Chapter 9 Hypothesis Testing: Single Population Statistics for Business and Economics 7th Edition Chapter 9 Hypothesis Testing: Single Population Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall What is a Hypothesis? 9.1 A hypothesis is a claim (assumption) about a population parameter: population mean population proportion Example: The mean monthly cell phone bill of this city is μ = $42 Example: The proportion of adults in this city with cell phones is p = .68 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall The Null Hypothesis, H0 States the assumption (numerical) to be tested Example: The average number of TV sets in U.S. Homes is equal to three ( ) Is always about a population parameter, not about a sample statistic Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall The Null Hypothesis, H0 (continued) Begin with the assumption that the null hypothesis is true Similar to the notion of innocent until proven guilty Always contains “=” , “≤” or “” sign May or may not be rejected Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

The Alternative Hypothesis, H1 Is the opposite of the null hypothesis e.g., The average number of TV sets in U.S. homes is not equal to 3 ( H1: μ ≠ 3 ) Never contains the “=” , “≤” or “” sign May or may not be supported Is generally the hypothesis that the researcher is trying to support Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Hypothesis Testing Process Claim: the population mean age is 50. (Null Hypothesis: Population H0: μ = 50 ) Now select a random sample X Is = 20 likely if μ = 50? Suppose the sample If not likely, REJECT mean age is 20: X = 20 Sample Null Hypothesis Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Level of Significance,  Defines the unlikely values of the sample statistic if the null hypothesis is true Defines rejection region of the sampling distribution Is designated by  , (level of significance) Typical values are .01, .05, or .10 Is selected by the researcher at the beginning Provides the critical value(s) of the test Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Level of Significance and the Rejection Region Represents critical value a a H0: μ = 3 H1: μ ≠ 3 /2 /2 Rejection region is shaded Two-tail test H0: μ ≤ 3 H1: μ > 3 a Upper-tail test H0: μ ≥ 3 H1: μ < 3 a Lower-tail test Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Hypothesis Tests for the Mean  Known  Unknown Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Test of Hypothesis for the Mean (σ Known) 9.2 Convert sample result ( ) to a z value Hypothesis Tests for  σ Known σ Unknown Consider the test The decision rule is: (Assume the population is normal) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Decision Rule H0: μ = μ0 H1: μ > μ0 Alternate rule: a Do not reject H0 Reject H0 Z zα μ0 Critical value Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Example: Upper-Tail Z Test for Mean ( Known) A phone industry manager thinks that customer monthly cell phone bill have increased, and now average over $52 per month. The company wishes to test this claim. (Assume  = 10 is known) Form hypothesis test: H0: μ ≤ 52 the average is not over $52 per month H1: μ > 52 the average is greater than $52 per month (i.e., sufficient evidence exists to support the manager’s claim) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Example: Find Rejection Region (continued) Suppose that  = .10 is chosen for this test Find the rejection region: Reject H0  = .10 Do not reject H0 Reject H0 1.28 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Example: Sample Results (continued) Obtain sample and compute the test statistic Suppose a sample is taken with the following results: n = 64, x = 53.1 ( = 10 was assumed known) Using the sample results, Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Example: Decision (continued) Reach a decision and interpret the result: Reject H0  = .10 Do not reject H0 Reject H0 1.28 z = 0.88 Do not reject H0 since z = 0.88 < 1.28 i.e.: there is not sufficient evidence that the mean bill is over $52 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall One-Tail Tests In many cases, the alternative hypothesis focuses on one particular direction This is an upper-tail test since the alternative hypothesis is focused on the upper tail above the mean of 3 H0: μ ≤ 3 H1: μ > 3 This is a lower-tail test since the alternative hypothesis is focused on the lower tail below the mean of 3 H0: μ ≥ 3 H1: μ < 3 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Upper-Tail Tests H0: μ ≤ 3 H1: μ > 3 There is only one critical value, since the rejection area is in only one tail a Do not reject H0 Reject H0 zα Z μ Critical value Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Lower-Tail Tests H0: μ ≥ 3 H1: μ < 3 There is only one critical value, since the rejection area is in only one tail a Reject H0 Do not reject H0 -z Z μ Critical value Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Two-Tail Tests In some settings, the alternative hypothesis does not specify a unique direction H0: μ = 3 H1: μ ¹ 3 /2 /2 There are two critical values, defining the two regions of rejection x 3 Reject H0 Do not reject H0 Reject H0 z -z/2 +z/2 Lower critical value Upper critical value Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall