MTH 165 Comparing Population Parameters 1. Suppose a random sample of 200 dogs found 98 preferred chicken-based food pellets versus beef-based food pellets. A random sample of 150 cats found 61 prefer chicken to beef. a. Explain in general terms why we should not conclude the proportion of dogs who prefer chicken is greater than that of cats by simply comparing the 49% we found in the sample of dogs to the 40.67% in the sample of cats. At the 5% level of significance can we conclude the proportion of women who prefer name-brand products is greater than that of men? Explain carefully the meaning of the P-value in this context. What is a Type I error in the context of the test? What is a Type II error in the context of the test? Estimate the difference of the population proportions with 90% confidence. What does it mean to be 90% confident in this context? If 0 was in the interval estimate, what does this suggest may be true about the population proportions? b. c. d. e. f. g. 2. A random sample of 12 students that completed an SAT review course had a mean SAT score of 725 with standard deviation 50; a random sample of 10 students not enrolled in the course had a mean SAT score of 680 with standard deviation 40. Assume the scores of all students who complete the review and who do not complete the review follow normal distributions. a. Does this provide good evidence that students who complete the review course score better than their peers who do not complete the course? Carefully explain what the P-value means in the context of this test. What is a Type I error in the context of this test? What is a Type II error in the context of this test? b. c. d. 3. A random sample of 50 autoworkers showed they worked an average of 42 hours per week with standard deviation 6 hours. A random sample of 30 insurance representatives showed they work an average of 38 hours per week with standard deviation 8 hours. a. Calculate a 95% confidence interval for the mean difference of hours worked by all autoworkers and insurance representatives. Does the interval suggest one population works more hours per week on average than the other? Which one? Explain. What was the margin of error associated with your estimate? Explain what the MOE tells us about the process. b. c. 4. A manufacturer proposes installing new machines in a factory that produces widgets for ACME, Inc. ACME would like to know if the new machines are more effective than the ones currently in use. The following table gives the number of items made in one hour by seven randomly selected workers on the two different machines. Worker Current equipment New equipment a. b. c. 1 15 16 2 18 20 3 14 13 4 20 23 5 16 19 6 18 18 7 21 20 Do the data suggest the employees are more efficient when using the new machine? In other words, do they produce more units on average each hour when using the new equipment? What is the meaning of the P-value in the context of this test? The new equipment is a more costly machine to operate than the previous equipment, but it would produce a greater profit if the employees produce more units each hour. Discuss the practical implications for the company if it commits a Type I error when testing the claim in part (a). 1 5. In a test of braking performance, a tire manufacturer measured the stopping distance for one its tire models. A car made repeated stops from a speed of 60 mph. The test was conducted on two different conditionsdry and wet pavement. Below are the data showing the stopping distances (ft) for 10 trials under both conditions: Dry Wet 145 211 152 191 141 220 143 207 131 198 148 208 126 206 140 177 135 183 133 223 a. b. c. Do the data suggest the average stopping distance increases on wet pavement? Explain the meaning of a Type II error in the context of this study. Estimate the actual increase in the average stopping distance on wet pavement. 6. Suppose a random sample of 120 registered Democrats found 82 cried on election night. Whereas, a random sample of 100 registered Republicans found 61 cried on election night. a. Do the data provide significant evidence that the proportion of Democrats and the proportion of Republicans who cried on election night were different? Test at the 5% level of significance. What would mean to commit a Type I error in the context of this test? Does a 95% confidence interval for the difference of the population proportions confirm your conclusion in part (a)? Explain. b. c. 7. It is a common belief that using premium gasoline over regular gasoline will improve fuel efficiency. A company tests this belief by using 10 cars from its fleet. Each car is filled first with either regular or premium gasoline, decided by a coin toss, and the mileage for that thankful is recorded. Then the mileage is recorded again for the same cars for a thankful of the other grade of gasoline. Car # Regular Premium a. b. c. d. 1 16 19 2 20 22 3 21 24 4 22 24 5 23 25 6 22 25 7 27 26 8 25 26 9 27 28 10 28 32 Is there evidence that cars get significantly better fuel economy with premium gasoline? Test at the 2.5% level of significance. Explain the meaning of a Type I error in the context of the study. What might be the result to the company of committing a Type I error? Even if the company believes there is a significant increase in fuel efficiency, why might it decide to stick with regular gasoline? (HINT: Consider a 95% confidence interval for the difference of the means) Suppose a researcher at the company had mistakenly treated the data as two independent samples. Would the results of the study be different? Explain. 2