
  measured a sixth specimen, what would you expect
  its viscosity to be?

Based on a problem in Devore Farnum, Applied
Statistics for Engineers Scientists, 1999
20
Example 5

• Measurements of stabilized viscosity were made on
  five asphalt specimens, resulting in values of
  2781, 2900, 3013, 2856, and 2888 cP. If you
  measured a sixth specimen, what would you expect
  its viscosity to be?
• Mean 2887.60 cP, sample std dev 84.03 cP
• n5, so use t-critical for ?4 and ?0.025
• T-critical 2.776 (Table B2 in text)
• PI 2.776 ? 84.03 ? sqrt(1 1/5) 256 cP
• The viscosity of the next asphalt will be 2890
  260 cP (95 prediction interval)

Based on a problem in Devore Farnum, Applied
Statistics for Engineers Scientists, 1999
21
Example 6

• Measurements of stabilized viscosity were made on
  five asphalt specimens, resulting in values of
  2781, 2900, 3013, 2856, and 2888 cP. Give the
  range of viscosities within which you would
  expect 75 of specimens from this manufacturer
  to lie.

Based on a problem in Devore Farnum, Applied
Statistics for Engineers Scientists, 1999
22
Example 6

• Measurements of stabilized viscosity were made on
  five asphalt specimens, resulting in values of
  2781, 2900, 3013, 2856, and 2888 cP. Give the
  range of viscosities within which you would
  expect 75 of specimens from this manufacturer
  to lie.
• Mean 2887.60 cP, sample std dev 84.03 cP
• n5. This sample is not large enough to derive a
  tolerance interval. At least two more specimens
  must be tested.

Based on a problem in Devore Farnum, Applied
Statistics for Engineers Scientists, 1999
23
Example 7

• In order to learn which brand of battery lasts
  longest in Kasey the Kinderbot, you purchase 50
  Duralife batteries and 45 Rayolife batteries.
  You find that the Duralife batteries last an
  average of 4.15 hours (standard deviation 0.92
  hours). You find that the Rayolife batteries
  last an average of 4.53 hours (standard deviation
  0.84 hours). How much longer do Rayolife
  batteries last?

Based on a problem in Devore Farnum, Applied
Statistics for Engineers Scientists, 1999
24
Example 7

• You find that the 50 Duralife batteries last an
  average of 4.15 hours (standard deviation 0.92
  hours). You find that the 45 Rayolife batteries
  last an average of 0.84 hours (standard deviation
  1.64 hours). How much longer do Rayolife
  batteries last?
• The difference between the means is 0.38.
• 95 CI for the difference is 1.96 ?
  sqrt((0.922/50)(0.842/45)) 0.3539
• Rayolife batteries last 0.38 0.35 hours longer
  than Duralife batteries (95 confidence level).

Based on a problem in Devore Farnum, Applied
Statistics for Engineers Scientists, 1999
25
Example 8

• You want to learn which brand of battery lasts
  longest in Kasey the Kinderbot, but youre on a
  budget, so you purchase 10 Duralife batteries and
  10 Rayolife batteries. You find that the
  Duralife batteries last an average of 4.15 hours
  (standard deviation 0.92 hours). You find that
  the Rayolife batteries last an average of 4.53
  hours (standard deviation 0.84 hours). How much
  longer do Rayolife batteries last?

Based on a problem in Devore Farnum, Applied
Statistics for Engineers Scientists, 1999
26
Example 8

• Youre on a budget, so you purchase 10 Duralife
  batteries and 10 Rayolife batteries. You find
  that the Duralife batteries last an average of
  4.15 hours (standard deviation 0.92 hours). You
  find that the Rayolife batteries last an average
  of 4.53 hours (standard deviation 0.84 hours).
  How much longer do Rayolife batteries last?
• CI 2.262 ? sqrt((0.922/10)(0.842/10)) 0.891
  hours
• Based on samples of 10, the difference in
  lifetime between the battery brands is not
  significant at the 95 confidence level.
• Note that if the sample sizes are not the same,
  there is a complex formula that must be used to
  calculate the degrees of freedom.

Based on a problem in Devore Farnum, Applied
Statistics for Engineers Scientists, 1999