 # Chapter 9 Title and Outline 1 9 Tests of Hypotheses for a Single Sample 9-1 Hypothesis Testing 9-1.1 Statistical Hypotheses 9-1.2 Tests of Statistical.

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Chapter 9 Title and Outline 1 9 Tests of Hypotheses for a Single Sample 9-1 Hypothesis Testing 9-1.1 Statistical Hypotheses 9-1.2 Tests of Statistical Hypotheses 9-1.3 1-Sided & 2-Sided Hypotheses 9-1.4 P-Values in Hypothesis Tests 9-1.5 Connection between Hypothesis Tests & Confidence Intervals 9-1.6 General Procedures for Hypothesis Tests 9-2 Tests on the Mean of a Normal Distribution, Variance Known 9-2.1 Hypothesis Tests on the Mean 9-2.2 Type II Error & Choice of Sample Size 9-2.3 Large-Sample Test 9-3 Tests on the Mean of a Normal Distribution, Variance Unknown 9-3.1 Hypothesis Tests on the Mean 9-3.2 Type II Error & Choice of Sample Size 9-4 Tests of the Variance & Standard Deviation of a Normal Distribution. 9-4.1 Hypothesis Test on the Variance 9-4.2 Type II Error & Choice of Sample Size 9-5 Tests on a Population Proportion 9-5.1 Large-Sample Test on a Proportion 9-5.2 Type II Error & Choice of Sample Size 9-6 Summary Table of Inference Procedures for a Single Sample 9-7 Testing for Goodness of Fit 9-8 Contingency Table Tests 9-9 Non-Parametric Procedures 9-9.1 Sign Test 9-9.2 Wilcoxen Signed-Rank Test 9-9.3 Comparison to the t-test CHAPTER OUTLINE

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Learning Objectives for Chapter 9 After careful study of this chapter, you should be able to do the following: 1.Structure engineering decision-making as hypothesis tests. 2.Test hypotheses on the mean of a normal distribution using either a Z-test or a t-test procedure. 3.Test hypotheses on the variance or standard deviation of a normal distribution. 4.Test hypotheses on a population proportion. 5.Use the P-value approach for making decisions in hypothesis tests. 6.Compute power & Type II error probability. Make sample size selection decisions for tests on means, variances & proportions. 7.Explain & use the relationship between confidence intervals & hypothesis tests. 8.Use the chi-square goodness-of-fit test to check distributional assumptions. 9.Use contingency table tests. 2Chapter 9 Learning Objectives

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-1 Hypothesis Testing 9-1.1 Statistical Hypotheses Definition Statistical hypothesis testing and confidence interval estimation of parameters are the fundamental methods used at the data analysis stage of a comparative experiment, in which the engineer is interested, for example, in comparing the mean of a population to a specified value.

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-1 Hypothesis Testing 9-1.1 Statistical Hypotheses For example, suppose that we are interested in the burning rate of a solid propellant used to power aircrew escape systems. Now burning rate is a random variable that can be described by a probability distribution. Suppose that our interest focuses on the mean burning rate (a parameter of this distribution). Specifically, we are interested in deciding whether or not the mean burning rate is 50 centimeters per second.

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-1 Hypothesis Testing 9-1.1 Statistical Hypotheses null hypothesis alternative hypothesis One-sided Alternative Hypotheses Two-sided Alternative Hypothesis

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-1 Hypothesis Testing 9-1.1 Statistical Hypotheses Test of a Hypothesis A procedure leading to a decision about a particular hypothesis Hypothesis-testing procedures rely on using the information in a random sample from the population of interest. If this information is consistent with the hypothesis, then we will conclude that the hypothesis is true; if this information is inconsistent with the hypothesis, we will conclude that the hypothesis is false.

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-1 Hypothesis Testing 9-1.2 Tests of Statistical Hypotheses Figure 9-1 Decision criteria for testing H 0 :  = 50 centimeters per second versus H 1 :   50 centimeters per second.

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-1 Hypothesis Testing 9-1.2 Tests of Statistical Hypotheses Definitions

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-1 Hypothesis Testing 9-1.2 Tests of Statistical Hypotheses Sometimes the type I error probability is called the significance level, or the  -error, or the size of the test.

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-1 Hypothesis Testing 9-1.2 Tests of Statistical Hypotheses

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-1 Hypothesis Testing

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-1 Hypothesis Testing Figure 9-3 The probability of type II error when  = 52 and n = 10.

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-1 Hypothesis Testing

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-1 Hypothesis Testing Figure 9-4 The probability of type II error when  = 50.5 and n = 10.

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-1 Hypothesis Testing

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-1 Hypothesis Testing Figure 9-5 The probability of type II error when  = 2 and n = 16.

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-1 Hypothesis Testing

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-1 Hypothesis Testing

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-1 Hypothesis Testing Definition The power is computed as 1 - , and power can be interpreted as the probability of correctly rejecting a false null hypothesis. We often compare statistical tests by comparing their power properties. For example, consider the propellant burning rate problem when we are testing H 0 :  = 50 centimeters per second against H 1 :  not equal 50 centimeters per second. Suppose that the true value of the mean is  = 52. When n = 10, we found that  = 0.2643, so the power of this test is 1 -  = 1 - 0.2643 = 0.7357 when  = 52.

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-1 Hypothesis Testing 9-1.3 One-Sided and Two-Sided Hypotheses Two-Sided Test: One-Sided Tests:

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-1 Hypothesis Testing Example 9-1

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-1 Hypothesis Testing The bottler wants to be sure that the bottles meet the specification on mean internal pressure or bursting strength, which for 10-ounce bottles is a minimum strength of 200 psi. The bottler has decided to formulate the decision procedure for a specific lot of bottles as a hypothesis testing problem. There are two possible formulations for this problem, either or

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-1 Hypothesis Testing 9-1.4 P-Values in Hypothesis Tests Definition

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-1 Hypothesis Testing 9-1.4 P-Values in Hypothesis Tests

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-1 Hypothesis Testing 9-1.4 P-Values in Hypothesis Tests

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-1 Hypothesis Testing 9-1.5 Connection between Hypothesis Tests and Confidence Intervals

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-1 Hypothesis Testing 9-1.6 General Procedure for Hypothesis Tests 1. From the problem context, identify the parameter of interest. 2. State the null hypothesis, H 0. 3. Specify an appropriate alternative hypothesis, H 1. 4. Choose a significance level, . 5. Determine an appropriate test statistic. 6. State the rejection region for the statistic. 7. Compute any necessary sample quantities, substitute these into the equation for the test statistic, and compute that value. 8. Decide whether or not H 0 should be rejected and report that in the problem context.

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-2 Tests on the Mean of a Normal Distribution, Variance Known 9-2.1 Hypothesis Tests on the Mean We wish to test : The test statistic is :

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-2 Tests on the Mean of a Normal Distribution, Variance Known 9-2.1 Hypothesis Tests on the Mean Reject H 0 if the observed value of the test statistic z 0 is either : z 0 > z  /2 or z 0 < -z  /2 Fail to reject H 0 if -z  /2 < z 0 < z  /2

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-2 Tests on the Mean of a Normal Distribution, Variance Known

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-2 Tests on the Mean of a Normal Distribution, Variance Known Example 9-2

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-2 Tests on the Mean of a Normal Distribution, Variance Known Example 9-2

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-2 Tests on the Mean of a Normal Distribution, Variance Known Example 9-2

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-2 Tests on the Mean of a Normal Distribution, Variance Known 9-2.1 Hypothesis Tests on the Mean (Eq. 9-10 & 11)

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-2 Tests on the Mean of a Normal Distribution, Variance Known 9-2.1 Hypothesis Tests on the Mean (Continued) (Eq. 9-12 & 18)

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-2 Tests on the Mean of a Normal Distribution, Variance Known 9-2.1 Hypothesis Tests on the Mean (Continued)

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-2 Tests on the Mean of a Normal Distribution, Variance Known P-Values in Hypothesis Tests

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-2 Tests on the Mean of a Normal Distribution, Variance Known 9-2.2 Type II Error and Choice of Sample Size Finding the Probability of Type II Error  (Eq. 9-19)

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-2 Tests on the Mean of a Normal Distribution, Variance Known 9-2.2 Type II Error and Choice of Sample Size Finding the Probability of Type II Error  (Eq. 9-20)

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-2 Tests on the Mean of a Normal Distribution, Variance Known 9-2.2 Type II Error and Choice of Sample Size Finding the Probability of Type II Error  (Figure 9-9) Figure 9-9 The distribution of Z 0 under H 0 and H 1

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-2 Tests on the Mean of a Normal Distribution, Variance Known 9-2.2 Type II Error and Choice of Sample Size Sample Size Formulas For a two-sided alternative hypothesis: (Eq. 9-22)

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-2 Tests on the Mean of a Normal Distribution, Variance Known 9-2.2 Type II Error and Choice of Sample Size Sample Size Formulas For a one-sided alternative hypothesis: ( Eq. 9-23)

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-2 Tests on the Mean of a Normal Distribution, Variance Known Example 9-3

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-2 Tests on the Mean of a Normal Distribution, Variance Known Example 9-3

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-2 Tests on the Mean of a Normal Distribution, Variance Known 9-2.2 Type II Error and Choice of Sample Size Using Operating Characteristic Curves (Eq. 9-24)

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-2 Tests on the Mean of a Normal Distribution, Variance Known 9-2.2 Type II Error and Choice of Sample Size Using Operating Characteristic Curves

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-2 Tests on the Mean of a Normal Distribution, Variance Known Example 9-4

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-2 Tests on the Mean of a Normal Distribution, Variance Known 9-2.3 Large Sample Test

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-3 Tests on the Mean of a Normal Distribution, Variance Unknown 9-3.1 Hypothesis Tests on the Mean One-Sample t-Test

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-3 Tests on the Mean of a Normal Distribution, Variance Unknown 9-3.1 Hypothesis Tests on the Mean (Figures 9-10, 11, 12) Figure 9-10 The reference distribution for H 0 :  =  0 with critical region for (a) H 1 :    0, (b) H 1 :  >  0, and (c) H 1 :  <  0.

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-3 Tests on the Mean of a Normal Distribution, Variance Unknown Example 9-6

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-3 Tests on the Mean of a Normal Distribution, Variance Unknown Example 9-6

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-3 Tests on the Mean of a Normal Distribution, Variance Unknown Example 9-13 Figure 9-13 Normal probability plot of the coefficient of restitution data from Example 9-6.

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-3 Tests on the Mean of a Normal Distribution, Variance Unknown Example 9-6

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-3 Tests on the Mean of a Normal Distribution, Variance Unknown 9-3.2 P-value for a t-Test The P -value for a t -test is just the smallest level of significance at which the null hypothesis would be rejected. Notice that t 0 = 2.72 in Example 9-6, and that this is between two tabulated values, 2.624 and 2.977. Therefore, the P -value must be between 0.01 and 0.005. These are effectively the upper and lower bounds on the P -value.

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-3 Tests on the Mean of a Normal Distribution, Variance Unknown 9-3.3 Type II Error and Choice of Sample Size The type II error of the two-sided alternative (for example) would be

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-3 Tests on the Mean of a Normal Distribution, Variance Unknown Example 9-7

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-4 Hypothesis Tests on the Variance and Standard Deviation of a Normal Distribution 9-4.1 Hypothesis Test on the Variance (Eq. 9-34, 35)

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-4 Hypothesis Tests on the Variance and Standard Deviation of a Normal Distribution 9-4.1 Hypothesis Test on the Variance

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-4 Hypothesis Tests on the Variance and Standard Deviation of a Normal Distribution 9-4.1 Hypothesis Test on the Variance (Eq. 9-33, 34)

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-4 Hypothesis Tests on the Variance and Standard Deviation of a Normal Distribution 9-4.1 Hypothesis Test on the Variance (Figure 9-14)

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-4 Hypothesis Tests on the Variance and Standard Deviation of a Normal Distribution Example 9-8

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-4 Hypothesis Tests on the Variance and Standard Deviation of a Normal Distribution Example 9-8

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-4 Hypothesis Tests on the Variance and Standard Deviation of a Normal Distribution 9-4.2 Type II Error and Choice of Sample Size For the two-sided alternative hypothesis : Operating characteristic curves are provided in Charts VIi and VIj :

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-4 Hypothesis Tests on the Variance and Standard Deviation of a Normal Distribution Example 9-9

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-5 Tests on a Population Proportion 9-5.1 Large-Sample Tests on a Proportion (Eq. 9-40) Many engineering decision problems include hypothesis testing about p. An appropriate test statistic is

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-5 Tests on a Population Proportion Example 9-10

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-5 Tests on a Population Proportion Example 9-10

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-5 Tests on a Population Proportion Another form of the test statistic Z 0 is or

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-5 Tests on a Population Proportion 9-5.2 Type II Error and Choice of Sample Size For a two-sided alternative (Eq. 9-42) If the alternative is p < p 0 (Eq. 9-43) If the alternative is p > p 0 (Eq. 9-44)

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-5 Tests on a Population Proportion 9-5.3 Type II Error and Choice of Sample Size For a two-sided alternative (Eq. 9-45) For a one-sided alternative (Eq. 9-46)

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-5 Tests on a Population Proportion Example 9-11

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-5 Tests on a Population Proportion Example 9-11

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-7 Testing for Goodness of Fit The test is based on the chi-square distribution. Assume there is a sample of size n from a population whose probability distribution is unknown. Let O i be the observed frequency in the ith class interval. Let E i be the expected frequency in the ith class interval. The test statistic is(Eq.9-47)

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Example 9-12 9-7 Testing for Goodness of Fit

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-7 Testing for Goodness of Fit Example 9-12

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-7 Testing for Goodness of Fit Example 9-12

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-7 Testing for Goodness of Fit Example 9-12

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-7 Testing for Goodness of Fit Example 9-12

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-7 Testing for Goodness of Fit Example 9-12

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-8 Contingency Table Tests Many times, the n elements of a sample from a population may be classified according to two different criteria. It is then of interest to know whether the two methods of classification are statistically independent;

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-8 Contingency Table Tests (Eq. 9-48)

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-8 Contingency Table Tests (Eq.9-49) (Eq. 9-50)

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-8 Contingency Table Tests Example 9-14

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-8 Contingency Table Tests Example 9-14

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-8 Contingency Table Tests Example 9-14

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-8 Contingency Table Tests Example 9-14

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Important Terms & Concepts of Chapter 9 α and β Connection between hypothesis tests & confidence intervals Critical region for a test statistic Goodness-of-fit test Homogeneity test Inference Independence test Non-parametric or distribution-free methods Normal approximation to non- parametric tests Null distribution Null hypothesis 1 & 2-sided alternative hypotheses Operating Characteristic (OC) curves Power of a test P-value Ranks Reference distribution for a test statistic Sample size determination for hypothesis tests Significance level of a test Sign test Statistical hypotheses Statistical vs. practical significance Test statistic Type I & Type II errors Wilcoxen signed-rank test Chapter 9 Summary88

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