Math 10 - Homework 6 – Answers to homework
1. What are the two types of hypotheses used in a hypothesis test? How are they related? Ho: Null Hypotheses – A statement about a population parameter that is assumed to be true for the purposes of testing Ha: Alternative Hypothesis - A statement about a population parameter that is assumed to be true is the Null Hypothesis is rejected during testing. These two Hypotheses are complements of each other. 2. Describe the two types of error possible in a hypothesis test decision. Type I error: Rejecting a true Ho Type II error: Failing to reject a false Ho True or False? In Exercises 3-8, determine whether the statement is true of false. If it is false, rewrite it as a true statement. 3. In a hypothesis test, you assume the alternative hypothesis is true. False, you assume the Null Hypothesis is true. 4. A statistical hypothesis is a statement about a sample. False, it is a statement about a population parameter. 5. If you decide to reject the null hypothesis, you can support the alternative hypothesis. True 6. The level of significance is the maximum probability you allow for rejecting a null hypothesis when it is actually true. True 7. A large P-value in a test will favor a rejection of the null hypothesis. False, a small p-value supports rejecting the null hypothesis. 8. If you want to support a claim, write it as your null hypothesis. False, to support a claim write it as the alternative hypothesis. Stating Hypotheses In Exercises 9-14, use the given statement to represent a claim. Write its complement and state which is Ho and which is Ha. 9. Ha: p >.65 Ho: pζ65 10. Ho: ζ 128 Ha: >128 11. Ha: 2 ≠ 5 Ho: 2 = 5 12. Ho: =1.2 Ha: ≠1.2 13. Ho: p η 0.45 Ha: p < 0.45 14. Ha: < 0.21 Ho: η 0.21 Think about the context of the claim. Determine whether you want to support or reject the claim. a. State the null and alternative hypotheses in words. b. Write the null and alternative hypotheses in appropriate symbols c. Describe in words Type I error (the consequence of rejecting a true null hypothesis.) d. Describe in words Type II error (the consequence of failing to reject a false null hypothesis.) 15. You represent a chemical company that is being sued for paint damage to automobiles. You want to support the claim that the mean repair cost per automobile is about $650. How would you write the null and alternative hypotheses? Ho: =650 (cost is $650) Ha: ≠ͷ0 (Cost is not $650) Type I Error – Claim cost is not $650, when it actually is $650 Type II Error – Cost is not $650, but fail to reject the claim that is $650. 16. You are on a research team that is investigating the mean temperature of adult humans. The commonly accepted claim is that the mean temperature is about 98.6°F. You want to show that this claim is false. How would you write the null and alternative hypotheses?= Ho: =98.6 (Normal Temp is 98.6F) Ha: ≠98.6 (Normal Temp is not 98.6F) Type I Error – Claim normal temperature is not 98.6F, when it actually is 98.6F Type II Error – Normal Temperature is not 98.6F, but fail to detect that. 17. A light bulb manufacturer claims that the mean life of a certain type of light bulb is at least 750 hours. You are skeptical of this claim and want to refute it. Ho: η50 (Bulbs last at least 750 hours) Ha: <750 (Bulbs last less than 750 hours) Type I Error – Incorrectly claim light bulbs last less than 750 hours Type II Error – Fail to detect that light bulbs last less than 750 hours 18. As stated by a company's shipping department, the number of shipping errors per million shipments has a standard deviation that is less than 3. Can you support this claim? Ho: η3 (Standard Deviation of shipping errors is at 3) Ha: <3 (Standard Deviation of errors is under 3) Type I Error – Incorrectly claim Std Dev of Shipping errors is under 3 Type II Error – Fail to detect that Std Dev of errors is under 3 19. A research organization reports that 33% of the residents in Ann Arbor, Michigan are college students. You want to reject this claim. Ho: p=0.33 (33% of residents are college students) Ha: p ≠0.33 (It’s not true 33% of residents are students) Type I Error – Incorrectly claim percentage of residents who are college students is not33% Type II Error – Fail to detect that residents who are college students is not33% 20. The results of a recent study show that the proportion of people in the western United States who use seat belts when riding in a car or truck is under 84%. You want to support this claim. Ho: pη0.84 (At least 84% of west people use seat belts) Ha: p <0.84 (Less than 84% use seat belts) Type I Error – Incorrectly claim less than 84% of people in west use seat belts Type II Error – Fail to detect that less than 84% of people in west use seat belts PART B – Hypothesis Testing Procedure 21. In your work for a national health organization, you are asked to monitor the amount of sodium in a certain brand of cereal. You find that a random sample of 82 cereal servings has a mean sodium content of 232 milligrams with a standard deviation of 10 milligrams. At = 0.01 , can you conclude that the mean sodium content per serving of cereal is more than 230 milligrams?