"Suggested Problem Set 3"
Suggested Problem Set 3 Instructions: 1) This Problem set covers Ch19-Ch21. It contains 14 exercise questions. 2) It is the preparation material for Mid-term 2 and Final Exam 3) Showing your work to the instructor in or before the lecture class on Aug 3rd (Fri) will help you to earn up to 5 points extra credit which can be added to your total score of 300 points 4) The answer key will be posted on the class page after the lecture class on Aug 3rd (Fri) 5) Before the answer key is posted, you are always welcomed to discuss your work with the instructor. 1. (Ch19) A TV talk show asks viewers to register their opinions on prayer in schools by logging on to a website. Of the 622 people who voted, 478 favored prayer in schools. We want to estimate the level of support among the general public. Please construct a 95% confidence interval for the population proportion of support a) What is the critical value you are going to use? b) Please construct a 95% confidence interval for the population proportion of support and interpret it. c) If we want to control the margin of the error of the 95% confidence interval to be within 0.01, how large a sample do we need? Use the original sample as a pilot study. 2. (Ch19) A consumer group hoping to assess customer experiences with auto dealers surveys 167 people who recently bought new cars; 3% of them expressed dissatisfaction with the salesperson. We want to learn about the general opinion towards the salesperson. a) Please construct a 90% confidence interval for the population proportion of dissatisfaction and interpret it. b) If we want to control the margin of the error of the 90% confidence interval to be within 0.01, how large a sample do we need? Please use the conservative method. 3. (Ch19)A catalog sales company promises to deliver orders placed on the internet within 3 days. Follow-up calls to 113 randomly selected customers show that a one-sample z-confidence interval for the proportion of all orders that arrive on time is (0.82, 0.94). a) In the follow-up survey, what is the proportion of the sampled orders that arrive on time? b) What is the margin of error (ME) of the confidence interval? c) What is the confidence level of the confidence interval? d) If we decrease the confidence level, the width of the confidence interval will be increased or decreased? Why? e) Several people try to interpret the confidence interval from different points of view. Are their conclusions correct? Explain 1) Between 82% and 94% of all orders arrive on time 2) 95% of all random samples of customers will show that 88% of orders arrive on time 3) 95% of all random samples of customers will show that 82% to 94% of the sampled orders arrive on time 4) We are 95% confident that between 82% and 94% of the orders placed by the customers in this sample arrived on time 5) On 95% of the days, between 82% and 94% of the orders will arrive on time 6) Among 100 95% confidence intervals generated from different samples, there are exactly 95 of them will capture the true population proportion 7) We are 95% confident that between 82% and 94% of all orders arrive on time 4. (Ch19) Several factors are involved in the creation of a confidence interval. Among them are the sample size, the confidence level, and margin of error. Which statements are true? a) For a given sample size, higher confidence means a smaller margin of error. b) For a specified confidence level, larger samples provide smaller margin of error c) For a fixed margin of error, larger samples provide greater confidence d) For a given confidence level, halving the margin of error requires a sample twice as large e) For a given sample size, reducing the margin of error will cause lower confidence f) For a given sample size, Higher confidence means less precision g) Reducing the margin of error will cause less precision 5. (Ch19) A state’s environmental agency worries that many cars may be violating clean air emissions standards. The agency hopes to check a sample of vehicles in order to estimate that percentage with a margin of error of 3% and 90% confidence. To gauge the size of the problem, the agency first picks 60 cars and finds 9 with faulty emissions systems. How many should be sampled for a full investigation? 6. (Ch19) In preparing a report on the economy, we need to estimate the percentage of businesses that plan to hire additional employees in the next 60 days. a) How many randomly selected employers must we contact in order to create an estimate in which we are 98% confident with a margin of error of 5%? b) Suppose we want to reduce the margin of error to 3% (still 98% confidence level). What sample size will suffice? 7. (Ch20~21)Write the null and alternative hypotheses you would use to test each of the following situations. And also identify whether the alternative hypothesis is one-sided or two sided. a) In the 1950s only about 40% of high school graduates went on to college. Has the percentage changed? b) 20% of cars of a certain model have needed costly transmission work after being driven between 50,000 and 100,000 miles. The manufacturer hopes that a redesign of a transmission component has solved this problem c) We field-test a new-flavor soft drink, planning to market it only if we are sure that over 60% of the people like the flavor. d) In recent years, 10% of college juniors have applied for study abroad. The dean’s office conducts a survey to see if that’s increased this year. e) A pharmaceutical company conducts a clinical trial to see if less patients who take a new drug experience headache relief than the 22%who claimed relief after taking the placebo f) Is a coin fair? 8. (Ch20~21)After the political ad campaign, pollsters check the governor’s negatives. They test the hypothesis that the ads produced no change against the alternative that the negatives are now below 30% and find a P-value of 0.22. Which conclusion is appropriate? Explain. a) There is a 22% chance that the ads worked b) There is a 78% chance that the ads worked c) There is a 22% chance that the poll they conducted is correct d) There is a 22% chance that natural sampling variation could produce poll results like these (or more extreme) if there is really a change in public opinion e) There is a 22% chance that natural sampling variation could produce poll results like these (or more extreme) if there’s really no change in public opinion 9. (Ch20~21)In Nov 2001, the Ag Globe Trotter newsletter reported that 90% of adults drink milk. A regional farmers’ organization planning a new marketing campaign across its multicounty area polls a random sample of 750 adults living there. In this sample, 657 people said that they drink milk. Do these responses provide strong evidence that the 90% figure is not accurate for this region? Correct the mistakes you find in a student’s attempt to test an appropriate hypothesis with . A student’s attempt: H0 : HA : P-value=P(z>1.99)=0.0233 There is 2.33% chance that the stated percentage is correct for this region Given alpha level , we reject H0 at this level. And the test is not statistically significant at this level. 10.(Ch20~21)The National Center for Education Statistics monitors many aspects of elementary and secondary education nationwide. Their 1996 numbers are often used as a baseline to assess changes. In 1996, 31% of students reported that their mothers had graduated from college. In 2000, responses from 8368 students found that this figure had grown to 32%. Is this evidence of a change in education level among mothers? Please conduct an appropriate hypotheses testing. a) Write appropriate hypotheses b) Compute the test statistic c) Compute the P-value and explain what the P-value means in this context. d) Given =0.05, make the conclusion. What possible error could you make, Type I or type II? What is the probability of making a type I error? e) Given =0.01, make the conclusion. What possible error could you make, Type I or type II? What is the probability of making a type I error? f) Given =0.10, make the conclusion. What possible error could you make, Type I or type II? What is the probability of making a type I error? 11.(Ch20~21)A research wants to study the effect of the TV ads to the candidate’s supporting rate. Let p be the supporting rate. He wanted to see whether the supporting rate was increased or not after a TV ad was released. So he set the hypotheses as H0 : p=0.31 vs. HA: p>0.31. And he got a P-value= 0.035. Question: Given =0.05, are we going to reject H0 or fail to reject H0 when we do a test for H0 : p=0.31 vs. HA: p≠0.31? 12.(Ch20~21)True or false a) If we reject H0 at =0.01, we will also reject it when =0.05 b) If we fail to reject H0 at =0.05, we will also fail to reject it when =0.10 c) If the test is statistically significant at =0.01, then it will not be statistically significant when =0.10 d) If the test is statistically significant at =0.05, then it will also be statistically significant when =0.01 13.(Ch20~21)Production managers on an assembly line must monitor the output to be sure that the level of defective products remains small. They periodically inspect a random sample of the items produced. If they find a significant increase in the proportion of items that must be rejected, they will halt the assembly process until the problem can be identified and repaired. a) In this context, what is Type I error? b) In this context, what is Type II error? 14.(Ch20~21) A company with a fleet of 1500 cars found that the emission systems of 7 out of the 22 they tested failed to meet the pollution control guidelines. Is this strong evidence that more than 20% of the fleet might be out of compliance? Conduct a test with =0.05