Introduction to Statistical Quality Control, 6 th Edition by Douglas C. Montgomery. Copyright (c) 2009 John Wiley & Sons, Inc.

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3 Learning Objectives Chapter 4 3

4 4.1 Statistics and Sampling Distributions Statistical inference is concerned with drawing conclusions about populations (or processes) based on sample data from that system Random sample a sample that is selected so that the observations are independent a random sample has the property that it has the same probability of selection as any other sample. Statistic any function of the observations in a sample that doesn't contain unknown parameters The sample mean, the sample variance, and the sample standard deviation are all statistics Chapter 4 4

5 Observations in a sample are used to draw conclusions about the population Chapter 4 5

6 Sampling Distributions A statistic is a random variable, because a different sample with produce a different observed value of the statistic Every statistic has a probability distribution The probability bili distribution ib i of a statistic i is called a sampling distribution Chapter 4 6

7 Sampling from a Normal Distribution Chapter 4 7

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12 Sampling from a Bernoulli Distribution Chapter 4 12

13 Sampling from a Poisson Distribution Chapter 4 13

14 4.2 Point Estimation of Process Parameters Distributions are described by their parameters Parameters are generally unknown and must be estimated Point estimator a statistic that a single numerical value that is the estimate of the parameter Examples, page 110 & 111 Chapter 4 14

15 Properties of Point Estimators Chapter 4 15

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18 4.3 Statistical Inference for a Single Sample Statistical inference = decision making Hypothesis testing Null llhypothesis, H 0 Alternative hypothesis, H 1 Confidence intervals These two techniques are closely related Chapter 4 18

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21 4.3.1 Inference on the Mean of a Population, Variance Known Chapter 4 21

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23 Minitab output Chapter 4 23

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26 Upper and lower confidence bounds Chapter 4 26

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28 432P-Values P Chapter 4 28

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30 4.3.3 Inference on the Mean of a Normal Distribution, Variance Unknown Chapter 4 30

31 Checking the normality assumption we will see how to do this in Example 4.3 Chapter 4 31

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36 4.3.4 Inference on the Variance of a Normal Distribution Chapter 4 36

37 Upper and lower confidence bounds Chapter 4 37

38 4.3.5 Inference on a Population Proportion Chapter 4 38

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42 4.3.6 Type II Error and Sample Size Chapter 4 42

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45 Minitab can perform sample size and power calculations. From Example 4.7: From Example 4.3: Chapter 4 45

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47 4.4 Statistical Inference for Two Samples Comparing means, comparing variances, comparing proportions Chapter 4 47

48 4.4.1 Inference on the Difference in Means, Variances Known Chapter 4 48

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51 This is a two-sided CI. The one-sided confidence bounds would be found by using only one of the limits in Equation (4.49) with α/2 replaced by α. Chapter 4 51

52 4.4.2 Inference on the Difference in Means of Two Normal Distributions, ib i Variances Unknown Chapter 4 52

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57 Confidence Intervals Case 1: Case 2: Chapter 4 57

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61 Paired Data: Chapter 4 61

62 4.4.3 Inference on the Variances of Two Normal Distributions Chapter 4 62

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64 The two-sided CI is: Chapter 4 64

65 4.4.4 Inference on Two Proportions Chapter 4 65

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68 4.5 What if There Are More Than Two Populations? The Analysis of Variance Example: Does changing the hardwood concentration in the pulp affect the mean tensile strength of paper? Chapter 4 68

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70 The Analysis of Variance (ANOVA) Chapter 4 70

71 The ANOVA is based on the following partitioning of the total sum of squares (which measures the total variability in the sample data): Chapter 4 71

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75 The ANOVA test statistic is: If F 0 is greater than the critical value F α,a-1, a(n-1) then the null hypothesis of equal treatment means is rejected. A P-value approach can also be used. The P-value would be the probability above F 0 in the F a-1, a(n-1) ) distribution. Chapter 4 75

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78 Graphical comparison of individual means 20% hardwood produces higher mean strength than the others; 5% hardwood produces lower strength; 10% and 15% hardwood don t differ but give lower strength than 20%. Chapter 4 78

79 4.5.3 Checking Assumptions: Residual Analysis e y yˆ y y ij ij ij ij i. Chapter 4 79

80 Residual Plots: Chapter 4 80

81 4.6 Linear Regression Models Chapter 4 81

82 Models that are not linear in the regressors can still be fit using linear regression techniques, so long as they are linear in the parameters. Important cases include models with interaction terms and polynomials. l Chapter 4 82

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84 The method of least squares: Least squares normal equations Chapter 4 84

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92 The test procedure is to calculate the test statistic Chapter 4 92

93 The regression sum of squares is Chapter 4 93

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95 See Table 4.12 (slide 91) for the t- tests on the individual regressors in the consumer finance model both variables are significant Chapter 4 95

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106 Other Diagnostic Tools Standardized and Studentized residuals R-student an outlier diagnostic The PRESS statistic i R 2 for prediction based on PRESS a measure of how well the model will predict new data Measure of leverage hat diagonals Cook s distance a measure of influence Chapter 4 106

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108 Learning Objectives Chapter 4 108