L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 8 1 MER301: Engineering Reliability LECTURE 8: Chapter 4: Statistical Inference, Point Estimation and Hypothesis Testing

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 8 2 Summary  Statistical Inference Estimates of Population Parameters  Point Estimation  Standard error Hypothesis Testing

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 8 3 Statistical Inference  Consists of methods used to make decisions draw conclusions about populations  Methods utilize information from Samples drawn from Populations  Two major areas parameter estimation hypothesis testing

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 8 4 Comment on Engineering Use  Statistical Inference has long been used to establish manufacturing process capability and to monitor manufacturing lines as part of quality programs  More recently, it has become part of the Statistical Design Methods toolkit to take account of variation in the design phase

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 8 5 Relationship Between Population and Sample  Parameters of populations are what we want to know  Estimates from sample data are what we get  Need to estimate parameters from the sample data

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 8 6 Point Estimate  An ESTIMATOR is the statistic (Random Variable) used to generate the estimate of a parameter  An ESTIMATE is a number derived from Sample Data “The Objective of Point Estimation is to Select a Single Number, Based on Sample Data, that is the Most Plausible Value for …..”

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 8 7 Point Estimators for a Population  Let X be the Random Variable of a Population Population Mean: Population Variance:  Estimators for Population Mean and Variance for a Random Sample of Size n

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 8 8 Parameters, Statistics, and Point Estimates

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 8 9 Example 8.1: Parameters, Statistics, and Point Estimates  In estimating the mean number of goals scored in a Division 1 (M’s) hockey game for a given year, the statistic used is a sample mean  is a POINT ESTIMATOR for the mean goals/game  The 59 Division 1 Men’s teams play about a thousand games per year and scores from a sample of 34 are as shown on the following page  So for this particular sample the value of 3.23 goals per per game could be taken as a POINT ESTIMATE

L Berkley Davis Copyright 2009 Example 8.1: Parameters, Statistics, and Point Estimates(continued) 10

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 8 11 How Good is the Estimate  Estimators should generate estimates that are close in value to θ  This is expected if the Estimator has the properties is UNBIASED for θ has small variance

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 8 12 Unbiased Estimator Definition

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 8 13 Unbiased Estimators for Population Parameters  Let X be the Random Variable of a Population Population Mean: Population Variance:  Unbiased Estimators of the Population Mean and Variance for a Random Sample of Size n

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 8 14 How Good is the Estimate  Estimators should generate estimates that are close in value to θ  This is expected if the Estimator has the properties is UNBIASED for θ has small variance -Standard Error of the Mean -How Samples are Drawn

L Berkley Davis Copyright 2009 Central Limit Theorem and the Standard Error of the Mean… Standard Error of the Mean: Distribution of 15

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 8 16 Standard Error

L Berkley Davis Copyright 2009 Example 8.1(continued) 17

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 8 18 How Good is the Estimate  Estimators should generate estimates that are close in value to θ  This is expected if the Estimator has the properties is UNBIASED for θ has small variance -Standard Error of the Mean -How Samples are Drawn

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 8 19 Example 7.3: Throwing Dice..

L Berkley Davis Copyright 2009 Example 7.3: Throwing Dice.. 20

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 8 21 Example 7.3: Throwing Dice..

L Berkley Davis Copyright 2009 Example 7.3: Throwing Dice.. 22

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 8 23 Relationship Between Population and Sample  Parameters of populations are what we want to know  Estimates from samples are what we get  Need to estimate parameters from the samples

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L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 8 25 Hypothesis Testing  What is it? Methods for making conclusions about the values of population parameters by using sample data and statistical techniques  Why are Engineers interested? For manufacturing processes, lines are sampled and the results assessed using hypothesis testing to establish whether the processes are producing product with the required CTQ’s In statistical design, hypothesis testing is used to compare the performance of two different designs to see whether there are real differences

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 8 26 Hypothesis Testing  The hypothesis consists of an either/or statement about parameters of the populations,for example this is a two sided hypothesis Null Hypothesis Alternative Hypothesis

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 8 27 Hypotheses for Two Sided and One Sided Hypothesis Tests  Two –Sided Test will detect the differences on either tail of the distribution  Right-tailed test Upper tail region  H 0 : µ=µ 0  H 1 : µ>µ 0  Left-tailed test Lower tail region  H 0 : µ=µ 0  H 1 : µ<µ 0

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 8 28 Hypothesis Testing Hypotheses are always statements about the population, not the sample from the population The sample is being used to estimate parameters of the population Values of population parameters are obtained from previous knowledge, from theoretical calculations, or are customer requirements Testing procedures use samples drawn from the population to make statistical predictions as to whether the hypothesis is true.

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 8 29 Hypothesis Testing  Data from the Sample will be used to calculate a Test Statistic. This may be of the form  A Critical Region for the test is established based on the purpose for the test. For example, power plants performance tests have a tolerance band- outside of that band is the critical region

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L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 8 31 Hypothesis Testing Error (4-7)

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 8 32 Summary of Hypothesis Testing  Comments on Hypothesis Testing Null Hypothesis is what is tested Rejection of the Null Hypothesis always leads to accepting the Alternative Hypothesis Test Statistic is computed from Sample data Critical region is the range of values for the test statistic where we reject the Null Hypothesis in favor of the Alternative Hypothesis Rejecting when it is true is a Type I error Failing to reject when it is false is a Type II error

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 8 33 Example 8.2 :Hypothesis Testing(1) Suppose we are interested in the burning rate of a solid propellant used to power aircrew escape systems. The burning rate of a solid propellant during manufacture is a random variable that,over the long term with a controlled manufacturing process, is characterized by population parameters mean and standard deviation Our interest focuses on the mean burning rate (a parameter of this population), a critical performance CTQ Specifically, we are interested in deciding whether or not the mean burning rate of current production is kept stable at the population mean of 50 centimeters per second.

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 8 34 Example 8.2 :Hypothesis Testing(2)  The mean burning rate of each production lot needs to be maintained within a narrow range of the population mean for the system to function properly. The manufacturing process can be designed to maintain the mean burning range for each lot within engineering tolerances  The Hypotheses are given by

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 8 35 Example 8.2 :Hypothesis Testing(3) Critical/Acceptance Regions

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 8 36 Example 8.2 :Hypothesis Testing(4)  The mean and standard deviation of the population are to be 50cm/sec and 2.5cm/sec, respectively.  A sample of ten(10) specimens from the propellant is tested and the mean sample burning rate calculated (there will be Gage R&R considerations, ie how well can the burning rate be measured- with what accuracy and precision?)

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 8 37 Example 8.2 :Hypothesis Testing(5)  The Z values corresponding to the critical values are  The probability of a Type I error is given by  The probability of a Type II error calculation is done for a specific alternative mean burning rate. The fraction of a distribution centered about that mean that is within the acceptable region for the hypothesis is the Type II error

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 8 38 Example 8.2 :Hypothesis Testing(6) Type I Error  The Type I error is the probability that a sample will be rejected when in fact the true mean of the product being sampled is really within the acceptance region or

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 8 39 Example 8.2 :Hypothesis Testing(7) Type II Error  The probability of a Type II error is shown by the shaded region

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 8 40 Summary of Errors  The probability of making a Type I error- rejecting a good sample or false negative- is mostly under the direct control of the experimenter. There is a direct effect of both sample size and the size of the acceptable tolerance band  The probability of making a Type II error-accepting a bad sample or a false positive-is affected by the same variables but is also strongly influenced by the exact values of the test variables

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 8 41 Sensitivity of a Statistical Test  This is 1-P(Type II error) and is descriptive of the sensitivity of a statistical test, ie, of the ability of the test to detect differences. The planning of the test- how samples will be drawn, the number of samples, establishment of the critical regions- is one factor that influences Power

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 8 42 Hypothesis Testing Procedure

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 8 43 Summary  Statistical Inference Estimates of Population Parameters  Point Estimation  Standard error Hypothesis Testing