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Chapter 9 Hypothesis Testing McGraw-Hill/Irwin Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter Outline 9.1 The Null and Alternative Hypotheses and Errors in Testing 9.2 z Tests about a Population Mean: σ Known 9.3 z Tests about a Population Mean: σ Unknown 9.4 z Tests about a Population Proportion 9.5 Type II Error Probabilities and Sample Size Determination (Optional) 9.6 The Chi-Square Distribution (Optional) 9.7 Statistical Inference for a Population Variance (Optional) 9-2 9.1 Null and Alternative Hypotheses and Errors in Hypothesis Testing One-Sided, “Greater Than” Alternative H0: 0 vs. Ha: > 0 One-Sided, “Less Than” Alternative H0 : 0 vs. Ha : < 0 Two-Sided, “Not Equal To” Alternative H0 : = 0 vs. Ha : 0 where 0 is a given constant value (with the appropriate units) that is a comparative value 9-3 The Idea of a Test Statistic z x 0 x x 0 n 9-4 Type I and Type II Errors Table 9.1 9-5 Typical Values Low alpha gives small chance of rejecting a true H0 Typically, = 0.05 Strong evidence is required to reject H0 Usually choose α between 0.01 and 0.05 = 0.01 requires very strong evidence is to reject H0 Tradeoff between and β For fixed n, the lower , the higher β 9-6 9.2 z Tests about a Population Mean: σ Known Test hypotheses about a population mean using the normal distribution Called z tests Require that the true value of the population standard deviation σ is known In most real-world situations, σ is not known When σ is unknown, test hypotheses about a population mean using the t distribution Here, assume that we know σ 9-7 Steps in Testing a “Greater Than” Alternative 1. 2. 3. 4. 5. State the null and alternative hypotheses Specify the significance level a Select the test statistic Determine the critical value rule for rejecting H0 Collect the sample data and calculate the value of the test statistic 6. Decide whether to reject H0 by using the test statistic and the critical value rule 7. Interpret the statistical results in managerial terms and assess their practical importance 9-8 Steps in Testing a “Greater Than” Alternative in Trash Bag Case #1 1. State the null and alternative hypotheses H0: 50 Ha: > 50 2. Specify the significance level α = 0.05 3. Select the test statistic z x 50 x x 50 n 9-9 Steps in Testing a “Greater Than” Alternative in Trash Bag Case #2 4. Determine the critical value rule for deciding whether or not to reject H0 Reject H0 in favor of Ha if the test statistic z is greater than the rejection point zα This is the critical value rule In the trash bag case, the critical value rule is to reject H0 if the calculated test statistic z is > 1.645 9-10 Steps in Testing a “Greater Than” Alternative in Trash Bag Case #3 z Figure 9.1 x 50 n 50 . 575 50 1 . 65 2 . 20 40 9-11 Steps in Testing a “Greater Than” Alternative in Trash Bag Case #4 6. Decide whether to reject H0 by using the test statistic and the rejection rule Compare the value of the test statistic to the critical value according to the critical value rule In the trash bag case, z = 2.20 is greater than z0.05 = 1.645 Therefore reject H0: μ ≤ 50 in favor of Ha: μ > 50 at the 0.05 significance level 7. Interpret the statistical results in managerial terms and assess their practical importance 9-12 Effect of α At α = 0.01, the rejection point is z0.01 = 2.33 In the trash example, the test statistic z = 2.20 is < z0.01 = 2.33 Therefore, cannot reject H0 in favor of Ha at the α = 0.01 significance level This is the opposite conclusion reached with α=0.05 So, the smaller we set α, the larger is the rejection point, and the stronger is the statistical evidence that is required to reject the null hypothesis H0 9-13 The p-Value The p-value is the probability of the obtaining the sample results if the null hypothesis H0 is true Sample results that are not likely if H0 is true have a low p-value and are evidence that H0 is not true The p-value is the smallest value of α for which we can reject H0 The p-value is an alternative to testing with a z test statistic 9-14 Steps Using a p-value to Test a “Greater Than” Alternative 4. Collect the sample data and compute the value of the test statistic 5. Calculate the p-value by corresponding to the test statistic value 6. Reject H0 if the p-value is less than α 9-15 Steps in Testing a “Less Than” Alternative in Payment Time Case #1 1. State the null and alternative hypotheses H0: ≥ 19.5 vs. Ha: < 19.5 2. Specify the significance level α = 0.01 3. Select the test statistic z x 19 . 5 x x 19 . 5 n 9-16 Steps in Testing a “Less Than” Alternative in Payment Time Case #2 4. Determine the rejection rule for deciding whether or not to reject H0 The rejection rule is to reject H0 if the calculated test statistic –z is less than –2.33 5. Collect the sample data and calculate the value of the test statistic z x 19 . 5 n 18 . 1077 19 . 5 4 .2 2 . 67 65 9-17 Steps in Testing a “Less Than” Alternative in Payment Time Case #3 6. Decide whether to reject H0 by using the test statistic and the rejection rule In the payment time case, z = –2.67 is less than z0.01 = –2.33 Therefore reject H0: μ ≥ 19.5 in favor of Ha: μ < 19.5 at the 0.01 significance level 7. Interpret the statistical results in managerial terms and assess their practical importance 9-18 Steps Using a p-value to Test a “Less Than” Alternative 4. Collect the sample data and compute the value of the test statistic 5. Calculate the p-value by corresponding to the test statistic value 6. Reject H0 if the p-value is less than α 9-19 Steps in Testing a “Not Equal To” Alternative in Valentine Day Case #1 1. State null and alternative hypotheses H0: = 330 vs. Ha: ≠ 330 2. Specify the significance level α = 0.05 3. Select the test statistic z x 330 x x 330 n 9-20 Steps in Testing a “Not Equal To” Alternative in Valentine Day Case #2 4. Determine the rejection rule for deciding whether or not to reject H0 Rejection points are zα = 1.96, –zα = – 1.96 Reject H0 in favor of Ha if the test statistic z satisfies either: z greater than the rejection point zα/2, or –z less than the rejection point –zα/2 9-21 Steps in Testing a “Not Equal To” Alternative in Valentine Day Case #3 5. Collect the sample data and calculate the value of the test statistic z x 330 n 326 330 40 1 . 00 100 6. Decide whether to reject H0 by using the test statistic and the rejection rule 7. Interpret the statistical results in managerial terms and assess their practical importance 9-22 Steps Using a p-value to Test a “Not Equal To” Alternative 4. Collect the sample data and compute the value of the test statistic 5. Calculate the p-value by corresponding to the test statistic value The p-value is 0.1587 · 2 = 0.3174 6. Reject H0 if the p-value is less than 9-23 Interpreting the Weight of Evidence Against the Null Hypothesis If p < 0.10, there is some evidence to reject H0 If p < 0.05, there is strong evidence to reject H0 If p < 0.01, there is very strong evidence to reject H0 If p < 0.001, there is extremely strong evidence to reject H0 9-24 9.3 t Tests about a Population Mean: σ Unknown Suppose the population being sampled is normally distributed The population standard deviation σ is unknown, as is the usual situation If the population standard deviation σ is unknown, then it will have to estimated from a sample standard deviation s Under these two conditions, have to use the t distribution to test hypotheses 9-25 Defining the t Random Variable: σ Unknown Define a new random variable t: t x s n The sampling distribution of this random variable is a t distribution with n – 1 degrees of freedom 9-26 Defining the t Statistic: σ Unknown Let x be the mean of a sample of size n with standard deviation s Also, µ0 is the claimed value of the population mean x 0 Define a new test statistic t s n If the population being sampled is normal, and s is used to estimate σ, then … The sampling distribution of the t statistic is a t distribution with n – 1 degrees of freedom 9-27 t Tests about a Population Mean: σ Unknown 9-28 9.4 z Tests Tests about a Population Proportion z ˆp p 0 p 0 1 p 0 n 9-29 Example 9.6: The Cheese Spread Case z pˆ p 0 p 0 1 p 0 n . 063 . 10 . 10 1 . 10 3 . 90 1, 000 9-30 9.5 Type II Error Probabilities and Sample Size Determination (Optional) Want the probability β of not rejecting a false null hypothesis That is, want the probability β of committing a Type II error 1 - β is called the power of the test 9-31 Calculating β Assume that the sampled population is normally distributed, or that a large sample is taken Test… H0: µ = µ0 vs Ha: µ < µ0 or Ha: µ > µ0 or Ha: µ ≠ µ0 Want to make the probability of a Type I error equal to α and randomly select a sample of size n 9-32 Calculating β Continued The probability β of a Type II error corresponding to the alternative value µa for µ is equal to the area under the standard normal curve to the left of z* 0 a n Here z* equals zα if the alternative hypothesis is one-sided (µ < µ0 or µ > µ0) Also z* ≠ zα/2 if the alternative hypothesis is two-sided (µ ≠ µ0) 9-33 Sample Size Assume that the sampled population is normally distributed, or that a large sample is taken Test H0: = 0 vs. Ha: < 0 or Ha: > 0 or Ha: ≠ 0 Want to make the probability of a Type I error equal to and the probability of a Type II error corresponding to the alternative value for equal to b 9-34 Sample Size n Continued z * z b 0 2 a 2 2 9-35 9.6 The Chi-Square Distribution (Optional) The chi-square ² distribution depends on the number of degrees of freedom A chi-square point ²α is the point under a chi-square distribution that gives right-hand tail area Figures 9.14 and 9.15 9-36 9.7 Statistical Inference for Population Variance (Optional) If s2 is the variance of a random sample of n measurements from a normal population with variance σ2 The sampling distribution of the statistic (n 1) s2 / σ2 is a chi-square distribution with (n – 1) degrees of freedom Can calculate confidence interval and perform hypothesis testing 9-37 Confidence Interval for Population Variance ( n 1) s ( n 1 ) s , 2 2 /2 1 / 2 2 2 9-38 Hypothesis Testing for Population Variance Test of H0: σ2 = σ20 Test Statistic: 2 ( n 1) s 2 0 2 Reject H0 in favor of Ha: σ2 > σ20 if 2 > 2α Ha: σ2 < σ20 if 2 < 21-α Ha: σ2 ≠ σ20 if 2 > 2α or 2 < 21-α 9-39 Selecting an Appropriate Test Statistic for a Test about a Population Mean Figure 9.18 9-40