McGraw-Hill/Irwin Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved. A PowerPoint Presentation Package to Accompany Applied Statistics in Business & Economics, 4 th edition David P. Doane and Lori E. Seward Prepared by Lloyd R. Jaisingh

9-2 One-Sample Hypothesis Tests Chapter Contents 9.1 Logic of Hypothesis Testing 9.2 Statistical Hypothesis Testing 9.3 Testing a Mean: Known Population Variance 9.4 Testing a Mean: Unknown Population Variance 9.5 Testing a Proportion 9.6 Power Curves and OC Curves (Optional) 9.7 Tests for One Variance (Optional) Chapter 9

9-3 Chapter Learning Objectives LO9-1: List the steps in testing hypotheses. LO9-2: Explain the difference between H 0 and H 1. LO9-3: Define Type I error, Type II error, and power. LO9-4: Formulate a null and alternative hypothesis for LO9-4: Formulate a null and alternative hypothesis for μ or π. LO9-5: Find critical values of z or t in tables or by using Excel. LO9-6: Do a hypothesis test for a mean with known LO9-6: Do a hypothesis test for a mean with known σ using z. LO9-7: Do a hypothesis test for a mean with unknown σ using t. LO9-8: Use tables or Excel to find the p-value in tests of μ. LO9-9: Perform a hypothesis test for a proportion and find the p-value. LO9-10: Interpret a power curve or OC curve (optional). LO9-11: Do a hypothesis test for a variance (optional). Chapter 9 One-Sample Hypothesis Tests

9-4 Steps in Hypothesis Testing Step 1: State the assumption to be tested. Step 2: Specify the decision rule. Step 3: Collect the data to test the hypothesis. Step 4: Make a decision. Step 5: Take action based on the decision. Chapter Logic of Hypothesis Testing LO9-1: List the steps in testing hypotheses. LO9-2: Explain the difference between H 0 and H 1. State the Hypothesis State the Hypothesis Hypotheses are a pair of mutually exclusive, collectively exhaustive statements about the world.Hypotheses are a pair of mutually exclusive, collectively exhaustive statements about the world. One statement or the other must be true, but they cannot both be true.One statement or the other must be true, but they cannot both be true. H 0 : Null Hypothesis. H 1 : Alternative Hypothesis.H 0 : Null Hypothesis. H 1 : Alternative Hypothesis. These two statements are hypotheses because the truth is unknown.These two statements are hypotheses because the truth is unknown. LO9-1

9-5 Efforts will be made to reject the null hypothesis.Efforts will be made to reject the null hypothesis. If H 0 is rejected, we tentatively conclude H 1 to be the case.If H 0 is rejected, we tentatively conclude H 1 to be the case. H 0 is sometimes called the maintained hypothesis.H 0 is sometimes called the maintained hypothesis. H 1 is called the action alternative because action may be required if we reject H 0 in favor of H 1.H 1 is called the action alternative because action may be required if we reject H 0 in favor of H 1. State the Hypothesis State the Hypothesis Chapter 9 Can Hypotheses be Proved? Can Hypotheses be Proved? We cannot prove a null hypothesis, we can only fail to reject it.We cannot prove a null hypothesis, we can only fail to reject it. Role of Evidence Role of Evidence The null hypothesis is assumed true and a contradiction is sought.The null hypothesis is assumed true and a contradiction is sought. 9.1 Logic of Hypothesis Testing LO9-2

9-6 Types of Error Types of Error Type I error: Rejecting the null hypothesis when it is true. This occurs with probability .Type I error: Rejecting the null hypothesis when it is true. This occurs with probability . Type II error: Failure to reject the null hypothesis when it is false. This occurs with probability .Type II error: Failure to reject the null hypothesis when it is false. This occurs with probability . Chapter 9 LO9-3: Define Type I error, Type II error, and power. 9.1 Logic of Hypothesis Testing LO9-3

9-7 A statistical hypothesis is a statement about the value of a population parameter.A statistical hypothesis is a statement about the value of a population parameter. A hypothesis test is a decision between two competing mutually exclusive and collectively exhaustive hypotheses about the value of the parameter.A hypothesis test is a decision between two competing mutually exclusive and collectively exhaustive hypotheses about the value of the parameter. When testing a mean we choose between three tests.When testing a mean we choose between three tests. Chapter Statistical Hypothesis Testing LO4LO9-4 LO9-4: Formulate a null and alternative hypothesis for μ or π.

9-8 The direction of the test is indicated by H 1 :The direction of the test is indicated by H 1 : Chapter 9 Decision Rule for Two-Tailed Test Decision Rule for Two-Tailed Test LO Statistical Hypothesis Testing > indicates a right-tailed test. < indicates a left-tailed test. ≠ indicates a two-tailed test.

9-9 When to use a One- or Two-Sided Test When to use a One- or Two-Sided Test A two-sided hypothesis test (i.e.,  ≠  0 ) is used when direction ( ) is of no interest to the decision makerA two-sided hypothesis test (i.e.,  ≠  0 ) is used when direction ( ) is of no interest to the decision maker A one-sided hypothesis test is used when - the consequences of rejecting H 0 are asymmetric, or - where one tail of the distribution is of special importance to the researcher.A one-sided hypothesis test is used when - the consequences of rejecting H 0 are asymmetric, or - where one tail of the distribution is of special importance to the researcher. Rejection in a two-sided test guarantees rejection in a one-sided test, other things being equal.Rejection in a two-sided test guarantees rejection in a one-sided test, other things being equal. Chapter Statistical Hypothesis Testing LO9-4

9-10 Decision Rule for a One- Sided Test Decision Rule for a One- Sided Test Chapter 9 Decision Rule for Right-Tailed Test Decision Rule for Right-Tailed Test Reject H 0 if the test statistic > right-tail critical value.Reject H 0 if the test statistic > right-tail critical value. 9.2 Statistical Hypothesis Testing LO9-4 Decision Rule for Left-Tailed Test Decision Rule for Left-Tailed Test Reject H 0 if the test statistic < left-tail critical value.Reject H 0 if the test statistic < left-tail critical value.

9-11 Chapter Testing a Mean: Known Population Variance The test statistic is compared with a critical value from a table. The critical value is the boundary between two regions (reject H 0, do not reject H 0 ) in the decision rule. The critical value shows the range of values for the test statistic that would be expected by chance if the null hypothesis were true. These z-values can be computed using EXCEL. LO9-5 LO9-5: Find critical values of z or t in tables or by using Excel.

9-12 The hypothesized mean  0 that we are testing is a benchmark.The hypothesized mean  0 that we are testing is a benchmark. The value of  0 does not come from a sample.The value of  0 does not come from a sample. The test statistic compares the sample mean with the hypothesized mean  0.The test statistic compares the sample mean with the hypothesized mean  0. Chapter Testing a Mean: Known Population Variance LO9-7: Do a hypothesis test for a mean with unknown σ using t. Using Student’s t When the population standard deviation  is unknown and the population may be assumed normal, the test statistic follows the Student’s t distribution with n – 1 degrees of freedom.When the population standard deviation  is unknown and the population may be assumed normal, the test statistic follows the Student’s t distribution with n – 1 degrees of freedom. LO9-6LO9-7 LO9-6: Do a hypothesis test for a mean with known σ using z.

9-13 Using the p-Value Approach Using the p-Value Approach The p-value is the probability of the sample result (or one more extreme) assuming that H 0 is true.The p-value is the probability of the sample result (or one more extreme) assuming that H 0 is true. The p-value can be obtained using Excel’s cumulative standard normal function =NORMSDIST(z)The p-value can be obtained using Excel’s cumulative standard normal function =NORMSDIST(z) The p-value can also be obtained from Appendix C-2.The p-value can also be obtained from Appendix C-2. Using the p-value, we reject H 0 if p-value < .Using the p-value, we reject H 0 if p-value < . Chapter Testing a Mean: Known Population Variance Analogy to Confidence Intervals Analogy to Confidence Intervals A two-tailed hypothesis test at the 5% level of significance (  =.05) is exactly equivalent to asking whether the 95% confidence interval for the mean includes the hypothesized mean.A two-tailed hypothesis test at the 5% level of significance (  =.05) is exactly equivalent to asking whether the 95% confidence interval for the mean includes the hypothesized mean. If the confidence interval includes the hypothesized mean, then we cannot reject the null hypothesis.If the confidence interval includes the hypothesized mean, then we cannot reject the null hypothesis. LO9-6

9-14 Chapter 9 Using the p-value LO9-8: Use tables or Excel to find the p-value in tests of μ. 9.4 Testing a Mean: Unknown Population Variance LO9-8

9-15 Chapter Testing a Proportion LO9-9 LO9-9 : Perform a hypothesis test for a proportion and find the p-value.

9-16 The value of  0 that we are testing is a benchmark such as past experience, an industry standard, or a product specification.The value of  0 that we are testing is a benchmark such as past experience, an industry standard, or a product specification. The value of  0 does not come from a sample.The value of  0 does not come from a sample. Chapter 9 Critical Value The test statistic is compared with a critical z value from a table.The test statistic is compared with a critical z value from a table. The critical value shows the range of values for the test statistic that would be expected by chance if the H 0 were true.The critical value shows the range of values for the test statistic that would be expected by chance if the H 0 were true. NOTE: In the case where n  0 < 10, use MINITAB to test the hypotheses by finding the exact binomial probability of a sample proportion p. For example,NOTE: In the case where n  0 < 10, use MINITAB to test the hypotheses by finding the exact binomial probability of a sample proportion p. For example, 9.5 Testing a Proportion LO9-9