Statistics

1
Statistics

• Sampling Intervals for a Single Sample

Contents, figures, and exercises come from the
textbook Applied Statistics and Probability for
Engineers, 5th Edition, by Douglas C. Montgomery,
John Wiley Sons, Inc., 2011.
2
Confidence Interval on the Mean of a Normal
Distribution, Variance Known

• If , ,, are normally and independently
  distributed with unknown mean and known
  variance
• has a standard normal
  distribution

3

• Confidence interval on the mean, variance known

Contents, figures, and exercises come from the
textbook Applied Statistics and Probability for
Engineers, 5th Edition, by Douglas C. Montgomery,
John Wiley Sons, Inc., 2011.
4

• From , we have
• If is used as an estimate of , we can be
  confident that the error
  will not exceed a specified amount when the
  sample size is

Contents, figures, and exercises come from the
textbook Applied Statistics and Probability for
Engineers, 5th Edition, by Douglas C. Montgomery,
John Wiley Sons, Inc., 2011.
5

• One-sided confidence bounds on the mean, variance
  known
• A upper-confidence bound for
  is
• A lower-confidence bound for
  is

Contents, figures, and exercises come from the
textbook Applied Statistics and Probability for
Engineers, 5th Edition, by Douglas C. Montgomery,
John Wiley Sons, Inc., 2011.
6

• General method to derive a confidence interval
• We find a statistic that
• 1. depends on both
  the sample and
• 2. The probability distribution of
  does not depend on and any other
  unknown parameter
• For example,
• Find constants and so that

7

• Large-sample confidence interval on the mean
• When is large, the quantity
• has an approximate standard normal distribution.
  Consequently,
• is a large-sample confidence interval for ,
  with confidence level of approximately
  .

Contents, figures, and exercises come from the
textbook Applied Statistics and Probability for
Engineers, 5th Edition, by Douglas C. Montgomery,
John Wiley Sons, Inc., 2011.
8

• Large-sample approximate confidence interval
• If the quantity
• has an approximate standard normal distribution.
  Consequently,
• is a large-sample approximate confidence interval
  for

Contents, figures, and exercises come from the
textbook Applied Statistics and Probability for
Engineers, 5th Edition, by Douglas C. Montgomery,
John Wiley Sons, Inc., 2011.
9

• Example 8-1 Metallic Material Transition
• Ten measurements 64.1, 64.7, 64.5, 64.6, 64.5,
  64.3, 64.6, 64.8, 64.2, 64.3
• Assume it is a normal distribution with
  . Find a 95 CI for .
• Example 8-2 Metallic Material Transition
• Determine how many specimens must be tested to
  ensure that the 95 CI for has a length of
  at most 1.0.
• Example 8-3 One-Sided Confidence Bound
• Determine a lower, one-sided 95 CI for .

Contents, figures, and exercises come from the
textbook Applied Statistics and Probability for
Engineers, 5th Edition, by Douglas C. Montgomery,
John Wiley Sons, Inc., 2011.
10

• Example 8-4 Mercury Contamination
• 53 measurements 1.230, 0.490,
• , , ,
  .
• Find a 95 CI for .

Contents, figures, and exercises come from the
textbook Applied Statistics and Probability for
Engineers, 5th Edition, by Douglas C. Montgomery,
John Wiley Sons, Inc., 2011.
11

• Exercise 8-14
• The life in hours of a 75-watt light bulb is
  known to be normally distributed with
  hours. A random sample of 20 bulbs has a mean
  life of hours.
• (a) Construct a 95 two-sided confidence interval
  on the mean life.
• (b) Construct a 95 lower-confidence bound on the
  mean life. Compare the lower bound of this
  confidence interval with the one in part (a).

Contents, figures, and exercises come from the
textbook Applied Statistics and Probability for
Engineers, 5th Edition, by Douglas C. Montgomery,
John Wiley Sons, Inc., 2011.
12
Confidence Interval on the Mean of a Normal
Distribution, Variance Unknown

• Distribution
• Let , ,, are normally and independently
  distributed with unknown mean and unknown
  variance . The random variable
• has a distribution with degrees of
  freedom.

13

• PDF of distribution

From Wikipedia, http//www.wikipedia.org.
14

• CDF of distribution

From Wikipedia, http//www.wikipedia.org.
15

• The probability density function
• is the number of degrees of freedom
• Mean
• Variance for
• Percentage points
• is a large-sample confidence interval for ,
  with confidence level of approximately
  .

Contents, figures, and exercises come from the
textbook Applied Statistics and Probability for
Engineers, 5th Edition, by Douglas C. Montgomery,
John Wiley Sons, Inc., 2011.
16

• confidence interval on

Contents, figures, and exercises come from the
textbook Applied Statistics and Probability for
Engineers, 5th Edition, by Douglas C. Montgomery,
John Wiley Sons, Inc., 2011.
17

• Confidence interval on the mean, variance unknown
• If and are the mean and standard
  deviation of a random sample from a normal
  distribution with unknown variance , a
  confidence interval on is
  given by
• where is the upper
  percentage point of the distribution with
  degrees of freedom

Contents, figures, and exercises come from the
textbook Applied Statistics and Probability for
Engineers, 5th Edition, by Douglas C. Montgomery,
John Wiley Sons, Inc., 2011.
18

• Normal probability plot
• The sample , ,, is arranged as
  , ,, ,where is the smallest
  observation, is the second-smallest
  observation, and so forth.
• The ordered observations are then plotted
  against their observed cumulative frequency
  on the appropriate probability paper.
• Or, plot the standardized normal scores
  against , where

Contents, figures, and exercises come from the
textbook Applied Statistics and Probability for
Engineers, 5th Edition, by Douglas C. Montgomery,
John Wiley Sons, Inc., 2011.
19

• Percent-percent plot

From Wikipedia, http//www.wikipedia.org.
20

• Example 8-5 Alloy Adhesion
• The load at specimen failure 19.8, 10.1,
• , , .
• Find a 95 CI on .

Contents, figures, and exercises come from the
textbook Applied Statistics and Probability for
Engineers, 5th Edition, by Douglas C. Montgomery,
John Wiley Sons, Inc., 2011.
21

• Exercise 8-41
• An article in Nuclear Engineering International
  (February 1988, p. 33) describes several
  characteristics of fuel rods used in a reactor
  owned by an electric utility in Norway.
  Measurements on the percentage of enrichment of
  12 rods were reported as follows 2.94, 3.00,
  2.90, 2.75, 3.00, 2.95, 2.90, 2.75, 2.95, 2.82,
  2.81, 3.05.
• (a) Use a normal probability plot to check the
  normality assumption.
• (b) Find a 99 two-sided confidence interval on
  the mean percentage of enrichment. Are you
  comfortable with the statement that the mean
  percentage of enrichment is 2.95? Why?

Contents, figures, and exercises come from the
textbook Applied Statistics and Probability for
Engineers, 5th Edition, by Douglas C. Montgomery,
John Wiley Sons, Inc., 2011.
22
Confidence Interval on the Variance and Standard
Deviation of a Normal Distribution

• Distribution
• Let , ,, are normally and independently
  distributed mean and variance , and let
  be the sample variance. The random variable
• has a chi-square distribution with
  degrees of freedom.

23

• PDF of distribution

From Wikipedia, http//www.wikipedia.org.
24

• CDF of distribution

From Wikipedia, http//www.wikipedia.org.
25

• The probability density function
• is the number of degrees of freedom
• Mean
• Variance
• Percentage points
• is a large-sample confidence interval for ,
  with confidence level of approximately
  .

Contents, figures, and exercises come from the
textbook Applied Statistics and Probability for
Engineers, 5th Edition, by Douglas C. Montgomery,
John Wiley Sons, Inc., 2011.
26

• Since
• is chi-square with degrees of freedom, we
  have

Contents, figures, and exercises come from the
textbook Applied Statistics and Probability for
Engineers, 5th Edition, by Douglas C. Montgomery,
John Wiley Sons, Inc., 2011.
27

• Confidence interval on the variance
• If is the sample variance from a random
  sample of observations from a normal
  distribution with unknown variance , then a
  confidence interval on is
  
• Where and are the upper
  and lower percentage points of the
  chi-square distribution with
• degrees of freedom, respectively. A
  confidence interval for has lower and upper
  limits that are the square roots of the
  corresponding limits in the above equation

Contents, figures, and exercises come from the
textbook Applied Statistics and Probability for
Engineers, 5th Edition, by Douglas C. Montgomery,
John Wiley Sons, Inc., 2011.
28

• One-sided confidence bounds on the variance
• The lower and upper
  confidence bounds on are
• respectively.

Contents, figures, and exercises come from the
textbook Applied Statistics and Probability for
Engineers, 5th Edition, by Douglas C. Montgomery,
John Wiley Sons, Inc., 2011.
29

• Example 8-6 Detergent Filling
• , .
• Find a 95 upper confidence bound on and
  .
• Exercise 8-44
• A rivet is to be inserted into a hole. A random
  sample of parts is selected, and the
  hole diameter is measured. The sample standard
  deviation of the hole diameter measurements is
  millimeters. Construct a 99 lower
  confidence bound for .

Contents, figures, and exercises come from the
textbook Applied Statistics and Probability for
Engineers, 5th Edition, by Douglas C. Montgomery,
John Wiley Sons, Inc., 2011.
30
Large-Sample Confidence Interval for a population
proportion

• Normal approximation for a binomial proportion
• If is large, the distribution of
• is approximately standard normal.

31

• PMF of binomial distribution

From Wikipedia, http//www.wikipedia.org.
32

• To construct the confidence interval on ,

Contents, figures, and exercises come from the
textbook Applied Statistics and Probability for
Engineers, 5th Edition, by Douglas C. Montgomery,
John Wiley Sons, Inc., 2011.
33

• Approximate confidence interval on a binomial
  proportion
• If is the proportion of observations in a
  random sample of size that belongs to a
  class of interest, an approximate
  confidence interval on the proportion of
  the population that belongs to this class is
• where is the upper percentage of
  the standard normal distribution.
• Required and

Contents, figures, and exercises come from the
textbook Applied Statistics and Probability for
Engineers, 5th Edition, by Douglas C. Montgomery,
John Wiley Sons, Inc., 2011.
34

• Sample size for a specified error on a binomial
  proportion
• Set
• Then
• Or

Contents, figures, and exercises come from the
textbook Applied Statistics and Probability for
Engineers, 5th Edition, by Douglas C. Montgomery,
John Wiley Sons, Inc., 2011.
35

• Approximate one-sided confidence bounds on a
  binomial proportion
• The approximate lower and
  upper confidence bounds are
• respectively.

Contents, figures, and exercises come from the
textbook Applied Statistics and Probability for
Engineers, 5th Edition, by Douglas C. Montgomery,
John Wiley Sons, Inc., 2011.
36

• Example 8-7 Crankshaft Bearings
• , , and
• Find a 95 two-sided confidence interval for
  .
• Example 8-8 Crankshaft Bearings
• How large a sample is required if we want to be
  95 confident that the error in using to
  estimate is less than 0.05?

Contents, figures, and exercises come from the
textbook Applied Statistics and Probability for
Engineers, 5th Edition, by Douglas C. Montgomery,
John Wiley Sons, Inc., 2011.
37

• Exercise 8-53
• The fraction of defective integrated circuits
  produced in a photolithography process is being
  studied. A random sample of 350 circuits is
  tested, revealing 15 defectives.
• (a) Calculate a 95 two-sided CI on the fraction
  of defective circuits produced by this particular
  tool.
• (b) Calculate a 95 upper confidence bound on the
  fraction of defective circuits.

Contents, figures, and exercises come from the
textbook Applied Statistics and Probability for
Engineers, 5th Edition, by Douglas C. Montgomery,
John Wiley Sons, Inc., 2011.
38
Tolerance and Prediction Intervals

• is a single future observation
• Then
• has a standard normal distribution and
• Has a distribution with degrees of
  freedom.

39

• Prediction interval
• A prediction interval (PI) on
  a single future observation from a normal
  distribution is given by

40

• Tolerance interval
• A tolerance interval for capturing at least
  of the values in a normal distribution with
  confidence level is
• where is a tolerance interval factor found
  in Appendix Tabel XII. Values are given for
  90, 95, and 99 and for 90, 95, and 99
  confidence.

41

• Example 8-9 Alloy Adhesion
• , , and
• Find a 95 prediction interval on the load at
  failure for a new specimen.
• Example 8-10 Alloy Adhesion
• Find a tolerance interval for the load at failure
  that includes 90 of the values in the population
  with 95 confidence.

42

• Exercise 8-77
• Consider the rainfall data in Exercise 8-33.
  Compute a 95 tolerance interval that has
  confidence level 95. Compare the length of the
  tolerance interval with the length of the 95 CI
  on the population mean. Discuss the difference in
  interpretation of these two intervals.