Lecture Business statistics in practice (7/e): Chapter 9 - Bowerman, O'Connell, Murphree

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Chapter 9 - Hypothesis testing. After mastering the material in this chapter, you will be able to: Set Up appropriate null and alternative hypotheses, describe Type I and Type II errors and their probabilities, use critical values and p-values to perform a z test about a population mean when s is known,...

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Nội dung Text: Lecture Business statistics in practice (7/e): Chapter 9 - Bowerman, O'Connell, Murphree

Chapter 9 Hypothesis Testing McGrawHill/Irwin Copyright © 2014 by The McGrawHill Companies, Inc. All rights reserved.
Hypothesis Testing 9.1 Null and Alternative Hypotheses and Errors in Testing 9.2 z Tests about a Population Mean σ Known 9.3 t 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) 92
LO9-1: Specify appropriate null and alternative hypotheses. 9.1 Null and Alternative Hypotheses and Errors in Hypothesis Testing  Null hypothesis, H0, is a statement of the basic proposition being tested ◦ Represents the status quo and is not rejected unless there is convincing sample evidence that it is false  Alternative hypothesis, Ha, is an alternative accepted only if there is convincing sample evidence it is true  OneSided, “Greater Than” H0: μ μ0 vs. Ha: μ > μ0  OneSided, “Less Than” H0 : μ μ0 vs. Ha : μ
LO9-1 Types of Decisions  As a result of testing H0 vs. Ha, will decide either of the following decisions for the null hypothesis H0: ◦Do not reject H0 or reject H0  To “test” H0 vs. Ha, use the “test statistic” x 0 x 0 z x n  z measures the distance between μ0 and  on the sampling distribution of the sample mean  If the population is normal or n is large*, the test statistic z follows a normal distribution 94
LO9-2: Describe Type I and Type II errors and their probabilities. Error Probabilities Type I Error: Rejecting H0 when it is true ◦ is the probability of making a Type I error ◦1 – is the probability of not making a Type I error Type II Error: Failing to reject H0 when it is false ◦β is the probability of making a Type II error ◦1 – β is the probability of not making a Type II error Table 9.1 95
LO9-2 Typical Values Usually set to a low value ◦So there is a 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 to reject H0 Tradeoff between and β ◦For fixed sample size, the lower , the higher β  And the higher , the lower β 96
LO9-3: Use critical values and p-values to perform a z test about a population mean when σ is known. 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 realworld situations, σ is not known  But often is estimated from s of a single sample  When σ is unknown, test hypotheses about a population mean using the t distribution ◦Here, assume that we know σ 97
LO9-3 Steps in Testing a “Greater Than” Alternative 1. State the null and alternative hypotheses 2. Specify the significance level α 3. Select the test statistic 4. Determine the critical value rule for deciding whether or not to reject H0 5. 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 rejection rule 7. Interpret the statistical results in managerial terms and assess their practical importance 98
LO9-4: Use critical values and p-values to perform a t test about a population mean when σ is unknown. 9.3 t Tests about a Population Mean: σ Unknown Assume 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 deviations Under these two conditions, have to use the t distribution to test hypotheses 99
LO9-5: Use critical 9.4 z Tests about a Population values and p-values to perform a large sample z test about a Proportion population proportion. Alternative Reject H0 if: pvalue Ha: ρ > ρ0 z > z Area under t distribution to right of z Ha: ρ zα/2 or z
LO9-6: Calculate Type II error probabilities and the power of a test, and determine sample size (Optional). 9.5 Type II Error Probabilities and Sample Size Determination (Optional)  Want the probability β of not rejecting a false null hypothesis ◦ Want the probability β of committing a Type II error  1 β is called the power of the test  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  Want to make the probability of a Type I error equal to α and randomly select a sample of size n 911
LO9-6 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 0 a z* n Here z* equals zα if the alternative hypothesis is onesided (µ µ0) Also z* ≠ zα/2 if the alternative hypothesis is twosided (µ ≠ µ0) 912