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2018. 7. 25. Homework 7 Chapter 7-YoonJee Choi Instructor: Norly GERMAIN Course: MATH2001: Statistics PB1 40427 Student: YoonJee Choi Date: 07/25/18 Assignment: Homework 7 Chapter 7 1. What are the two types of hypotheses used in a hypothesis test? How are they related? What are the two types of hypotheses used in a hypothesis test? left­tailed and right­tailed null and alternative type I and type II population and sample How are they related? One is a subset of the other. They are complements. They sum to zero. They are equal. 2. What are the two decisions that you can make from performing a hypothesis test? What are the two decisions that you can make from performing a hypothesis test? Select all that apply. A. accept the alternative hypothesis B. reject the alternative hypothesis C. reject the null hypothesis D. accept the null hypothesis E. make a type II error F. G. fail to reject the null hypothesis H. make a type I error fail to reject the alternative hypothesis 3. Use the given statement to represent a claim. Write its complement and state which is H0 and which is Ha. μ ≥ 464 Find the complement of the claim. μ (1) 464 Which is H0 and which is Ha? (1) A. H0 : μ ≤ 464 Ha: μ ≥ 464 B. H0 : μ ≥ 464 Ha: μ = 464 C. H0 : μ ≥ 464 Ha: μ < 464 D. H0 : μ ≥ 464 Ha: μ ≠ 464 E. H0 : μ = 464 Ha: μ ≥ 464 F. G. H0 : μ < 464 Ha: μ ≥ 464 H. H0 : μ ≥ 464 Ha: μ > 464 I. ≠ = < ≤ H0 : μ ≥ 464 Ha: μ ≤ 464 H0 : μ > 464 Ha: μ ≥ 464 ≥ > https://xlitemprod.pearsoncmg.com/api/v1/print/math 1/19 2018. 7. 25. Homework 7 Chapter 7-YoonJee Choi 4. Use the given statement to represent a claim. Write its complement and state which is H0 and which is Ha . p > 0.45 Find the complement of the claim. p (1) 0.45 Which is H0 and which is Ha? A. H0 : p > 0.45 B. H0 : p > 0.45 C. H0 : p ≤ 0.45 Ha : p ≤ 0.45 Ha : p ≥ 0.45 Ha : p > 0.45 D. H0 : p > 0.45 Ha : p ≠ 0.45 E. H0 : p > 0.45 Ha : p > 0.45 F. G. H0 : p < 0.45 Ha : p > 0.45 H. H0 : p > 0.45 Ha : p < 0.45 I. (1) < H0 : p ≥ 0.45 Ha : p > 0.45 H0 : p ≠ 0.45 Ha : p > 0.45 ≤ > ≥ ≠ 5. A null and alternative hypothesis are given. Determine whether the hypothesis test is left­tailed, right­tailed, or two­tailed. H0 : p = 0.7 Ha : p ≠ 0.7 What type of test is being conducted in this problem? A. Left­tailed test B. Right­tailed test C. Two­tailed test 6. A null and alternative hypothesis are given. Determine whether the hypothesis test is left­tailed, right­tailed, or two­tailed. H0 : σ = 5 Ha : σ ≠ 5 What type of test is being conducted in this problem? A. Left­tailed test B. Two­tailed test C. Right­tailed test https://xlitemprod.pearsoncmg.com/api/v1/print/math 2/19 2018. 7. 25. Homework 7 Chapter 7-YoonJee Choi 7. Find the P­value for a left­tailed hypothesis test with a test statistic of z = − 1.08. Decide whether to reject H0 if the level of significance is α = 0.05. P­value = (Round to four decimal places as needed.) State your conclusion. Choose the correct answer below. Since P > α, reject H0 . Since P ≤ α, reject H0 . Since P ≤ α, fail to reject H0 . Since P > α, fail to reject H0 . 8. Find the P­value for the indicated hypothesis test with the given standardized test statistic, z. Decide whether to reject H0 for the given level of significance α. Right­tailed test with test statistic z = 1.14 and α = 0.05 P­value = (Round to four decimal places as needed.) State your conclusion. Fail to reject H0 Reject H0 9. Find the P­value for the indicated hypothesis test with the given standardized test statistic, z. Decide whether to reject H0 for the given level of significance α. Two­tailed test with test statistic z = − 2.23 and α = 0.04 P­value = (Round to four decimal places as needed.) State your conclusion. Reject H0 Fail to reject H0 10. Find the critical value(s) for a left­tailed z­test with α = 0.07. Include a graph with your answer. The critical value(s) is(are) . (Round to two decimal places as needed. Use a comma to separate answers as needed.) Draw a graph of the rejection region. Choose the correct graph below. A. B. C. D. α 1 2 1 α 2 α α α z ­3 0 3 z ­3 https://xlitemprod.pearsoncmg.com/api/v1/print/math 0 3 z ­3 0 3 z ­3 0 3 3/19 2018. 7. 25. Homework 7 Chapter 7-YoonJee Choi 11. Find the critical value(s) and rejection region(s) for the type of z­test with level of significance α. Include a graph with your answer. Right­tailed test, α = 0.04 The critical value(s) is/are z = . (Round to two decimal places as needed. Use a comma to separate answers as needed.) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. (Round to two decimal places as needed.) A. The rejection region is z > . B. The rejection region is z < . C. The rejection regions are z < and z > . Choose the correct graph of the rejection region below. A. B. 0 z C. −z 0 D. z 0 z −z 0 z 12. Find the critical value(s) and rejection region(s) for the type of z­test with level of significance α. Include a graph with your answer. Two­tailed test, α = 0.07 The critical value(s) is/are z = (Round to two decimal places as needed. Use a comma to separate answers as needed.) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. (Round to two decimal places as needed.) A. The rejection regions are z < and z > B. The rejection region is z < . C. The rejection region is z > . . Choose the correct graph of the rejection region below. A. B. −z 0 z https://xlitemprod.pearsoncmg.com/api/v1/print/math C. z 0 D. −z 0 z 0 z 4/19 2018. 7. 25. Homework 7 Chapter 7-YoonJee Choi 13. State whether the standardized test statistic z indicates that you should reject the null hypothesis. (a) z = 2.544 (b) z = 2.636 (c) z = − 2.381 (d) z = − 2.675 z ­4 − z0 = − 2.575 0 z0 = 2.575 4 (a) For z = 2.544, should you reject or fail to reject the null hypothesis? A. Fail to reject H0 because − 2.575 < z < 2.575. B. Reject H0 because z > 2.575. C. Reject H0 because − 2.575 < z < 2.575. D. Fail to reject H0 because z > 2.575. (b) For z = 2.636, should you reject or fail to reject the null hypothesis? A. Fail to reject H0 because z > 2.575. B. Reject H0 because − 2.575 < z < 2.575. C. Fail to reject H0 because − 2.575 < z < 2.575. D. Reject H0 because z > 2.575. (c) For z = − 2.381, should you reject or fail to reject the null hypothesis? A. Fail to reject H0 because z < − 2.575. B. Reject H0 because z < − 2.575. C. Reject H0 because − 2.575 < z < 2.575. D. Fail to reject H0 because − 2.575 < z < 2.575. (d) For z = − 2.675, should you reject or fail to reject the null hypothesis? A. Reject H0 because − 2.575 < z < 2.575. B. Reject H0 because z < − 2.575. C. Fail to reject H0 because z < − 2.575. D. Fail to reject H0 because − 2.575 < z < 2.575. https://xlitemprod.pearsoncmg.com/api/v1/print/math 5/19 2018. 7. 25. Homework 7 Chapter 7-YoonJee Choi 14. Test the claim about the population mean, μ, at the given level of significance using the given sample statistics. Claim: μ = 50; α = 0.07; σ = 3.46. Sample statistics: x = 49.7, n = 69 Identify the null and alternative hypotheses. Choose the correct answer below. A. H0 : μ < 50 Ha : μ = 50 B. H0 : μ = 50 Ha : μ > 50 C. H0 : μ > 50 Ha : μ = 50 D. H0 : μ = 50 Ha : μ ≠ 50 E. H0 : μ ≠ 50 Ha : μ = 50 F. H0 : μ = 50 Ha : μ < 50 Calculate the standardized test statistic. The standardized test statistic is (Round to two decimal places as needed.) . Determine the critical value(s). Select the correct choice below and fill in the answer box to complete your choice. (Round to two decimal places as needed.) A. The critical value is B. The critical values are ± . . Determine the outcome and conclusion of the test. Choose the correct answer below. A. Fail to reject H0 . At the 7% significance level, there is not enough evidence to support the claim. B. Fail to reject H0 . At the 7% significance level, there is not enough evidence to reject the claim. C. Reject H0 . At the 7% significance level, there is enough evidence to reject the claim. D. Reject H0 . At the 7% significance level, there is enough evidence to support the claim. https://xlitemprod.pearsoncmg.com/api/v1/print/math 6/19 2018. 7. 25. Homework 7 Chapter 7-YoonJee Choi 15. Use technology to help you test the claim about the population mean, μ, at the given level of significance, α, using the given sample statistics. Assume the population is normally distributed. Claim: μ ≤ 1240; α = 0.09; σ = 197.94. Sample statistics: x = 1259.62, n = 300 Identify the null and alternative hypotheses. Choose the correct answer below. A. H0 : μ > 1259.62 Ha : μ ≤ 1259.62 B. H0 : μ > 1240 Ha : μ ≤ 1240 C. H0 : μ ≤ 1240 Ha : μ > 1240 D. H0 : μ ≥ 1240 Ha : μ < 1240 E. H0 : μ ≥ 1259.62 F. H0 : μ ≤ 1259.62 Ha : μ < 1259.62 Ha : μ > 1259.62 Calculate the standardized test statistic. The standardized test statistic is (Round to two decimal places as needed.) . Determine the P­value. P= (Round to three decimal places as needed.) Determine the outcome and conclusion of the test. (1) H0 . At the 9% significance level, there (2) (3) the claim. (1) Fail to reject (2) Reject https://xlitemprod.pearsoncmg.com/api/v1/print/math is not is (3) enough evidence to support reject 7/19 2018. 7. 25. Homework 7 Chapter 7-YoonJee Choi 16. A light bulb manufacturer guarantees that the mean life of a certain type of light bulb is at least 769 hours. A random sample of 30 light bulbs has a mean life of 759 hours. Assume the population is normally distributed and the population standard deviation is 56 hours. At α = 0.08, do you have enough evidence to reject the manufacturer's claim? Complete parts (a) through (e). (a) Identify the null hypothesis and alternative hypothesis. A. H0 : μ > 769 Ha : μ ≤ 769 (claim) B. H0 : μ ≤ 759 Ha : μ > 759 (claim) C. H0 : μ ≠ 769(claim) Ha : μ = 769 D. H0 : μ < 759 (claim) E. H0 : μ ≥ 769 (claim) F. H0 : μ = 759 Ha : μ ≥ 759 Ha : μ < 769 Ha : μ ≠ 759 (claim) (b) Identify the critical value(s). Use technology. z0 = (Use a comma to separate answers as needed. Round to two decimal places as needed.) Identify the rejection region(s). Choose the correct answer below. A. B. Fail to reject H0 . Reject H0 . C. Fail to reject H0 . Reject H0 . Fail to reject H0 . Reject H0 . Reject H0 . z ­4 0 4 z ­4 0 4 z ­4 0 4 (c) Identify the standardized test statistic. Use technology. z= (Round to two decimal places as needed.) (d) Decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. A. Reject H0 . There is sufficient evidence to reject the claim that mean bulb life is at least 769 hours. C. Fail to reject H0 . There is sufficient evidence to reject the claim that mean bulb life is at least 769 hours. https://xlitemprod.pearsoncmg.com/api/v1/print/math B. Fail to reject H0 . There is not sufficient evidence to reject the claim that mean bulb life is at least 769 hours. D. Reject H0 . There is not sufficient evidence to reject the claim that mean bulb life is at least 769 hours. 8/19 2018. 7. 25. Homework 7 Chapter 7-YoonJee Choi 17. A scientist estimates that the mean nitrogen dioxide level in a city is greater than 29 parts per billion. To test this estimate, you determine the nitrogen dioxide 43 26 22 36 31 36 20 35 44 34 37 levels for 31 randomly selected days. The results (in parts per billion) are listed 19 17 17 25 42 24 27 32 42 42 23 to the right. Assume that the population standard deviation is 11. At 28 41 14 18 37 21 41 28 35 α = 0.06, can you support the scientist's estimate? Complete parts (a) through (e). (a) Write the claim mathematically and identify H0 and Ha . Choose from the following. A. H0 : μ = 29 (claim) Ha : μ > 29 B. H0 : μ ≤ 29 (claim) Ha : μ > 29 C. H0 : μ < 29 Ha : μ ≥ 29 (claim) D. H0 : μ ≤ 29 Ha : μ > 29 (claim) E. H0 : μ ≥ 29 (claim) Ha : μ < 29 F. H0 : μ = 29 Ha : μ > 29 (claim) (b) Find the critical value and identify the rejection region. z0 = (Round to two decimal places as needed.) Rejection region: z (1) (c) Find the standardized test statistic. z= (Round to two decimal places as needed.) (d) Decide whether to reject or fail to reject the null hypothesis. Fail to reject H0 Reject H0 (e) Interpret the decision in the context of the original claim. At the 6% significance level, there (2) enough evidence to (3) that the mean nitrogen dioxide level in the city is greater than 29 parts per billion. (1) > < (2) is not is https://xlitemprod.pearsoncmg.com/api/v1/print/math (3) the scientist's claim support reject 9/19 2018. 7. 25. Homework 7 Chapter 7-YoonJee Choi 18. Find the critical value(s) and rejection region(s) for the indicated t­test, level of significance α, and sample size n. Left­tailed test, α = 0.005, n = 17 1 Click the icon to view the t­distribution table. The critical value(s) is/are . (Round to the nearest thousandth as needed. Use a comma to separate answers as needed.) Determine the rejection region(s). Select the correct choice below and fill in the answer box(es) within your choice. (Round to the nearest thousandth as needed.) A. t < C. and t > 1: t-Distribution Table https://xlitemprod.pearsoncmg.com/api/v1/print/math 10/19 2018. 7. 25. https://xlitemprod.pearsoncmg.com/api/v1/print/math Homework 7 Chapter 7-YoonJee Choi 11/19 2018. 7. 25. Homework 7 Chapter 7-YoonJee Choi 19. Find the critical value(s) and rejection region(s) for the indicated t­test, level of significance α, and sample size n. Two­tailed test, α = 0.02, n = 14 2 Click the icon to view the t­distribution table. The critical value(s) is/are . (Round to the nearest thousandth as needed. Use a comma to separate answers as needed.) Determine the rejection region(s). Select the correct choice below and fill in the answer box(es) within your choice. (Round to the nearest thousandth as needed.) A. t < C. t < B. and t > 2: t-Distribution Table https://xlitemprod.pearsoncmg.com/api/v1/print/math 12/19 2018. 7. 25. https://xlitemprod.pearsoncmg.com/api/v1/print/math Homework 7 Chapter 7-YoonJee Choi 13/19 2018. 7. 25. Homework 7 Chapter 7-YoonJee Choi 20. State whether the standardized test statistic t indicates that you should reject the null hypothesis. Explain. (a) t = 1.537 (b) t = 0 (c) t = − 1.403 (d) t = − 1.525 t ­4 0 4 t0 = − 1.489 (a) For t = 1.537, should you reject or fail to reject the null hypothesis? A. Fail to reject H0 , because t > − 1.489. B. Reject H0 , because t > − 1.489. C. Fail to reject H0 , because t < − 1.489. D. Reject H0 , because t < − 1.489. (b) For t = 0, should you reject or fail to reject the null hypothesis? A. Fail to reject H0 , because t < − 1.489. B. Reject H0 , because t > − 1.489. C. Fail to reject H0 , because t > − 1.489. D. Reject H0 , because t < − 1.489. (c) For t = − 1.403, should you reject or fail to reject the null hypothesis? A. Fail to reject H0 , because t < − 1.489. B. Reject H0 , because t < − 1.489. C. Reject H0 , because t > − 1.489. D. Fail to reject H0 , because t > − 1.489. (d) For t = − 1.525, should you reject or fail to reject the null hypothesis? A. Reject H0 , because t > − 1.489. B. Fail to reject H0 , because t > − 1.489. C. Fail to reject H0 , because t < − 1.489. D. Reject H0 , because t < − 1.489. https://xlitemprod.pearsoncmg.com/api/v1/print/math 14/19 2018. 7. 25. Homework 7 Chapter 7-YoonJee Choi 21. State whether the standardized test statistic t indicates that you should reject the null hypothesis. Explain. (a) t = − 1.744 (b) t = 1.619 (c) t = 1.765 (d) t = − 1.741 t ­4 0 − t0 = − 1.672 4 t0 = 1.672 (a) For t = − 1.744, should you reject or fail to reject the null hypothesis? A. Reject H0 , because − 1.672 < t < 1.672. B. Fail to reject H0 , because t > 1.672. C. Fail to reject H0 , because t < − 1.672. D. Reject H0 , because t < − 1.672. (b) For t = 1.619, should you reject or fail to reject the null hypothesis? A. Reject H0 , because t < − 1.672. B. Fail to reject H0 , because − 1.672 < t < 1.672. C. Fail to reject H0 , because t > 1.672. D. Reject H0 , because − 1.672 < t < 1.672. (c) For t = 1.765, should you reject or fail to reject the null hypothesis? A. Fail to reject H0 , because t > 1.672. B. Reject H0 , because − 1.672 < t < 1.672. C. Reject H0 , because t > 1.672. D. Fail to reject H0 , because t < − 1.672. (d) For t = − 1.741, should you reject or fail to reject the null hypothesis? A. Reject H0 , because − 1.672 < t < 1.672. B. Fail to reject H0 , because t < − 1.672. C. Reject H0 , because t < − 1.672. D. Fail to reject H0 , because t > 1.672. https://xlitemprod.pearsoncmg.com/api/v1/print/math 15/19 2018. 7. 25. Homework 7 Chapter 7-YoonJee Choi 22. Use a t­test to test the claim about the population mean μ at the given level of significance α using the given sample statistics. Assume the population is normally distributed. Claim: μ = 52,400; α = 0.01 3 Sample statistics: x = 52,167, s = 1800, n = 17 Click the icon to view the t­distribution table. What are the null and alternative hypotheses? Choose the correct answer below. A. H0: μ ≥ 52,400 Ha: μ < 52,400 B. H0: μ = 52,400 Ha: μ ≠ 52,400 C. H0: μ ≠ 52,400 Ha: μ = 52,400 D. H0: μ ≤ 52,400 Ha: μ > 52,400 What is the value of the standardized test statistic? The standardized test statistic is . (Round to two decimal places as needed.) What is(are) the critical value(s)? The critical value(s) is(are) . (Round to three decimal places as needed. Use a comma to separate answers as needed.) Decide whether to reject or fail to reject the null hypothesis. A. Reject H0 . There is not enough evidence to reject the claim. B. Fail to reject H0 . There is enough evidence to reject the claim. C. Fail to reject H0 . There is not enough evidence to reject the claim. D. Reject H0 . There is enough evidence to reject the claim. 3: t-Distribution Table https://xlitemprod.pearsoncmg.com/api/v1/print/math 16/19 2018. 7. 25. https://xlitemprod.pearsoncmg.com/api/v1/print/math Homework 7 Chapter 7-YoonJee Choi 17/19 2018. 7. 25. Homework 7 Chapter 7-YoonJee Choi 23. An employment information service claims the mean annual salary for senior level product engineers is $97,000. The annual salaries (in dollars) for a random sample of 16 senior level product engineers are shown in the table to the right. At α = 0.05, test the claim that the mean salary is $97,000. Complete parts (a) through (e) below. Assume the population is normally distributed. Annual Salaries 96,237 93,507 74,165 76,995 76,217 103,919 82,061 85,076 100,618 82,475 102,436 91,074 112,793 80,961 103,997 110,406 (a) Identify the claim and state H0 and Ha . H0 : (1) (2) Ha : (3) (4) (Type integers or decimals. Do not round.) The claim is the (5) hypothesis. (b) Use technology to find the critical value(s) and identify the rejection region(s). The critical value(s) is/are t0 = . (Use a comma to separate answers as needed. Round to two decimal places as needed.) Choose the graph which shows the rejection region. A. B. t < − t0 , t > t0 C. t > t0 − t0 < t < t0 t ­4 − t0 0 t0 D. t < t0 t 4 ­4 0 t0 t 4 ­4 − t0 0 t0 t 4 ­4 t0 0 4 (c) Find the standardized test statistic, t. The standardized test statistic is t = . (Use a comma to separate answers as needed. Round to two decimal places as needed.) (d) Decide whether to reject or fail to reject the null hypothesis. (6) H0 because the standardized test statistic (7) in the rejection region. (e) Interpret the decision in the context of the original claim. There (8) enough evidence at the % level of significance to (9) claim that the mean annual salary for senior level product engineers is (10) integers or decimals. Do not round.) (1) = ≥ > ≤ p ≠ σ (6) (2) μ σ 2 (3) (7) Reject https://xlitemprod.pearsoncmg.com/api/v1/print/math > ≥ p < ≠ μ ≤ σ < Fail to reject σ is not is (4) (8) $ 2 (5) the . (Type alternative null = is not is (9) reject support 18/19 2018. 7. 25. (10) Homework 7 Chapter 7-YoonJee Choi not equal to equal to greater than or equal to less than or equal to less than greater than https://xlitemprod.pearsoncmg.com/api/v1/print/math 19/19
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