Section 7.3 Hypothesis Testing for the Mean (Small Samples) © 2012 Pearson Education, Inc. All rights reserved. 1 of 101

Section 7.3 Objectives Find critical values in a t-distribution Use the t-test to test a mean μ © 2012 Pearson Education, Inc. All rights reserved. 2 of 101

Finding Critical Values in a t- Distribution 1.Identify the level of significance . 2.Identify the degrees of freedom d.f. = n – 1. 3.Find the critical value(s) using Table 5 in Appendix B in the row with n – 1 degrees of freedom. If the hypothesis test is a.left-tailed, use “One Tail,  ” column with a negative sign, b.right-tailed, use “One Tail,  ” column with a positive sign, c.two-tailed, use “Two Tails,  ” column with a negative and a positive sign. © 2012 Pearson Education, Inc. All rights reserved. 3 of 101

Example: Finding Critical Values for t Find the critical value t 0 for a left-tailed test given  = 0.05 and n = 21. Solution: t 0- © 2012 Pearson Education, Inc. All rights reserved. 4 of 101

Example: Finding Critical Values for t Find the critical values -t 0 and t 0 for a two-tailed test given  = 0.10 and n = 26. Solution: © 2012 Pearson Education, Inc. All rights reserved. 5 of 101

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t-Test for a Mean μ (n < 30,  Unknown) t-Test for a Mean A statistical test for a population mean. The t-test can be used when the population is normal or nearly normal,  is unknown, and n < 30. The test statistic is the sample mean The standardized test statistic is t. The degrees of freedom are d.f. = n – 1. © 2012 Pearson Education, Inc. All rights reserved. 7 of 101

Using the t-Test for a Mean μ (Small Sample) 1.State the claim mathematically and verbally. Identify the null and alternative hypotheses. 2.Specify the level of significance. 3.Identify the degrees of freedom and sketch the sampling distribution. 4.Determine any critical value(s). State H 0 and H a. Identify . Use Table 5 in Appendix B. d.f. = n – 1. In WordsIn Symbols © 2012 Pearson Education, Inc. All rights reserved. 8 of 101

Using the t-Test for a Mean μ (Small Sample) 5.Determine any rejection region(s). 6.Find the standardized test statistic. 7.Make a decision to reject or fail to reject the null hypothesis. 8.Interpret the decision in the context of the original claim. If t is in the rejection region, reject H 0. Otherwise, fail to reject H 0. In WordsIn Symbols © 2012 Pearson Education, Inc. All rights reserved. 9 of 101

Example: Testing μ with a Small Sample A used car dealer says that the mean price of a 2008 Honda CR-V is at least $20,500. You suspect this claim is incorrect and find that a random sample of 14 similar vehicles has a mean price of $19,850 and a standard deviation of $1084. Is there enough evidence to reject the dealer’s claim at α = 0.05? Assume the population is normally distributed. (Adapted from Kelley Blue Book) © 2012 Pearson Education, Inc. All rights reserved. 10 of 101

Solution: Testing μ with a Small Sample H 0 : H a : α = df = Rejection Region: Test Statistic: Decision: © 2012 Pearson Education, Inc. All rights reserved. 11 of 101

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Example: Testing μ with a Small Sample An industrial company claims that the mean pH level of the water in a nearby river is 6.8. You randomly select 19 water samples and measure the pH of each. The sample mean and standard deviation are 6.7 and 0.24, respectively. Is there enough evidence to reject the company’s claim at α = 0.05? Assume the population is normally distributed. © 2012 Pearson Education, Inc. All rights reserved. 13 of 101

Solution: Testing μ with a Small Sample H 0 : H a : α = df = Rejection Region: Test Statistic: Decision: © 2012 Pearson Education, Inc. All rights reserved. 14 of 101

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Section 7.3 Summary Found critical values in a t-distribution Used the t-test to test a mean μ © 2012 Pearson Education, Inc. All rights reserved. 16 of 101