Inferences About Process Quality

1
Chapter 3

• Inferences About Process Quality

2
3-1. Statistics and Sampling Distributions

• Statistical methods are used to make decisions
  about a process
• Is the process out of control?
• Is the process average you were given the true
  value?
• What is the true process variability?

3
3-1. Statistics and Sampling Distributions

• Statistics are quantities calculated from a
  random sample taken from a population of
  interest.
• The probability distribution of a statistic is
  called a sampling distribution.

4
3-1.1 Sampling from a Normal Distribution

• Let X represent measurements taken from a normal
  distribution. X
• Select a sample of size n, at random, and
  calculate the sample mean,
• Then

5
3-1.1 Sampling from a Normal Distribution

• Probability example
• The life of an automotive battery is normally
  distributed with mean 900 days and standard
  deviation 35 days. What is the probability that
  a random sample of 25 batteries will have an
  average life of more than 910 days?

6
Example

• Z (910-900)/(35/SQRT(25) 1.429
• P(Xbar gt 910) 1 - .9235 .0765

7
3-1.1 Sampling from a Normal Distribution

• Chi-square (?2) Distribution
• If x1, x2, , xn are normally and independently
  distributed random variables with mean zero and
  variance one, then the random variable
• is distributed as chi-square with n degrees of
  freedom

8
3-1.1 Sampling from a Normal Distribution

• Chi-square (?2) Distribution
• Furthermore, the sampling distribution of
• is chi-square with n 1 degrees of freedom
  when sampling from a normal population.

9
3-1.1 Sampling from a Normal Distribution

• Chi-square (?2) Distribution for various degrees
  of freedom.

5
10
20
10
3-1.1 Sampling from a Normal Distribution

• t-distribution
• If x is a standard normal random variable and if
  y is a chi-square random variable with k degrees
  of freedom, then
• is distributed as t with k degrees of freedom.

11
3-1.1 Sampling from a Normal Distribution

• F-distribution
• If w and y are two independent chi-square random
  variables with u and v degrees of freedom,
  respectively, then
• is distributed as F with u numerator degrees
  of freedom and v denominator degrees of freedom.

12
3-1.2 Sampling from a Bernoulli
Distribution

• A random variable, x, with probability function
• is called a Bernoulli random variable.
• The sum of a sample from a Bernoulli process has
  a binomial distribution with parameters n and p.

13
3-1.2 Sampling from a Bernoulli
Distribution

• x1, x2, , xn taken from a Bernoulli process
• The sample mean is a discrete random variable
  given by
• The mean and variance of are

14
3-1.3 Sampling from a Poisson
Distribution

• Consider a random sample of size n x1, x2, ,
  xn, taken from a Poisson process with parameter ?
  
• The sum, x x1 x2 xn is also Poisson
  with parameter n?.
• The sample mean is a discrete random variable
  given by
• The mean and variance of are

15
3-2. Point Estimation of Process Parameters

• Parameters are values representing the
  population. (Ex.) The population mean
  and variance, respectively.
• Parameters in reality are often unknown and must
  be estimated.
• Statistics are estimates of parameters.
• (Ex.) are the sample mean and
  sample variance, respectively.

16
3-2. Point Estimation of Process Parameters

• Two properties of good point estimators
• The point estimator should be unbiased.
• E(Q) Q
• The point estimator should have minimum variance.

17
Unbiased estimators

• The sample mean (Xbar) and variance (S2) are
  unbiased estimators of the population mean (m)
  and variance (s2)
• The sample standard deviation is not an unbiased
  estimator (S) of the standard deviation (s)
• E(S) c4s
• So, sest S/c4 (sest is s with a hat on it)
• App. Table VI gives values of c4 for 2 lt n lt 25

18
Range Method

• The range of a sample is often used in QC
• R xmax xmin
• The relative range is given by
• W R/s
• W has been well studied
• E(W) d2

19
Range method

• Since W R/s
• Then, sW R
• And s R/W
• And sest R/d2
• Values of d2 for 2 lt n lt 25 are in App. Table VI

20
Range method

• Using S is better than using the range method
• But, for small sample sizes (n lt 10) the range
  method is acceptable
• Often times, sample sizes are n 5 or n 6
• The relative efficiency of the range method is
  shown on the next frame

21
Range method
22
3-3. Statistical Inference for a Single
Sample

• Two categories of statistical inference
• Parameter Estimation
• Hypothesis Testing

23
3-3. Statistical Inference for a Single
Sample

• A statistical hypothesis is a statement about the
  values of the parameters of a probability
  distribution.

24
3-3. Statistical Inference for a Single
Sample

• Steps in Hypothesis Testing
• Identify the parameter of interest
• State the null hypothesis, H0 and alternative
  hypotheses, H1
• Choose a significance level
• State the appropriate test statistic
• State the rejection region
• Compare the value of the test statistic to the
  rejection region. Can the null hypothesis be
  rejected?

25
3-3. Statistical Inference for a Single
Sample

• Example An automobile manufacturer claims a
  particular automobile averages 35 mpg (highway).
• Suppose we are interested in testing this claim.
  We will sample 25 of these particular autos and
  under identical conditions calculate the average
  mpg for this sample.
• Before actually collecting the data, we decide
  that if we get a sample average less than 33 mpg
  or more than 37 mpg, we will reject the makers
  claim. (Critical Values)

26
3-3. Statistical Inference for a Single
Sample

• Example (continued)
• H0
• H1
• From the sample of 25 cars, the average mpg was
  found to be 31.5. What is your conclusion?

27
3-3. Statistical Inference for a Single
Sample

• Choice of Critical Values
• How are the critical values chosen?
• Wouldnt it be easier to decide how much room
  for error you will allow instead of finding the
  exact critical values for every problem you
  encounter?
• OR
• Wouldnt it be easier to set the size of the
  rejection region, rather than setting the
  critical values for every problem?

28
3-3. Statistical Inference for a Single
Sample

• Significance Level
• The level of significance, ? determines the size
  of the rejection region.
• The level of significance is a probability. It is
  also known as the probability of a Type I error
  (want this to be small)
• Type I error - rejecting the null hypothesis when
  it is true.
• How small? Usually want

29
3-3. Statistical Inference for a Single
Sample

• Types of Error
• Type I error - rejecting the null hypothesis when
  it is true.
• Pr(Type I error) ?. Sometimes called the
  producers risk.
• Type II error - not rejecting the null hypothesis
  when it is false.
• Pr(Type II error) ?. Sometimes called the
  consumers risk.

30
3-3. Statistical Inference for a Single
Sample

• Power of a Test
• The Power of a test of hypothesis is given by 1 -
  ?
• That is, 1 - ? is the probability of correctly
  rejecting the null hypothesis, or the probability
  of rejecting the null hypothesis when the
  alternative is true

31
3-3.1 Inference on the Mean of a
Population, Variance Known

• Hypothesis Testing
• Hypotheses H0 H1
• Test Statistic
• Significance Level, ?
• Rejection Region
• If Z0 falls into either of the two regions above,
  reject H0

32
3-3.1 Inference on the Mean of a
Population, Variance Known

• Example 3-1
• Hypotheses H0 H1
• Test Statistic
• Significance Level, ? 0.05
• Rejection Region
• Since 3.50 gt 1.645, reject H0 and conclude that
  the lot mean pressure strength exceeds 175 psi

33
3-3.1 Inference on the Mean of a
Population, Variance Known

• Confidence Intervals
• A general 100(1- ?) two-sided confidence
  interval on the true population mean, ? is
• 100(1- ?) One-sided confidence intervals are
• Upper
  Lower

34
3-3.1 Inference on the Mean of a
Population, Variance Known

• Confidence Interval on the Mean with Variance
  Known
• Two-Sided
• See the text for one-sided confidence intervals

35
3-3.1 Inference on the Mean of a
Population, Variance Known

• Example 3-2
• Reconsider Example 3-1. Suppose a 95 two-sided
  confidence interval is specified. Using Equation
  (3-28) we compute
• Our estimate of the mean bursting strength is 182
  psi ? 3.92 psi with 95 confidence

36
3-3.2 The Use of P-Values in
Hypothesis Testing

• If it is not enough to know if your test
  statistic, Z0 falls into a rejection region, then
  a measure of just how significant your test
  statistic is can be computed - P-value.
• P-values are probabilities associated with the
  test statistic, Z0.

37
3-3.2 The Use of P-Values in Hypothesis
Testing

• Definition
• The P-value is the smallest level of significance
  that would lead to rejection of the null
  hypothesis H0.

38
3-3.2 The Use of P-Values in Hypothesis
Testing

• Example
• Reconsider Example 3-1. The test statistic was
  calculated to be Z0 3.50 for a right-tailed
  hypothesis test. The P-value for this problem is
  then
• P 1 - ?(3.50) 0.00023
• Thus, H0 ? 175 would be rejected at any level
  of significance ? ? P 0.00023

39
3-3.3 Inference on the Mean of a
Population, Variance Unknown

• Hypothesis Testing
• Hypotheses H0 H1
• Test Statistic
• Significance Level, ?
• Rejection Region
• Reject H0 if

40
3-3.3 Inference on the Mean of a
Population, Variance Unknown

• Confidence Interval on the Mean with Variance
  Unknown
• Two-Sided
• See the text for the one-sided confidence
  intervals.

41
3-3.3 Inference on the Mean of a
Population, Variance Unknown

• Computer Output

42
3-3.4 Inference on the Variance of a
Normal Distribution

• Hypothesis Testing
• Hypotheses H0 H1
• Test Statistic
• Significance Level, ?
• Rejection Region

43
3-3.4 Inference on the Variance of a
Normal Distribution

• Confidence Interval on the Variance
• Two-Sided
• See the text for the one-sided confidence
  intervals.

44
Example

• Compute a 95 2-sided CI on the data in Table
  3.1.
• S2 2.76
• Find c2.025, 15 27.49 and c2.975,15 6.27
• 15(2.76)/27.49 lt s2 lt 15(2.76)/6.27
• 1.51 lt s2 lt 6.60

45
3-3.5 Inference on a Population
Proportion

• Hypothesis Testing
• Hypotheses H0 p p0 H1 p ? p0
• Test Statistic
• Significance Level, ?
• Rejection Region

46
3-3.5 Inference on a Population
Proportion

• Confidence Interval on the Population Proportion
• Two-Sided
• See the text for the one-sided confidence
  intervals.

47
Example

• A foundry produces castings used in the
  automotive industry. We wish to test the
  hypothesis that the fraction nonconforming from
  this process is 10. In a random sample of 250
  castings, 41 were found to be nonconforming.
• H0 p .10
• H1 p .10

48
Example, cont.

• np0 250(.10) 25
• So, x 41 gt np0
• Use Z0 (x - .5) - np0/SQRTnp0(1 p0)
• (41 - .5) - 25/SQRT25(.1)(1 .1) 3.27
• At a .05, Z.025 1.96
• So, reject H0
• P 2(1 - .99946) 2(.00054) .00108

49
3-3.6 The Probability of Type II Error

• Calculation of P(Type II Error)
• Assume the test of interest is H0 H1
• P(Type II Error) is found to be
• The Power of the test is then 1 - ?

50
Explanation

• Test statistic is
• Z0 (xbar m0)/s/SQRT(n) N(0,1)
• Assume H0 is false
• Then find the distribution of Z
• Suppose that the mean is really m1 m0 d where
  d gt 0
• Under this assumption Z0N(d/s/SQRT(n),1)
• Or, Z0Nd(SQRT(n)/s,1

51
Explanation, cont.

• Now, take a look at Fig. 3-6
• We are trying to calculate the shaded area
• We must calculate F(Za/2) F(-Za/2)
• So, we use the equation shown previously

52
Example

• Mean contents of coffee cans
• Given that s .1 oz.
• H0 m 16.0
• H1 m 16.0
• Suppose we want to find b if m 16.1 oz.

53
Example, cont.

• Now, d 16.1 16.0 .1
• Z.025 1.96
• Then b F1.96(.1)(3)/.1 -
  F-1.96(.1)(3)/.1
• Or, F(-1.04) F(-4.96)
• .1492

54
3-3.6 The Probability of Type II Error

• Operating Characteristic (OC) Curves
• Operating Characteristic (OC) curve is a graph
  representing the relationship between ?, ?, ? and
  n.
• OC curves are useful in determining how large a
  sample is required to detect a specified
  difference with a particular probability.

55
3-3.6 The Probability of Type II Error

• Operating Characteristic (OC) Curves

Previous example d .1/.1 1 b .1492
d/s
56
3-3.7 Probability Plotting

• Probability plotting is a graphical method for
  determining whether sample data conform to a
  hypothesized distribution based on a subjective
  visual examination of the data.
• Probability plotting uses special graph paper
  known as probability paper. Probability paper is
  available for the normal, lognormal, and Weibull
  distributions among others.
• Can also use the computer.

57
3-3.7 Probability Plotting

• Example 3-8

58
Tensile strength data
Note m and s can be estimated from the
probability plot as shown
59
3-4. Statistical Inference for Two Samples

• Previous section presented hypothesis testing and
  confidence intervals for a single population
  parameter.
• Results are extended to the case of two
  independent populations
• Statistical inference on the difference in
  population means,

60
3-4.1 Inference For a Difference in
Means, Variances Known

• Assumptions
• X11, X12, , X1n1 is a random sample from
  population 1.
• X21, X22, , X2n2 is a random sample from
  population 2.
• The two populations represented by X1 and X2 are
  independent
• Both populations are normal, or if they are not
  normal, the conditions of the central limit
  theorem apply

61
3-4.1 Inference For a Difference in
Means, Variances Known

• Point estimator for is
• where

62
3-4.1 Inference For a Difference in
Means, Variances Known

• Hypothesis Tests for a Difference in Means,
  Variances Known
• Null Hypothesis
• Test Statistic

63
3-4.1 Inference For a Difference in
Means, Variances Known

• Hypothesis Tests for a Difference in Means,
  Variances Known
• Alternative Hypotheses Rejection
  Criterion

64
3-4.1 Inference For a Difference in
Means, Variances Known

• Confidence Interval on a Difference in Means,
  Variances Known
• 100(1 - ?) confidence interval on the
  difference in means is given by

65
Example 3-9

• Drying time is being studied
• 10 samples of each paint
• Xbar1 121 and Xbar2 112
• Standard deviation, s 8, unaffected by paint
  formulation
• H0 m1 - m2 0
• H1 m1 - m2 gt 0

66
Example, cont.

• Z0 2.52 gt Z.05 1.645
• Therefore, reject H0
• P-value 1 F(2.52) .0059
• H0 would be rejected at any significance level a
  gt .0059

67
Confidence Interval

• On a difference in means, variances known
• See Eq. 3-49

68
3-4.2 Inference For a Difference in
Means, Variances Unknown

• Hypothesis Tests for a Difference in Means,
• Case I
• Point estimator for is
• where

69
3-4.2 Inference For a Difference in
Means, Variances Unknown

• Hypothesis Tests for a Difference in Means,
• Case I
• The pooled estimate of , denoted by is
  defined by

70
3-4.2 Inference For a Difference in
Means, Variances Unknown

• Hypothesis Tests for a Difference in Means,
• Case I
• Null Hypothesis
• Test Statistic

71
3-4.2 Inference For a Difference in
Means, Variances Unknown

• Hypothesis Tests for a Difference in Means,
  Variances Unknown
• Alternative Hypotheses Rejection
  Criterion

72
Example 3-10

• Discuss this example on pgs. 120-121
• The variances are assumed to be equal, so that
  they are pooled
• t.025,14 2.145

73
3-4.2 Inference For a Difference in
Means, Variances Unknown

• Hypothesis Tests for a Difference in Means,
• Case II
• Null Hypothesis
• Test Statistic

74
3-4.2 Inference For a Difference in
Means, Variances Unknown

• Hypothesis Tests for a Difference in Means,
• Case II
• The degrees of freedom for are given by

75
3-4.2 Inference For a Difference in
Means, Variances Unknown

• Confidence Intervals on a Difference in Means,
  Case I
• 100(1 - ?) confidence interval on the
  difference in means is given by

76
3-4.2 Inference For a Difference in
Means, Variances Unknown

• Confidence Intervals on a Difference in Means,
  Case II
• 100(1 - ?) confidence interval on the
  difference in means is given by

77
Example 3-11

• Discuss this example beginning on pg. 122
• The variances are assumed to be equal, so that
  they are pooled
• t.025,23 2.069
• Note that the 95 CI contains zero, so that we
  cannot conclude that there is a difference in the
  means

78
Minitab solution
Two-sample T for Catalyst 1 vs Catalyst 2
N Mean StDev SE Mean Catalyst 8
92.26 2.39 0.84 Catalyst 8 92.73
2.99 1.1 Difference mu Catalyst 1 -
mu Catalyst 2 Estimate for difference -0.47 95
CI for difference (-3.37, 2.42) T-Test of
difference 0 (vs not ) T-Value -0.35
P-Value 0.731 DF 14 Both use Pooled StDev
2.70
Note 0 is included in the CI
79
Minitab solution
No obvious differences
80
Minitab solution
Normality and equal variances are reasonable
81
3-4.2 Paired Data

• Observations in an experiment are often paired to
  prevent extraneous factors from inflating the
  estimate of the variance.
• Difference is obtained on each pair of
  observations, dj x1j x2j, where j 1, 2, ,
  n.
• Test the hypothesis that the mean of the
  difference, ?d, is zero.

82
3-4.2 Paired Data

• The differences, dj, represent the new set of
  data with the summary statistics

83
3-4.2 Paired Data

• Hypothesis Testing
• Hypotheses H0 ?d 0 H1 ?d ? 0
• Test Statistic
• Significance Level, ?
• Rejection Region t0 ? t?/2,n-1

84
Example 3-12

• dbar -1.38
• t0 -1.46
• t.025, 7 2.365
• Conclusion No strong evidence to indicate the
  the two machines differ in their tensile strength
• P-value .1877

85
Solution using Minitab
Paired T for Mach 1 - Mach 2
N Mean StDev SE Mean Mach 1
8 69.13 5.96 2.11 Mach 2
8 70.50 6.07 2.15 Difference
8 -1.375 2.669 0.944 95 CI for mean
difference (-3.608, 0.858) T-Test of mean
difference 0 (vs not 0) T-Value -1.46
P-Value 0.188
86
3-4.3 Inferences on the Variances of Two
Normal Distributions

• Hypothesis Testing
• Consider testing the hypothesis that the
  variances of two independent normal distributions
  are equal.
• Assume random samples of sizes n1 and n2 are
  taken from populations 1 and 2, respectively

87
3-4.3 Inferences on the Variances of Two
Normal Distributions

• Hypothesis Testing
• Hypotheses
• Test Statistic
• Significance Level, ?
• Rejection Region

88
3-4.3 Inferences on the Variances of Two
Normal Distributions

• Alternative Test Rejection
• Hypothesis Statistic Region

89
3-4.3 Inferences on the Variances of Two
Normal Distributions

• Confidence Intervals on Ratio of the Variances of
  Two Normal Distributions
• 100(1 - ?) two-sided confidence interval
  on the ratio of variances is given by

90
3-4.4 Inference on Two Population
Proportions

• Large-Sample Hypothesis Testing
• Hypotheses H0 p1 p2 H1 p1 ? p2
• Test Statistic
• Significance Level, ?
• Rejection Region

91
3-4.4 Inference on Two Population
Proportions

• Alternative Hypothesis Rejection Region

92
3-4.4 Inference on Two Population
Proportions

• Confidence Interval on the Difference in Two
  Population Proportions
• Two-Sided
• See the text for the one-sided confidence
  intervals.

93
3-5. What If We Have More Than Two
Populations?

• Example
• Investigating the effect of one factor (with
  several levels) on
• some response. See Table 3-5
• Hardwood Observations
• Concentration 1 2 3 4 5 6
  Totals Avg
• 5 7 8 15 11 9 10
  60 10
• 10 12 17 13 18 19 15
  94 15.67
• 15 14 18 19 17 16 18
  102 17
• 20 19 25 22 23 18 20
  127 21.17
• Overall 383 15.96

94
3-5. What If We Have More Than Two
Populations?

• Analysis of Variance
• Always a good practice to compare the levels of
  the factor using graphical methods such as
  boxplots.
• Comparative boxplots show the variability of the
  observations within a factor level and the
  variability between factor levels.

95
3-5. What If We Have More Than Two
Populations?

• Figure 3-14 (a)

96
3-5. What If We Have More Than Two
Populations?

• The observations yij can be modeled by
• a number of factor levels
• n number of replicates of observations
• per treatment (factor) level.

97
3-5. What If We Have More Than Two
Populations?

• The hypotheses being tested are
• H0
• H1 for at least one i
• Total variability can be measured by the total
  corrected sum of squares

98
3-5. What If We Have More Than Two
Populations?

• The sum of squares identity is
• Notationally, this is often written as
• SST SSTreatments SSE

99
3-5. What If We Have More Than Two
Populations?

• The expected value of the treatment sum of
  squares is
• If the null hypothesis is true, then

100
3-5. What If We Have More Than Two
Populations?

• The error mean square
• If the null hypothesis is true, the ratio
• has an F-distribution with a 1 and a(n 1)
  degrees of freedom.

101
3-5. What If We Have More Than Two
Populations?

• The following formulas can be used to calculate
  the sums of squares.
• Total Sum of Squares (SST)
• Sum of Squares for the Treatments (SSTreatment)
• Sum of Squares for error (SSE)
• SSE SST -SSTreatment

102
3-5. What If We Have More Than Two
Populations?

• Analysis of Variance Table 3-7

103
Example 3-13

• Four different hardwood concentrations
• Trts
• Six observations of each
• Completely randomized
• Hypotheses
• H0 t1 t2 t3 t4 0
• H1 ti 0 for at least one i

104
Example 3-13

• SST 7282 202 (383)2/24 512.96
• SStrts 60294210221272/6 (383)2/24
• 382.79
• SSE SST SStrts 512.96 382.79 130.17

105
Example 3-13, cont.
106
Example 3-13 cont.

• Since F.05,3,20 3.10, reject H0
• (H0 No difference in concentrations)
• Since F.01,3,20 4.94, reject H0
• (H0 No difference in concentrations)

107
3-5. What If We Have More Than Two
Populations?

• Analysis of Variance Table 3-8

108
3-5. What If We Have More Than Two
Populations?

• Residual Analysis
• Assumptions model errors are normally and
  independently distributed with equal variance.
• Check the assumptions by looking at residual
  plots.

109
3-5. What If We Have More Than Two
Populations?

• Residual Analysis
• Residual is given by eij yij ybari.
• ybar1. 10
• e11 7 10 3, and so on

110
3-5. What If We Have More Than Two
Populations?

• Residual Analysis
• Plot of residuals versus factor levels

111
3-5. What If We Have More Than Two
Populations?

• Residual Analysis
• Normal probability plot of residuals

112
Exercises

• Work as many odd-numbered exercises as necessary
  to make sure that you understand this chapter

113
End