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Chapter 7 Hypothesis Testing with One Sample Larson/Farber 4th ed. 1 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. Larson/Farber 4th ed. 2 Stating a Hypothesis Null hypothesis • A statistical hypothesis that contains a statement of equality such as ≤, =, or ≥. • Denoted H0 read “H subzero” or “H naught.” Alternative hypothesis • A statement of inequality such as >, ≠, or <. • Must be true if H0 is false. • Denoted Ha read “H sub-a.” complementary statements Larson/Farber 4th ed. 3 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%. Solution: H0: p = 0.82 Equality condition (Claim) Ha: p ≠ 0.82 Complement of H0 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. 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. Solution: H0: μ ≥ 2.5 gallons per minute Ha: μ < 2.5 gallons per minute Larson/Farber 4th ed. Complement of Ha Inequality (Claim) condition 5 Types of Errors Actual Truth of H0 Decision Do not reject H0 Reject H0 H0 is true Correct Decision Type I Error H0 is false Type II Error Correct Decision • 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. Larson/Farber 4th ed. 6 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) Larson/Farber 4th ed. 7 Solution: Identifying Type I and Type II Errors Let p represent the proportion of chicken that is contaminated. Hypotheses: H0: p ≤ 0.2 Ha: p > 0.2 (Claim) Chicken meets USDA limits. H0: p ≤ 0.20 Chicken exceeds USDA limits. H0: p > 0.20 p 0.16 Larson/Farber 4th ed. 0.18 0.20 0.22 0.24 8 Solution: Identifying Type I and Type II Errors Hypotheses: H0: p ≤ 0.2 Ha: p > 0.2 (Claim) A type I error is rejecting H0 when it is true. The actual proportion of contaminated chicken is less than or equal to 0.2, but you decide to reject H0. A type II error is failing to reject H0 when it is false. The actual proportion of contaminated chicken is greater than 0.2, but you do not reject H0. Larson/Farber 4th ed. 9 Solution: Identifying Type I and Type II Errors Hypotheses: H0: p ≤ 0.2 Ha: p > 0.2 (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. Larson/Farber 4th ed. 10 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) Larson/Farber 4th ed. 11 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. Larson/Farber 4th ed. 12 Making a Decision Decision Rule Based on P-value • Compare the P-value with α. If P ≤ α, then reject H0. If P > α, then fail to reject H0. Claim Decision Claim is H0 Claim is Ha Reject H0 There is enough evidence to reject the claim There is enough evidence to support the claim Fail to reject H0 There is not enough evidence to reject the claim There is not enough evidence to support the claim Larson/Farber 4th ed. 13 Example: Interpreting a Decision You perform a hypothesis test for the following claim. How should you interpret your decision if you reject H0? If you fail to reject H0? 2. Ha (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 Ha. • H0 is “the mean stopping distance…is greater than or equal to 136 feet.” Larson/Farber 4th ed. 14 Solution: Interpreting a Decision • If you reject H0 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 H0, 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.” Larson/Farber 4th ed. 15 Steps for Hypothesis Testing 1. State the claim mathematically and verbally. Identify the null and alternative hypotheses. H0: ? Ha: ? 2. Specify the level of significance. This sampling distribution is based on the assumption α= ? that H0 is true. 3. Determine the standardized sampling distribution and draw its graph. z 0 4. Calculate the test statistic and its standardized value. Add it to your sketch. z 0 Test statistic Larson/Farber 4th ed. 16 Steps for Hypothesis Testing 5. Find the P-value. 6. Use the following decision rule. Is the P-value less than or equal to the level of significance? No Fail to reject H0. Yes Reject H0. 7. Write a statement to interpret the decision in the context of the original claim. Larson/Farber 4th ed. 17 Section 7.2 Hypothesis Testing for the Mean (Large Samples) Larson/Farber 4th ed. 18 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 H0. 2. If P > α, then fail to reject H0. Larson/Farber 4th ed. 19 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 x • The standardized test statistic is z x standard error z x n n • When n ≥ 30, the sample standard deviation s can be substituted for σ. Larson/Farber 4th ed. 20 Using P-values for a z-Test for Mean μ In Words 1. State the claim mathematically and verbally. Identify the null and alternative hypotheses. In Symbols State H0 and Ha. 2. Specify the level of significance. Identify α. 3. Determine the standardized test statistic. z 4. Find the area that corresponds to z. Larson/Farber 4th ed. x n Use Table 4 in Appendix B. 21 Using P-values for a z-Test for Mean μ In Words In Symbols 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). Reject H0 if P-value 6. Make a decision to reject or is less than or equal fail to reject the null hypothesis. to α. Otherwise, fail to reject H0. 7. Interpret the decision in the context of the original claim. Larson/Farber 4th ed. 22 Example: Hypothesis Testing Using Pvalues 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. Larson/Farber 4th ed. 23 Solution: Hypothesis Testing Using Pvalues • • • • H0: μ ≥ 30 min Ha: μ < 30 min = 0.01 Test Statistic: z x n 28.5 30 3.5 36 2.57 Larson/Farber 4th ed. • P-value 0.0051 -2.57 0 z • Decision: 0.0051 < 0.01 Reject H0 At the 1% level of significance, you have sufficient evidence to conclude the mean delivery time is less than 30 minutes. 24 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 z0 separates the rejection region from the nonrejection region. Larson/Farber 4th ed. 25 Using Rejection Regions for a z-Test for a Mean μ In Words 1. State the claim mathematically and verbally. Identify the null and alternative hypotheses. 2. Specify the level of significance. In Symbols State H0 and Ha. Identify . 3. Sketch the sampling distribution. 4. Determine the critical value(s). Use Table 4 in Appendix B. 5. Determine the rejection region(s). Larson/Farber 4th ed. 26 Using Rejection Regions for a z-Test for a Mean μ In Words 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. Larson/Farber 4th ed. In Symbols x or if n 30 n use s. z If z is in the rejection region, reject H0. Otherwise, fail to reject H0. 27 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. Larson/Farber 4th ed. 28 Solution: Testing with Rejection Regions • • • • H0: μ ≥ $45,000 Ha: μ < $45,000 = 0.05 Rejection Region: • Test Statistic x 43, 500 45, 000 z n 5200 30 0.05 -1.645 0 -1.58 Larson/Farber 4th ed. z 1.58 • Decision: Fail to reject H0 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. 29 Section 7.3 Hypothesis Testing for the Mean (Small Samples) Larson/Farber 4th ed. 30 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. Larson/Farber 4th ed. 31 Example: Finding Critical Values for t Find the critical value t0 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 lefttailed, the critical value is negative. Larson/Farber 4th ed. 0.05 -1.725 0 t 32 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 x • The standardized test statistic is t. x t s n • The degrees of freedom are d.f. = n – 1. Larson/Farber 4th ed. 33 Using the t-Test for a Mean μ (Small Sample) In Words 1. State the claim mathematically and verbally. Identify the null and alternative hypotheses. In Symbols State H0 and Ha. 2. Specify the level of significance. Identify α. 3. Identify the degrees of freedom and sketch the sampling distribution. d.f. = n – 1. 4. Determine any critical value(s). Use Table 5 in Appendix B. Larson/Farber 4th ed. 34 Using the t-Test for a Mean μ (Small Sample) In Words 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. Larson/Farber 4th ed. In Symbols x t s n If t is in the rejection region, reject H0. Otherwise, fail to reject H0. 35 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) Larson/Farber 4th ed. 36 Solution: Testing μ with a Small Sample • • • • • H0: μ ≥ $23,900 Ha: μ < $23,900 α = 0.05 df = 14 – 1 = 13 Rejection Region: • Test Statistic: t -3.026 Larson/Farber 4th ed. s n 23, 000 23, 900 1113 14 3.026 • Decision: Reject H0 0.05 -1.771 0 x t 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 37 Section 7.4 Hypothesis Testing for Proportions Larson/Farber 4th ed. 38 z-Test for a Population Proportion 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 p̂ . • The standardized test statistic is z. z Larson/Farber 4th ed. pˆ pˆ pˆ pˆ p pq n 39 Using a z-Test for a Proportion p Verify that np ≥ 5 and nq ≥ 5 In Words 1. State the claim mathematically and verbally. Identify the null and alternative hypotheses. 2. Specify the level of significance. In Symbols State H0 and Ha. Identify α. 3. Sketch the sampling distribution. 4. Determine any critical value(s). Larson/Farber 4th ed. Use Table 5 in Appendix B. 40 Using a z-Test for a Proportion p In Words In Symbols 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. Larson/Farber 4th ed. p̂ p z pq n If z is in the rejection region, reject H0. Otherwise, fail to reject H0. 41 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 Larson/Farber 4th ed. 42 Solution: Hypothesis Test for Proportions • • • • • Test Statistic pˆ p 0.49 0.45 z pq n (0.45)(0.55) 200 H0: p = 0.45 Ha: p ≠ 0.45 = 0.05 Rejection Region: 0.025 -1.96 0.025 0 1.96 1.14 Larson/Farber 4th ed. z 1.14 • Decision: Fail to reject H0 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. 43