chapter 4 introduction to statistical quality control, 6th ...noordin/s/ch04 rev.pdftitle: microsoft...

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Learning Objectives

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4.1 Statistics and Sampling Distributions4.1 Statistics and Sampling Distributions

• Statistical inference is concerned with drawing gconclusions about populations (or processes) based on sample data from that system

• Random sample – a sample that is selected so that the• 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 sampleselection 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

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Observations in a sample are used to draw pconclusions about the population

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Sampling DistributionsSampling Distributions

• A statistic is a random variable because aA statistic is a random variable, because a different sample with produce a different observed value of the statisticobserved value of the statistic

• Every statistic has a probability distributionTh b bili di ib i f i i i• The probability distribution of a statistic is called a sampling distribution

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Sampling from a Normal Distribution

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Sampling from a Bernoulli Distribution

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Sampling from a Poisson Distribution

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4.2 Point Estimation of Process Parameters4.2 Point Estimation of Process Parameters

• Distributions are described by their parametersDistributions are described by their parameters• Parameters are generally unknown and must be

estimatedestimated• Point estimator – a statistic that a single

i l l h i h i f hnumerical value that is the estimate of the parameter

• Examples, page 110 & 111

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Properties of Point Estimators

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4.3 Statistical Inference for a Single Sample4.3 Statistical Inference for a Single Sample

• Statistical inference = decision makingStatistical inference decision making• Hypothesis testing

N ll h th i H– Null hypothesis, H0

– Alternative hypothesis, H1

• Confidence intervals• These two techniques are closely relatedq y

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4.3.1 Inference on the Mean of a Population, Variance Known

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Minitab output

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Upper and lower

confidence bounds

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4 3 2 P-Values4.3.2 P Values

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4.3.3 Inference on the Mean of a Normal Distribution, Variance UnknownDistribution, Variance Unknown

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Checking the normality assumption – we will see how to do this in Example 4.3

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4.3.4 Inference on the Variance of a Normal Distribution

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Upper and lower fidconfidence

bounds

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4.3.5 Inference on a Population Proportion

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4.3.6 Type II Error and Sample Size4.3.6 Type II Error and Sample Size

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Minitab can perform sample size and power calculations. From Example 4.7:

From Example 4.3:

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4.4 Statistical Inference for Two Samples4.4 Statistical Inference for Two Samples

Comparing means, comparing variances, comparing proportions

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4.4.1 Inference on the Difference in Means, Variances Known

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

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4.4.2 Inference on the Difference in Means of T N l Di ib i V i U kTwo Normal Distributions, Variances Unknown

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Confidence Intervals – Case 1:

Case 2:

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Paired Data:

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4.4.3 Inference on the Variances of Two Normal Distributions

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The two-sided CI is:

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4.4.4 Inference on Two Proportions

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4.5 What if There Are More Than Two Populations? Th A l i f V iThe Analysis of Variance

Example: Does changing the hardwood concentration in the pulp affectExample: Does changing the hardwood concentration in the pulp affect the mean tensile strength of paper?

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The Analysis of Variance (ANOVA)

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The ANOVA is based on the following partitioning of the total sum of squares (which measures the total variability in the sample data):q ( y p )

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The ANOVA test statistic is:

If F is greater than the critical value F then the nullIf F0 is greater than the critical value Fα,a-1, a(n-1) then the nullhypothesis of equal treatment means is rejected. A P-value approach can also be used. The P-value would be the probability above F0 in the F 1 ( 1) distribution

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above F0 in the Fa-1, a(n-1) distribution.

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Graphical comparison of individual means

20% hardwood produces higher mean strength than the p g gothers; 5% hardwood produces lower strength; 10% and 15% hardwood don’t differ but give lower strength than 20%.

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4.5.3 Checking Assumptions: Residual Analysis4.5.3 Checking Assumptions: Residual Analysis

.ˆij ij ij ij ie y y y y j j j j

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Resid al PlotsResidual Plots:

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4.6 Linear Regression Models

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Models that are not linear in the regressors can still be fit using linearModels 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 l i lpolynomials.

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The method of least squares:

Least squares

lnormal equations

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The test procedure is to calculate the test statistic

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The regression sum of squares is

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See Table 4.12 ( lid 91) f th t(slide 91) for the t-

tests on the individual

regressors in the ficonsumer finance

model – both variables are

significant

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Other Diagnostic ToolsOther Diagnostic Tools

• Standardized and Studentized residualsStandardized and Studentized residuals• R-student – an outlier diagnostic

h SS i i• The PRESS statistic• R2 for prediction based on PRESS – a measure

of how well the model will predict new data• Measure of leverage – hat diagonalsg g• Cook’s distance – a measure of influence

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Learning Objectivesg j

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chapter 4 introduction to statistical quality control, 6th ...noordin/s/ch04 rev.pdftitle: microsoft...

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