1. Inferences About Process Quality
Chapter 3
Inferences About Process Quality
Introduction to Statistical Quality Control, 4th Edition

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?
Introduction to Statistical Quality Control, 4th Edition

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.
Introduction to Statistical Quality Control, 4th Edition

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
Introduction to Statistical Quality Control, 4th Edition

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 1000 days?
Introduction to Statistical Quality Control, 4th Edition

6. 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
Introduction to Statistical Quality Control, 4th Edition

7. 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.
Introduction to Statistical Quality Control, 4th Edition

8. 3-1.1 Sampling from a Normal Distribution
Chi-square (2) Distribution for various degrees of freedom.
Introduction to Statistical Quality Control, 4th Edition

9. 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.
Introduction to Statistical Quality Control, 4th Edition

10. 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.
Introduction to Statistical Quality Control, 4th Edition

11. 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.
Introduction to Statistical Quality Control, 4th Edition

12. 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
Introduction to Statistical Quality Control, 4th Edition

13. 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
Introduction to Statistical Quality Control, 4th Edition

14. 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) The sample mean and sample variance, respectively.
Introduction to Statistical Quality Control, 4th Edition

15. 3-2. Point Estimation of Process Parameters
Two properties of good point estimators
The point estimator should be unbiased.
The point estimator should have minimum variance.
Introduction to Statistical Quality Control, 4th Edition

16. 3-3. Statistical Inference for a Single Sample
Two categories of statistical inference:
Parameter Estimation
Hypothesis Testing
Introduction to Statistical Quality Control, 4th Edition

17. 3-3. Statistical Inference for a Single Sample
A statistical hypothesis is a statement about the values of the parameters of a probability distribution.
Introduction to Statistical Quality Control, 4th Edition

18. 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 test statistic to the rejection region. Can the null hypothesis be rejected?
Introduction to Statistical Quality Control, 4th Edition

19. 3-3. Statistical Inference for a Single Sample
Example: An automobile manufacturer claims a particular automobile can average 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)
Introduction to Statistical Quality Control, 4th Edition

20. 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 What is your conclusion?
Introduction to Statistical Quality Control, 4th Edition

21. 3-3. Statistical Inference for a Single Sample
Choice of Critical Values
How are the critical values chosen?
Wouldn’t 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
Wouldn’t be easier to set the size of the rejection region, rather than setting the critical values for every problem?
Introduction to Statistical Quality Control, 4th Edition

22. 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
Introduction to Statistical Quality Control, 4th Edition

23. 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 producer’s risk.
Type II error - not rejecting the null hypothesis when it is false.
Pr(Type II error) = . Sometimes called the consumer’s risk.
Introduction to Statistical Quality Control, 4th Edition

24. 3-3. Statistical Inference for a Single Sample
An Engine Explodes
H0: An automobile engine explodes when started.
H1: An automobile engine does not explode when started.
Which error would you take action to avoid? Whose risk is higher, the producer’s or the consumer’s?
Introduction to Statistical Quality Control, 4th Edition

25. 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.
Introduction to Statistical Quality Control, 4th Edition

26. 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
Introduction to Statistical Quality Control, 4th Edition

27. 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 > 1.645, reject H0 and conclude that the lot mean pressure strength exceeds 175 psi.
Introduction to Statistical Quality Control, 4th Edition

28. 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
Introduction to Statistical Quality Control, 4th Edition

29. 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.
Introduction to Statistical Quality Control, 4th Edition

30. 3-3.1 Inference on the Mean of a Population, Variance Known
Example 3-2
Reconsider Example 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
Introduction to Statistical Quality Control, 4th Edition

31. 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.
Introduction to Statistical Quality Control, 4th Edition

32. 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.
Introduction to Statistical Quality Control, 4th Edition

33. 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) =
Thus, H0:  = 175 would be rejected at any level of significance   P =
Introduction to Statistical Quality Control, 4th Edition

34. 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
Introduction to Statistical Quality Control, 4th Edition

35. 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.
Introduction to Statistical Quality Control, 4th Edition

36. 3-3.3 Inference on the Mean of a Population, Variance Unknown
Computer Output
Introduction to Statistical Quality Control, 4th Edition

37. 3-3.4 Inference on the Variance of a Normal Distribution
Hypothesis Testing
Hypotheses: H0: H1:
Test Statistic:
Significance Level, 
Rejection Region:
Introduction to Statistical Quality Control, 4th Edition

38. 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.
Introduction to Statistical Quality Control, 4th Edition

39. 3-3.5 Inference on a Population Proportion
Hypothesis Testing
Hypotheses: H0: p = p0 H1: p  p0
Test Statistic:
Significance Level, 
Rejection Region:
Introduction to Statistical Quality Control, 4th Edition

40. 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.
Introduction to Statistical Quality Control, 4th Edition

41. 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 - 
Introduction to Statistical Quality Control, 4th Edition

42. 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.
Introduction to Statistical Quality Control, 4th Edition

43. 3-3.6 The Probability of Type II Error
Operating Characteristic (OC) Curves
Introduction to Statistical Quality Control, 4th Edition

44. 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.
Introduction to Statistical Quality Control, 4th Edition

45. 3-3.7 Probability Plotting
Example 3-8
Introduction to Statistical Quality Control, 4th Edition

46. 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,
Introduction to Statistical Quality Control, 4th Edition

47. 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
Introduction to Statistical Quality Control, 4th Edition

48. 3-4.1 Inference For a Difference in Means, Variances Known
Point estimator for is
where
Introduction to Statistical Quality Control, 4th Edition

49. 3-4.1 Inference For a Difference in Means, Variances Known
Hypothesis Tests for a Difference in Means, Variances Known
Null Hypothesis:
Test Statistic:
Introduction to Statistical Quality Control, 4th Edition

50. 3-4.1 Inference For a Difference in Means, Variances Known
Hypothesis Tests for a Difference in Means, Variances Known
Alternative Hypotheses Rejection Criterion
Introduction to Statistical Quality Control, 4th Edition

51. 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
Introduction to Statistical Quality Control, 4th Edition

52. 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
Introduction to Statistical Quality Control, 4th Edition

53. 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
Introduction to Statistical Quality Control, 4th Edition

54. 3-4.2 Inference For a Difference in Means, Variances Unknown
Hypothesis Tests for a Difference in Means,
Case I:
Null Hypothesis:
Test Statistic:
Introduction to Statistical Quality Control, 4th Edition

55. 3-4.2 Inference For a Difference in Means, Variances Unknown
Hypothesis Tests for a Difference in Means, Variances Unknown
Alternative Hypotheses Rejection Criterion
Introduction to Statistical Quality Control, 4th Edition

56. 3-4.2 Inference For a Difference in Means, Variances Unknown
Hypothesis Tests for a Difference in Means,
Case II:
Null Hypothesis:
Test Statistic:
Introduction to Statistical Quality Control, 4th Edition

57. 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
Introduction to Statistical Quality Control, 4th Edition

58. 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
Introduction to Statistical Quality Control, 4th Edition

59. 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
Introduction to Statistical Quality Control, 4th Edition

60. Introduction to Statistical Quality Control, 4th Edition
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.
Introduction to Statistical Quality Control, 4th Edition

61. Introduction to Statistical Quality Control, 4th Edition
3-4.2 Paired Data
The differences, dj, represent the “new” set of data with the summary statistics:
Introduction to Statistical Quality Control, 4th Edition

62. Introduction to Statistical Quality Control, 4th Edition
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
Introduction to Statistical Quality Control, 4th Edition

63. 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
Introduction to Statistical Quality Control, 4th Edition

64. 3-4.3 Inferences on the Variances of Two Normal Distributions
Hypothesis Testing
Hypotheses:
Test Statistic:
Significance Level, 
Rejection Region:
Introduction to Statistical Quality Control, 4th Edition

65. 3-4.3 Inferences on the Variances of Two Normal Distributions
Alternative Test Rejection
Hypothesis Statistic Region
Introduction to Statistical Quality Control, 4th Edition

66. 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
Introduction to Statistical Quality Control, 4th Edition

67. 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:
Introduction to Statistical Quality Control, 4th Edition

68. 3-4.4 Inference on Two Population Proportions
Alternative Hypothesis Rejection Region
Introduction to Statistical Quality Control, 4th Edition

69. 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.
Introduction to Statistical Quality Control, 4th Edition

70. 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 Totals Avg 5%
Overall
Introduction to Statistical Quality Control, 4th Edition

71. 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.
Introduction to Statistical Quality Control, 4th Edition

72. 3-5. What If We Have More Than Two Populations?
Figure 3-14 (a)
Introduction to Statistical Quality Control, 4th Edition

73. 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.)
Introduction to Statistical Quality Control, 4th Edition

74. 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”:
Introduction to Statistical Quality Control, 4th Edition

75. 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
Introduction to Statistical Quality Control, 4th Edition

76. 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
Introduction to Statistical Quality Control, 4th Edition

77. 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.
Introduction to Statistical Quality Control, 4th Edition

78. 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
Introduction to Statistical Quality Control, 4th Edition

79. 3-5. What If We Have More Than Two Populations?
Analysis of Variance Table 3-7
Introduction to Statistical Quality Control, 4th Edition

80. 3-5. What If We Have More Than Two Populations?
Analysis of Variance Table 3-8
Introduction to Statistical Quality Control, 4th Edition

81. 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.
Introduction to Statistical Quality Control, 4th Edition

82. 3-5. What If We Have More Than Two Populations?
Residual Analysis
Plot of residuals versus factor levels
Introduction to Statistical Quality Control, 4th Edition

83. 3-5. What If We Have More Than Two Populations?
Residual Analysis
Normal probability plot of residuals
Introduction to Statistical Quality Control, 4th Edition