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Chapter Eight Hypothesis Testing 8-2 of 23 McGraw-Hill/Irwin Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved. 8-2 of 23

Hypothesis Testing 8.1 Null and Alternative Hypotheses and Errors in Testing 8.2 Large Sample Tests about a Mean: Testing a One-Sided Alternative Hypothesis 8.3 Large Sample Tests about a Mean: Testing a Two-Sided Alternative Hypothesis 8.4 Small Sample Tests about a Population Mean 8.5 Hypothesis Tests about a Population Proportion 8.6* Type II Error Probabilities and Sample Size Determination 8.7* The Chi-Square Distribution 8.8* Statistical Inference for a Population Variance 8-3 of 23

8.1 Null and Alternative Hypotheses The null hypothesis, denoted H0, is a statement of the basic proposition being tested. The statement generally represents the status quo and is not rejected unless there is convincing sample evidence that it is false. The alternative or research hypothesis, denoted Ha, is an alternative (to the null hypothesis) statement that will be accepted only if there is convincing sample evidence that it is true. 8-4 of 23

Types of Hypothesis One-Sided, Greater Than H0:   50 Ha:  > 50 (Trash Bag) One-Sided, Less Than H0:   19.5 Ha:  < 19.5 (Accounts Receivable) Two-Sided, Not Equal To H0:  = 4.5 Ha:   4.5 (Camshaft) 8-5 of 23

Type I and Type II Errors Type I Error Rejecting H0 when it is true Type II Error Failing to reject H0 when it is false 8-6 of 23

8. 2 Large Sample Tests about a Mean: 8.2 Large Sample Tests about a Mean: Testing a One-Sided Alternative Hypothesis If the sampled population is normal or if n is large, we can reject H0:  = 0 at the  level of significance (probability of Type I error equal to ) if and only if the appropriate rejection point condition holds. Test Statistic Alternative Reject H0 if: : z > z : m < > a H a z < - z a If  unknown and n is large, estimate  by s. 8-7 of 23

Example: One-Sided, Greater Than Testing H0:   50 versus Trash Bag Ha:  > 50 for  = 0.05 and  = 0.01 8-8 of 23

Example: The p-Value for “Greater Than” Testing H0:   50 vs Ha:  > 50 using rejection points and p-value. Trash Bag The p-value or the observed level of significance is the probability of observing a value of the test statistic greater than or equal to z when H0 is true. It measures the weight of the evidence against the null hypothesis and is also the smallest value of  for which we can reject H0. 8-9 of 23

Example: One-Sided, Less Than Testing H0:   19.5 versus Accts Rec Ha:  < 19.5 for  = 0.01 8-10 of 23

Large Sample Tests about Mean: p-Values If the sampled population is normal or if n is large, we can reject H0:  = 0 at the  level of significance (probability of Type I error equal to ) if and only if the appropriate rejection point condition holds or, equivalently, if the corresponding p-value is less than . Alternative Reject H0 if: p-Value Test Statistic If  unknown and n is large, estimate  by s. 8-11 of 23

Five Steps of Hypothesis Testing Determine null and alternative hypotheses Specify level of significance (probability of Type I error)  Select the test statistic that will be used. Collect the sample data and compute the value of the test statistic. Use the value of the test statistic to make a decision using a rejection point or a p-value. Interpret statistical result in (real-world) managerial terms 8-12 of 23

Example: Two-Sided, Not Equal to Testing H0:  = 4.5 versus Camshaft Ha:   4.5 for  = 0.05 8-13 of 23

8.5 Small Sample Tests about a Population Mean If the sampled population is normal, we can reject H0:  = 0 at the  level of significance (probability of Type I error equal to ) if and only if the appropriate rejection point condition holds or, equivalently, if the corresponding p-value is less than . Alternative Reject H0 if: p-Value Test Statistic t, t/2 and p-values are based on n – 1 degrees of freedom. 8-14 of 23

Example: Small Sample Test about a Mean Testing H0:   18.8 vs Ha:  < 18.8 using rejection points and p-value. Credit Card Interest Rates 8-15 of 23

8.5 Hypothesis Tests about a Population Proportion If the sample size n is large, we can reject H0: p = p0 at the  level of significance (probability of Type I error equal to ) if and only if the appropriate rejection point condition holds or, equivalently, if the corresponding p-value is less than . Alternative Reject H0 if: p-Value Test Statistic 8-16 of 23

Example: Hypothesis Tests about a Proportion Testing H0: p  0.70 versus Ha: p > 0.70 using rejection points and p-value. Using Phantol, proportion of patients with reduced severity and duration of viral infections. 8-17 of 23

*8.6 Type II Error Probabilities Testing H0:   vs Ha:  < 3 (Amount of Coffee in 3-Pound Can) , Probability of Type II Error, Given  = 2.995,  = 0.05. 8-18 of 23

How Type II Error Varies Against Alternatives Testing H0:   vs Ha:  < 3 (Amount of Coffee in 3-Pound Can) , Probability of Type II Error ( = 0.05) Given  = 2.995, Given  = 2.990, Given  = 2.985, 8-19 of 23

*8.7 The Chi-Square Distribution The chi-square distribution depends on the number of degrees of freedom. A chi-square point is the point under a chi-square distribution that gives right-hand tail area . 8-20 of 23

*8.8 Statistical Inference for Population Variance If s2 is the variance of a random sample of n measurements from a normal population with variance 2, then the sampling distribution of the statistic (n - 1) s2 / 2 is a chi-square distribution with (n – 1) degrees of freedom and 100(1-)% confidence interval for 2 All chi-square points are based on n – 1 degrees of freedom Test of H0: 2 = 20 Test Statistic Reject H0 in favor of 8-21 of 23

Summary: Selecting an Appropriate Test Statistic for a Test about a Population Mean 8-22 of 23

Hypothesis Testing Summary: 8.1 Null and Alternative Hypotheses and Errors in Testing 8.2 Large Sample Tests about a Mean: Testing a One-Sided Alternative Hypothesis 8.3 Large Sample Tests about a Mean: Testing a Two-Sided Alternative Hypothesis 8.4 Small Sample Tests about a Population Mean 8.5 Hypothesis Tests about a Population Proportion 8.6* Type II Error Probabilities and Sample Size Determination 8.7* The Chi-Square Distribution 8.8* Statistical Inference for a Population Variance Summary: 8-23 of 23