1. Chapter 7 Hypothesis Testing with One Sample 1 Larson/Farber 4th ed.

2. Hypothesis Tests Hypothesis test A process that uses sample statistics to test a claim about the value of a population parameter. For example: An automobile manufacturer advertises that its new hybrid car has a mean mileage of 50 miles per gallon. To test this claim, a sample would be taken. If the sample mean differs enough from the advertised mean, you can decide the advertisement is wrong. 2 Larson/Farber 4th ed.

3. Stating a Hypothesis Null hypothesis A statistical hypothesis that contains a statement of equality such as ≤, =, or ≥. Denoted H 0 read “H subzero” or “H naught.” Alternative hypothesis A statement of inequality such as >, ≠, or <. Must be true if H 0 is false. Denoted H a read “H sub-a.” complementary statements 3 Larson/Farber 4th ed.

4. Example: Stating the Null and Alternative Hypotheses Write the claim as a mathematical sentence. State the null and alternative hypotheses and identify which represents the claim. 1.A university publicizes that the proportion of its students who graduate in 4 years is 82%. Equality condition Complement of H 0 H0:Ha:H0:Ha: (Claim) p = 0.82 p ≠ 0.82 Solution: 4 Larson/Farber 4th ed.

5. μ ≥ 2.5 gallons per minute Example: Stating the Null and Alternative Hypotheses Write the claim as a mathematical sentence. State the null and alternative hypotheses and identify which represents the claim. 2.A water faucet manufacturer announces that the mean flow rate of a certain type of faucet is less than 2.5 gallons per minute. Inequality condition Complement of H a H0:Ha:H0:Ha: (Claim) μ < 2.5 gallons per minute Solution: 5 Larson/Farber 4th ed.

6. Types of Errors A type I error occurs if the null hypothesis is rejected when it is true. A type II error occurs if the null hypothesis is not rejected when it is false. Actual Truth of H 0 DecisionH 0 is trueH 0 is false Do not reject H 0 Correct DecisionType II Error Reject H 0 Type I ErrorCorrect Decision 6 Larson/Farber 4th ed.

7. Example: Identifying Type I and Type II Errors The USDA limit for salmonella contamination for chicken is 20%. A meat inspector reports that the chicken produced by a company exceeds the USDA limit. You perform a hypothesis test to determine whether the meat inspector’s claim is true. When will a type I or type II error occur? Which is more serious? (Source: United States Department of Agriculture) 7 Larson/Farber 4th ed.

8. Let p represent the proportion of chicken that is contaminated. Solution: Identifying Type I and Type II Errors H0:Ha:H0:Ha: p ≤ 0.2 p > 0.2 Hypotheses: (Claim) 0.160.180.200.220.24 p H 0 : p ≤ 0.20H 0 : p > 0.20 Chicken meets USDA limits. Chicken exceeds USDA limits. 8 Larson/Farber 4th ed.

9. Solution: Identifying Type I and Type II Errors A type I error is rejecting H 0 when it is true. The actual proportion of contaminated chicken is less than or equal to 0.2, but you decide to reject H 0. A type II error is failing to reject H 0 when it is false. The actual proportion of contaminated chicken is greater than 0.2, but you do not reject H 0. H0:Ha:H0:Ha: p ≤ 0.2 p > 0.2 Hypotheses: (Claim) 9 Larson/Farber 4th ed.

10. Solution: Identifying Type I and Type II Errors H0:Ha:H0:Ha: p ≤ 0.2 p > 0.2 Hypotheses: (Claim) With a type I error, you might create a health scare and hurt the sales of chicken producers who were actually meeting the USDA limits. With a type II error, you could be allowing chicken that exceeded the USDA contamination limit to be sold to consumers. A type II error could result in sickness or even death. 10 Larson/Farber 4th ed.

11. Level of Significance Level of significance Your maximum allowable probability of making a type I error.  Denoted by , the lowercase Greek letter alpha. By setting the level of significance at a small value, you are saying that you want the probability of rejecting a true null hypothesis to be small. Commonly used levels of significance:   = 0.10  = 0.05  = 0.01 P(type II error) = β (beta) 11 Larson/Farber 4th ed.

12. P-values P-value (or probability value) The probability, if the null hypothesis is true, of obtaining a sample statistic with a value as extreme or more extreme than the one determined from the sample data. Depends on the nature of the test. 12 Larson/Farber 4th ed.

13. Making a Decision Decision Rule Based on P-value Compare the P-value with α.  If P ≤ α, then reject H 0.  If P > α, then fail to reject H 0. Claim DecisionClaim is H 0 Claim is H a Reject H 0 Fail to reject H 0 There is enough evidence to reject the claim There is not enough evidence to reject the claim There is enough evidence to support the claim There is not enough evidence to support the claim 13 Larson/Farber 4th ed.

14. Example: Interpreting a Decision You perform a hypothesis test for the following claim. How should you interpret your decision if you reject H 0 ? If you fail to reject H 0 ? 2.H a (Claim): Consumer Reports states that the mean stopping distance (on a dry surface) for a Honda Civic is less than 136 feet. Solution: The claim is represented by H a. H 0 is “the mean stopping distance…is greater than or equal to 136 feet.” 14 Larson/Farber 4th ed.

15. Solution: Interpreting a Decision If you reject H 0 you should conclude “there is enough evidence to support Consumer Reports’ claim that the stopping distance for a Honda Civic is less than 136 feet.” If you fail to reject H 0, you should conclude “there is not enough evidence to support Consumer Reports’ claim that the stopping distance for a Honda Civic is less than 136 feet.” 15 Larson/Farber 4th ed.

16. z 0 Steps for Hypothesis Testing 1.State the claim mathematically and verbally. Identify the null and alternative hypotheses. H 0 : ? H a : ? 2.Specify the level of significance. α = ? 3.Determine the standardized sampling distribution and draw its graph. 4.Calculate the test statistic and its standardized value. Add it to your sketch. z 0 Test statistic This sampling distribution is based on the assumption that H 0 is true. 16 Larson/Farber 4th ed.

17. Steps for Hypothesis Testing 5.Find the P-value. 6.Use the following decision rule. 7.Write a statement to interpret the decision in the context of the original claim. Is the P-value less than or equal to the level of significance? Fail to reject H 0. Yes Reject H 0. No 17 Larson/Farber 4th ed.

18. Section 7.2 Hypothesis Testing for the Mean (Large Samples) 18 Larson/Farber 4th ed.

19. Using P-values to Make a Decision Decision Rule Based on P-value To use a P-value to make a conclusion in a hypothesis test, compare the P-value with α. 1.If P ≤ α, then reject H 0. 2.If P > α, then fail to reject H 0. 19 Larson/Farber 4th ed.

20. Z-Test for a Mean μ Can be used when the population is normal and  is known, or for any population when the sample size n is at least 30. The test statistic is the sample mean The standardized test statistic is z When n ≥ 30, the sample standard deviation s can be substituted for σ. 20 Larson/Farber 4th ed.

21. Using P-values for a z-Test for Mean μ 1.State the claim mathematically and verbally. Identify the null and alternative hypotheses. 2.Specify the level of significance. 3.Determine the standardized test statistic. 4.Find the area that corresponds to z. State H 0 and H a. Identify α. Use Table 4 in Appendix B. 21 Larson/Farber 4th ed. In WordsIn Symbols

22. Using P-values for a z-Test for Mean μ Reject H 0 if P-value is less than or equal to α. Otherwise, fail to reject H 0. 5.Find the P-value. a.For a left-tailed test, P = (Area in left tail). b.For a right-tailed test, P = (Area in right tail). c.For a two-tailed test, P = 2(Area in tail of test statistic). 6.Make a decision to reject or fail to reject the null hypothesis. 7.Interpret the decision in the context of the original claim. 22 Larson/Farber 4th ed. In WordsIn Symbols

23. Example: Hypothesis Testing Using P- values In an advertisement, a pizza shop claims that its mean delivery time is less than 30 minutes. A random selection of 36 delivery times has a sample mean of 28.5 minutes and a standard deviation of 3.5 minutes. Is there enough evidence to support the claim at α = 0.01? Use a P-value. 23 Larson/Farber 4th ed.

24. Solution: Hypothesis Testing Using P- values H 0 : H a :  = Test Statistic: μ ≥ 30 min μ < 30 min 0.01 Decision: At the 1% level of significance, you have sufficient evidence to conclude the mean delivery time is less than 30 minutes. z 0-2.57 0.0051 P-value 0.0051 < 0.01 Reject H 0 24 Larson/Farber 4th ed.

25. Rejection Regions and Critical Values Rejection region (or critical region) The range of values for which the null hypothesis is not probable. If a test statistic falls in this region, the null hypothesis is rejected. A critical value z 0 separates the rejection region from the nonrejection region. 25 Larson/Farber 4th ed.

26. Using Rejection Regions for a z-Test for a Mean μ 1.State the claim mathematically and verbally. Identify the null and alternative hypotheses. 2.Specify the level of significance. 3.Sketch the sampling distribution. 4.Determine the critical value(s). 5.Determine the rejection region(s). State H 0 and H a. Identify. Use Table 4 in Appendix B. 26 Larson/Farber 4th ed. In WordsIn Symbols

27. Using Rejection Regions for a z-Test for a Mean μ 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 z is in the rejection region, reject H 0. Otherwise, fail to reject H 0. 27 Larson/Farber 4th ed. In WordsIn Symbols

28. Example: Testing with Rejection Regions Employees in a large accounting firm claim that the mean salary of the firm’s accountants is less than that of its competitor’s, which is $45,000. A random sample of 30 of the firm’s accountants has a mean salary of $43,500 with a standard deviation of $5200. At α = 0.05, test the employees’ claim. 28 Larson/Farber 4th ed.

29. Solution: Testing with Rejection Regions H 0 : H a :  = Rejection Region: μ ≥ $45,000 μ < $45,000 0.05 Decision: At the 5% level of significance, there is not sufficient evidence to support the employees’ claim that the mean salary is less than $45,000. Test Statistic z 0-1.645 0.05 -1.58 Fail to reject H 0 29 Larson/Farber 4th ed.

30. Section 7.3 Hypothesis Testing for the Mean (Small Samples) 30 Larson/Farber 4th ed.

31. Finding Critical Values in a t-Distribution 1.Identify the level of significance α. 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, α ” column with a negative sign, b.right-tailed, use “One Tail, α ” column with a positive sign, c.two-tailed, use “Two Tails, α ” column with a negative and a positive sign. 31 Larson/Farber 4th ed.

32. Example: Finding Critical Values for t Find the critical value t 0 for a left-tailed test given α = 0.05 and n = 21. Solution: The degrees of freedom are d.f. = n – 1 = 21 – 1 = 20. Look at α = 0.05 in the “One Tail, α” column. Because the test is left- tailed, the critical value is negative. t 0-1.725 0.05 32 Larson/Farber 4th ed.

33. t-Test for a Mean μ (n < 30, σ 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, σ 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. 33 Larson/Farber 4th ed.

34. 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 α. Use Table 5 in Appendix B. d.f. = n – 1. 34 Larson/Farber 4th ed. In WordsIn Symbols

35. 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. 35 Larson/Farber 4th ed. In WordsIn Symbols

36. Example: Testing μ with a Small Sample A used car dealer says that the mean price of a 2005 Honda Pilot LX is at least $23,900. You suspect this claim is incorrect and find that a random sample of 14 similar vehicles has a mean price of $23,000 and a standard deviation of $1113. 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) 36 Larson/Farber 4th ed.

37. Solution: Testing μ with a Small Sample H 0 : H a : α = df = Rejection Region: Test Statistic: Decision: μ ≥ $23,900 μ < $23,900 0.05 14 – 1 = 13 t 0-1.771 0.05 -3.026 At the 0.05 level of significance, there is enough evidence to reject the claim that the mean price of a 2005 Honda Pilot LX is at least $23,900 Reject H 0 37 Larson/Farber 4th ed.

38. Section 7.4 Hypothesis Testing for Proportions 38 Larson/Farber 4th ed.

39. z-Test for a Population Proportion A statistical test for a population proportion. Can be used when a binomial distribution is given such that np ≥ 5 and nq ≥ 5. The test statistic is the sample proportion. The standardized test statistic is z. 39 Larson/Farber 4th ed.

40. Using a z-Test for a Proportion p 1.State the claim mathematically and verbally. Identify the null and alternative hypotheses. 2.Specify the level of significance. 3.Sketch the sampling distribution. 4.Determine any critical value(s). State H 0 and H a. Identify α. Use Table 5 in Appendix B. Verify that np ≥ 5 and nq ≥ 5 40 Larson/Farber 4th ed. In WordsIn Symbols

41. Using a z-Test for a Proportion p 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 z is in the rejection region, reject H 0. Otherwise, fail to reject H 0. 41 Larson/Farber 4th ed. In WordsIn Symbols

42. Example: Hypothesis Test for Proportions Zogby International claims that 45% of people in the United States support making cigarettes illegal within the next 5 to 10 years. You decide to test this claim and ask a random sample of 200 people in the United States whether they support making cigarettes illegal within the next 5 to 10 years. Of the 200 people, 49% support this law. At α = 0.05 is there enough evidence to reject the claim? Solution: Verify that np ≥ 5 and nq ≥ 5. np = 200(0.45) = 90 and nq = 200(0.55) = 110 42 Larson/Farber 4th ed.

43. Solution: Hypothesis Test for Proportions H 0 : H a :  = Rejection Region: p = 0.45 p ≠ 0.45 0.05 Decision: At the 5% level of significance, there is not enough evidence to reject the claim that 45% of people in the U.S. support making cigarettes illegal within the next 5 to 10 years. Test Statistic z 0-1.96 0.025 1.96 0.025 1.14 Fail to reject H 0 43 Larson/Farber 4th ed.