STA 222Exercise 4 1. An electrical firm manufactures light bulbs that have a lifetime that is approximately normally distributed with a mean of 8000 hours and a standard deviation of 40 hours. Test the hypothesis that µ = 800 hours against the alternative µ ≠ 800 hours if a random sample of 30 bulbs has an average life of 788 hours. Use a 0.04 level of significance. (Z = -1.643 Do not reject H 0) A taxi driver claims to make an average of $12.00 on each fare, but the Taxation Office believes that average is higher than that. To test the driver’s claim, the Taxation Office takes a random sample of 30 fares. The amounts that the taxi driver made on the fare in the sample had a mean of $13.30 with a standard deviation of 2.50. Test the driver’s claim at α = 0.01. ( Z = 2.848 Reject H 0) In a research report by Richard H. Weindruch of the UCLA Medical School, it is claimed that mice with an average life span of 32 months will live to be about 40 months old when 40% of the calories in their food are replaced by vitamins and protein. Is there any reason to believe that the average life span is less than 40 months if 64 mice that are placed on this diet have an average life of 38 months with a standard deviation of 5.8 months? Use a 0.025 level of significance. (Z = -2.76 reject H0) A random sample of n = 2000 observations from a binomial population produced r = 1238 If your research hypothesis is that π is greater than 0.6, what should you choose for your alternative hypothesis? Your null hypothesis? Does your alternative hypothesis in part (a) imply a one- or two-tailed statistical test? Do the data provide sufficient evidence to indicate that π is greater than 0.60? Test using α = 0.05. ( Z = 1.734 Reject H 0) The service manager of an appliance sales company asserts that 6% of the appliances sold are returned to the service department for repair under the warranty, and the sales manager believes that this claim is too high. Test the service manager’s assertion at the 0.05 level of significance if 56 out of a random sample of 1000 appliance sales are returned to the service department for repair under the warranty. ( Z = -0.533 Do not reject H0) A publisher of a news magazine has found through past experience that 60% of its subscribers renew their subscriptions. Because it was heading into a business recession, the company decided to randomly select a small sample of subscribers and, via telephone questioning to determine whether they planned to renew their subscriptions. One hundred eight of a sample of 200 indicates that they planned to renew their subscriptions. If you want to detect whether the data provide sufficient evidence of a reduction in the proportion of all subscribers who will renew, what will you choose for your alternative hypothesis? Null hypothesis? Conduct the test using a 5% level of significance. State the results. (Z = -1.732 reject H0) How many subscribers would have to be included in the publisher’s sample in order π to estimate to within 0.01, with 95% confidence? (9220) 2. 3. 4. (a) (b) (c) 5. 6. (a) (b) (c) 752. A coin is thrown 500 times and 267 heads are obtained. Of these. 12 salespeople were selected at random.6 gallons with standard deviation of 1. With the currently marketed drug. 65% of users say that their headache is relieved by it. 75% said that their headache was relieved by the drug. a chain of discount furniture stores in Nilai. that the population mean is less than 750ml. Test whether the coin is unbiased.5. ( t = 1.5 gallons of water with standard deviation of 2. Test.112 Do not reject H 0) The management of Discount Furniture. designed an incentive plan for salespeople. ( Z =-1.0.521 do not reject H0) A drug company tested a new pain-relieving drug on a random sample of 100 headache suffers. If 35 randomly selected purchases made in station A average 9. 11. 36 gallons of water with standard deviation of 2.7. 35. test at the level of significance 0. sample data are collected which show that 50 showers taken by students in dormitory A used. on the average. at the 5% level. Her results are: 747. ( Z = 3. Test at α = 5%.01. 13. Test whether students in dormitory A use more water than dormitory B.3 gallons with a standard deviation of 2.0. ( Z = 1.5 gallons.130 Do not reject H0) Following is a random sample of grades achieved on a statistics examination by nine men students in a very large class and a random sample of grades achieved in the same examination by six women students: Men students: 79 88 64 91 83 66 89 74 68 Women students: 70 51 82 72 90 61 Use the level of significance of 1% to test whether the difference between the means of these two samples is significant. 10. and 51 randomly selected sales at station B average 8. at the 4% level. To evaluate this innovative plan. . using a 10% significance level.18. An inspector takes a random sample of six bottles of wine and determines the volumes of their contents.0 Determine whether these results provide significant evidence.5. Salesperso n Tan C W Lee M Y Khoo W L Leong S S Lee C C Chan K L Weekly income Before After Salesperso n RM320 RM340 Khoo K C 290 285 Chong C S 421 475 Chuah C S 510 510 Toh Y H 210 210 Kan K M 402 500 Mong Y K Weekly income Before After RM625 RM631 560 560 360 365 431 431 506 525 505 619 9. 748. 12. whether the difference between these two sample means is significant.06 reject H0) To collect information for a water conservation drive on a college campus. ( Z = 2. ( t = -1. on the average.3 gallons.75. 748. whether the new drug will have a greater proportion of satisfied users.041 do not reject H 0) Bottles of wine are supposed to contain 750ml of wine.097 reject H0) A study was made of the number of gallons of gasoline purchased by customers for their automobiles at two gasoline stations. correct to the nearest half milliliter. 8.0. while 50 showers taken by students in dormitory B used. and their weekly incomes before and after the plan were recorded. 751. 747. Test at a significance level of 3% that the true percentage of the helmets damaged is at most 30%.00075) Test at α = 10% that the true mean is 8. At 5% level of significance.1095 Do not reject H0) 18. the wear is measured on a ten-point scale (higher is better) with the following results. Eight persons randomly chosen used the product for 2 months. A machine produces metal rods used in an automobile suspension system. Pair number Old material New material 1 5 4 2 8 7 3 4 5 4 7 9 5 6 8 6 6 7 7 4 2 8 3 5 16. A random sample of eight pairs of shoes is selected for the test.25 8. and the diameter (in mm) of each rod is measured. A random sample of 9 rods is selected. Person 1 Final weight (lb) 146 Initial weight (lb) 150 .23 8. is there enough evidence to conclude that the average wear of the new material is better than that of the old material? (-0.33 1094. A new dietary product claims in its advertising that the use of the product for two months will result in an average weight loss of at least 5 pounds.20 8. the new material is placed on one shoe and the old material is placed on the other shoe.830 Reject H0) A random sample of 60 suspension helmets used by motorcycle riders was subjected to an impact test. at 5% significance level. (t = 2.20 t reject H0) 14. and the results were as following. 8.95 Control group 170 947 Can we conclude.Was there a significant increase in the salesperson’s income due to the innovative incentive plan? Use the 0. A training was conducted to determine whether 6 weeks of training is effective in reducing the cholesterol levels of the participants. and 15 of the helmets were damaged in the test.0.26 Find the unbiased estimates of the mean and variance of the diameters of the rod. (-0.20 8. All participants’ cholesterol level was recoded at the end of the 6 weeks and the results were as following: Mean Variance Treatment group 156. For each pair of shoes. ( 24. that the 6 weeks training program is effective in decreasing the cholesterol level of the participants? (-1. A treatment group consisting of 12 people was given lectures thrice a week on how to reduce their cholesterol level. ( 8.8819 Do not reject H0) 17. After a period of wear testing and mechanical abrasion.24 (i) (ii) 15. Another group of 15 people were randomly selected as a control group.845 Do not reject H0) A shoe company intends to test material for the soles shoes.2267.05 significance level to test the doubt.19 8. The results are: 8. 0.26 8.21 8. 66. of the 3 × 3 contingency table shown below. A random sample of 350 inexpensive electronic toys which are produced in Hong Kong. and Korea are examined by an importer in the United States to determine the quality of the toys. A total of 1000 automobiles were observed during the heavy early-morning traffic.327 Reject H0) 22. with the following results: Geographical location Hong Kong Japan Korea 104 64 79 29 17 24 A A1 A2 A3 Acceptable Imperfect. B1 40 63 31 B B2 72 53 38 B3 42 70 30 ( 12. The responses given in the table were obtained.2 193 198 3 161 165 4 183 187 5 195 201 6 141 143 7 195 201 8 150 156 Can we conclude that the claim of the advertisement is false at 1% significance level? (-19.05. Reject H 0) 21. and Japan? Test using α = 0.05. One of the many questions put to a sample of n = 100 entrepreneurs about their job characteristics work habits. and loners whose rough edges and uncompromising need to do it their own way set them in sharp contrast to senior executives in major American corporations’. Japan. Europe. social activities. etc.05. The results were as follows: Lane 1 2 3 4 Observed count 294 276 238 192 Do the data present sufficient evidence to indicate that some lanes are preferred over others? Test using α = 0.48. Test the null hypothesis of independence of the two classifications A and B. and their respective lanes were recorded. United States 45 Europe 46 Japan 9 Do these data provide evidence of a difference in the preference of entrepreneurs for the cars of the United States. concerned the origin of the car they personally drive most frequently. A 1985 Gallup survey portrays United States entrepreneurs as ‘……the mavericks dreamers. ( 26. ( 24. but salable . A city expressway using four lanes in each direction was studied to see whether drivers preferred to drive on the inside lanes.337 Reject H0) 19. Test using α = 0. Reject H 0) 20. which has 5 lanes after the tollgate. The results were as follows: Lane Observed count 1 96 2 154 3 275 4 5 225 171 Do the data provide sufficient evidence at 5% level of significance to indicate that some lanes are preferred over others? ( 101. Reject H0) . Subang Jaya. and the number of cars on respective lanes were recorded. ( 1.922. A tile company was interested in comparing the fraction of new house builders favoring three types of tiles as floor coverings for their houses in three different areas of Klang Valley. A total of 921 automobiles were observed during the early morning traffic.93. The LDP Expressway. Use a 5% significance level. Do not reject H0) 26. A survey was conducted and the data were as follows: Floor Coverings Type I Type II Type III Subang Jaya 224 196 80 Area Puchong 165 152 83 Petaling Jaya 36 44 20 Test at 5% level of significance whether there is any association between types of tiles used and the areas concerned. the following data were taken on 180 individuals: Hypertension No hypertension Nonsmoker 15 43 Moderate smoker 36 27 Heavy smoker 35 24 Test the hypothesis that the presence or absence of hypertension is independent of smoking habits.41. (5.e. ( 16. i. Do not reject H 0) 23. A die is tossed 180 times with the following results: Number Frequency 1 27 2 36 3 36 4 31 5 28 6 22 Is this a balanced die? Conduct a test of goodness-of-fit at 1% significance level. Do not reject H0) 25. was studied to see whether drivers preferred to drive on the inside lanes.53.Defective 16 10 7 Test at the level of significance α = 0. In an experiment to study the dependence of hypertension on smoking habits. Puchong and Petaling Jaya. ( 5.01 whether geographical location of the producer and quality of the toys are independent (no relationship). Reject H 0) 24. ( χ2 = 19. Growing conditions Seedling’s growth No fertilizer Fertilizer A Fertilizer B Strong growth 142 181 190 Medium growth 111 174 95 Dead 48 63 71 Test at 1% significance level whether there is an association between the seedling’s growth and types of fertilizer applied. 19 red-haired and 66 dark-haired. Do not reject H0) 30. a seedling’s growth is classified into one of three categories. as shown in the following table. 10% is redhaired and the remainder is dark-haired. The results are as follows. ( 0. and those he rejects. and has formulated a theory that overall. each of whom is asked to keep a record of the number of items he tests.549. Do not reject H0) An anthropologist is researching into hereditary factors in a certain country. B and C. contain 65 fair-haired. Number of breakdowns per shift Frequency: Expected number of shifts Actual number of shifts 0 14 10 1 27 23 2 27 25 3 18 22 4 9 10 5 5 10 29. in the course of a day. After a certain period of time. Do not reject H0) 31. Do not reject H0) The number of breakdowns that have occurred during the last 100 shifts is as follows. It has been customary to use a significance level of 0. Reject H0…) . A sample from one particular tribe. A factory employs there quality control inspectors. 40% of the population is fair-haired. Show whether the manager is justified in his claim that the difference between the number of actual and expected breakdowns is due to chance. Are these proportions consistent with the theory? Use α = 5%.5052. (2. however. ( 7.4635. Do the inspectors differ significantly in the proportion of items they reject? Use α = 5%. as follows.05.27. Inspector A Inspector B Inspector C Accepted 75 83 92 Rejected 15 19 16 28. Output in tones A B Factory: X Y 42 20 13 8 C 33 25 Do these figures show a significant difference at the 5% level? (1.88360. Two factories using materials purchased from the same supplier and closely controlled to an agreed specification produce output for a given period classified into three quality grades. Seedlings are grown without fertilizer or with one of two kinds of fertilizer. A.5633. Day Monday Tuesday Wednesday Thursday Friday Number of visitors 108 120 148 140 84 Test at 5% significance level whether the number of visitors is independent of the day of the week. the following table was produced. The following table summarizes data on compression strength (in pounds) for a sample of 12-oz aluminum cans filled with orange drinks and another sample filled with cola drinks. A manufacturer of nickel-hydrogen batteries randomly selects 100-nickel plates for test cells and found that 12 of the plates have blistered.8402. ( χ2 = 21. 23 16 19 21 20 23 20 18 ( t = -0. (t =0. . Does the result provide sufficient evidence to conclude that less than 15% of all plates blister under such circumstances? Test at 1% significance level. In a survey concerned with changes in working procedures. Opinion on changes in working procedures In favor Opposed Undecided Skilled workers 21 36 30 Unskilled workers 48 26 19 Test. were obtained. (Z = -0.4636. The table below shows the number of visitors of a university library during a week. ( χ2 = 14. 259. corrected to the nearest minute. that the opinion on working procedure is independent of the type of workers.867. Do the marks give evidence that the students have benefited by the extra coaching? Test at 1% level of significance. Student s Test 1 Test 2 1 23 24 2 20 19 3 19 22 4 18 10 5 20 20 6 18 22 7 17 20 8 9 10 11 35. Reject H0…) (ii) 10% significance level? ( Reject H0…) A company would like to estimate the length of time required to complete a service call. Beverage Sample size Sample mean Sample standard deviation Cola 45 1800 63 Orange 50 1750 70 Does the data suggest that average compression strength of cola drinks is higher than that of the orange drinks at the (i) 1% significance level? ( Z = 3. ( 51. They were given a month’s tuition and a second test was held at the end of it. Do not reject H0…) Eleven BMAUH students were given a test in Quantitative Methods.1863.2955. Do not reject H0) 37.6642. 48 51 28 66 81 36 40 59 50 (i) Estimate the population mean and variance for the length of time to complete a service call.25) (ii) It is believed that the true mean time to complete a service call is at most 50 minutes. Reject H0…) 34. Test at 5% significance level whether the belief is false. Reject H0…) 33. Do not reject H0…) 36. at level of significance of 5%.32. The following sample data. 40.3309. He recently has been informed that the government is considering a new law that would require him to alter the way in which the chemical is manufactured. Unsure about whether he should lobby against the legislation.50 Standard deviation $6.. The manager is concerned if the filling machine is now dispensing significantly less salad dressing into the bottles. Do not reject H0…) 39. The company investigates the complaints by examining the pay of 70 workers from each production line. The Imaginex Co. The data are shown below. used primarily in automobile batteries. The workers on production line A complained that they receive less pay than those on production line B. Reject H0…) (ii) State any assumption that you might have used in your test. (The differences are normally distributed) 41. The company’s managing director has observed that the output is normally distributed with a mean of 8200 litres per hour.7g. Reject H0…) A dispute exists between workers on two production lines. test whether the training is effective. Do not reject H0…) The times taken (in minutes) to complete a task of a particular type were recorded for a sample of eight employees before and after a period of training. ( t = -1.38. Test at 5% significance level.00 $7.00 $394. owns a factory that produces sulphuric acid. The production manager has taken a random sample of 50 bottles from a recent batch and found that the sample mean is 228. The machine is known to dispense.25g. 8283 8121 7905 8097 7969 8101 8069 8240 8410 8483 8510 7480 8097 8237 7682 8076 Do these data indicate that the new legislation will reduce output? Test at 5% significance level. on average.3066. Ltd.1629. The data collected are as follows. ( Z = -2. (Z = -1. the plant’s output is quite variable. Because of changing conditions. He reorganizes production facilities to comply with the proposed law and observed that the hourly output for two working days (16 hours). The results were as follows: Production line Sample statistics A B Mean $393. (t = 1. 230g of salad dressing per bottle with a standard deviation of 4. he undertakes an experiment. .50 Are workers on production line A received significantly less pay than those on production line B? Test at α = 5%.6977. Liquid salad dressing is dispensed into bottles by a filling machine. Time taken to complete task Employee Before training After training 1 15 13 2 14 15 3 18 15 4 14 13 5 15 13 6 17 16 7 13 14 8 12 12 (i) At 10% significance level. ) 44. A pharmaceutical company claims that not more than 2% of the vitamin bottles produced are incorrectly sealed. test whether we can infer that engine A gives a lower mean mileage compared to that of engine B. on average. ( 0.360) The manager of the company claims that less than 30% of the plastic bottles had scratch effects.42. The manufacturer of a certain brand of cigarette claims that its cigarettes contain less than 15mg of tar content per cigarette. A random check on the tar content. (i) (ii) Construct a 95% confidence interval for the proportion of bottles with scratch defects.2 16.1404.) A doctor in a private hospital wishes to evaluate two new treatments for arthritis.0 385.2 Assume that the population sampled is normally distributed.7 At 10% significance level.60 Assuming that the time taken for the patients to get relief after the treatment are normally distributed with equal variances..4 23. whether we can infer that the claim of the manufacturer is true. Test at 5% significance level.) The quality officer of a company has randomly selected 60 plastic bottles from the production line and found that 15 had scratch defects. Treatmen Number of arthritis patients Sample mean Sample variance t A 8 53. (Do not reject H0…. 45. at 5% significance level.3 Engine B 14. The time until the patient experienced relief after the treatment was then recorded. 46.2 21. test at 10% significance level. (i) (ii) Calculate the unbiased estimates of the mean and variance of tar content of the cigarettes. (Do not reject H0…. He randomly selected two groups of arthritis patients and assigned them to the two treatments randomly.. whether we can infer that the claim is true. (Do not reject H0…..190666667) Test.845 Do not reject H0…. whether the claim is false. xmg of 10 cigarettes revealed the following: Σx = 147.3 20.4 19.8 21. A random check on a sample of 200 vitamin bottles revealed that 6 of them are incorrectly sealed.6 455. Car 1 2 3 4 5 6 Engine A 14.14 B 10 49.8 Σx2 = 2186.4 29. ( Z = -0. The table below shows the distance traveled on 1 gallon of gasoline. ( 14.7 18.) 43.2 28.7 23. at 5% significance level. whether we can infer that treatment B is better than treatment A? (Do not reject H0…. Base on the sample result. 0.78 & 0. test.. .) An experiment was done on six randomly selected cars to compare the fuel consumption of two different car engines. ( t = -1. At 5% significance level.9065 Reject H0) 51.8683 Reject H0…) 48. Test using α = 5%. whether the bakery has overstated the mean number of chocolate chips per cookie.89 Assuming that the samples are drawn form two populations with equal variances.1004 Do not reject H0…) 50. 62% of the employees commute to work by public transport. The results are as following: Number of containers Sample mean Sample variance Machine A 20 128 1. Eight randomly selected bags of tin ore are taken from a smelter. The results are 38 24 18 21 35 23 (i) Calculate the unbiased estimates of the mean and variance of number of chocolate chips per cookie. at 10% significance level. .9 4.5 3. 8 Test whether the average percentage impurities is higher than 4%. ( 2. Random samples of containers are taken from the two machines and the weight of each container (measured in oz) is determined. using a 10% significance level. ( 26. A random sample of 12 male students and 15 females. whether we can infer that the campaign is effective in reducing the number of car accidents. using a 5% significance level. to fill ice-cream containers.8 4. A road safety campaign was conducted in order to reduce the number of car accidents in a big city. . . . on average. was taken and the information is summarized in the table below. .06256 Do not reject H0…) A factory uses two machines.9 4. A and B. 49. (0. The percentage impurities of the ore found in the sample are: 3.85 Machine B 15 126 2. The table below shows the number of car accidents for the first six months of year 2003 (before the campaign) and year 2004 (after the campaign).1) (ii) Based on the sample result. A teacher suspects that male students perform better than the females in a science subject.5. Month January February March April May June 2003 25 38 22 28 15 36 2004 20 29 35 18 32 27 Test. test. .2 3.88273 Do not reject H0) In a modern city. 65. A bakery advertises that the chocolate cookies produced had 30 chocolate chips per cookie. 3.4 4. can we infer that there is a difference in the mean weight of the containers processed by the two machines? (t = 3. Sample size Sample mean marks Sample variance Males 12 76 103 Females 15 64 122 Assume that samples are taken from populations that are normally distributed.6. .47. respectively. A recent survey among 200 employees in the city indicated that only 60% of 52. ( t = 0. A random sample of 6 cookies is selected and the number of chocolate chips is counted. using a 3% significance level. Test if the proportion of employees who commute by public transport has been reduced significantly. A sample of 40 packets from machine A gives a mean of 180 g and standard deviation 14 g.8380 Do not reject H0) . (7. (-0. Another sample of 40 packets from machine B gives a mean of 175 g and standard deviation 10 g.5827 Do not reject H0) 53. using a 5% significance level. Test if packets from machine A are significantly heavier than those from machine B. The following table summarizes the Location of cerebral tumours in 141 neurosurgical patients: Type of tumour Location of Benign Malignant Others tumour Frontal Lobes 23 9 6 Temporal Lobes 21 4 3 Elsewhere 34 24 17 Perform an appropriate test at 10% level of significance for the above data. Peanuts are packed into packets by two machines.844. (1.them commute to work by public transport. Reject H0) 54. STA222.Ex4 .