## DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1306/STAT1603

THE UNIVERSITY OF HONG KONGDEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCESTAT1306/STAT1603 Introductory Statistics (2015-2016)Assignment 5Hand in Questions 4, 5, 8 and 9 on or before November 30, 20151. A plaque beside an elevator reads âCapacity: 950 pounds, 6 personsâ. Assume the weights ofpeople entering the elevator are independently and identically distributed with Âµ = 150 poundsand standard deviation σ = 30 pounds.(a) A person is carrying a suitcase that weighs exactly ﬁve times his own weight. What is theprobability that the weight limit is exceeded?(b) If there are six people in the elevator, what is the probability that the weight limit isexceeded?(c) Repeat (b) if seven people crowd in.(d) What would be the answer to (c) if you are one of the seven and your weight is 135 pounds?2. A paint manufacturer wants to determine the average drying time of a new interior wall paint.In 12 test areas of equal size, he obtained a mean drying time of 66.3 minutes and a standarddeviation of 8.4 minutes. Construct a 95% conﬁdence interval for the true mean drying time.What assumption is needed in order to carry out the appropriate estimation procedure?3. A random sample of 6 students in a physical education class had their pulse rates (beats perminute) measured before and after the 50-yard dash, with the following results:Student123456Before748774969582After83969711012296Construct a 95% conﬁdence interval for the mean increase in pulse rate. Also construct a 90%conﬁdence interval for the population standard deviation for the increase in pulse rate.4. A process producing bricks is known to give an output whose weights are normally distributedwith population standard deviation 0.12 kg. A random sample of sixteen bricks from todayâsoutput had a mean weight of 4.07 kg.(a) Find a 99% conﬁdence interval for the mean weight of all bricks produced this day.(b) It is decided that a sample of twenty bricks will be taken tomorrow. Without doing thecalculations, state whether a 99% conﬁdence interval for the mean weight of tomorrowâsoutput would be wider than, narrower than or the same width as that found in (a).(c) How many more bricks should be sampled so that the sample mean will be within 0.01 kgof the population mean with a probability of at least 0.99?5. A study is being made to estimate the proportion of voters in a sizeable community who favorthe construction of a nuclear power plant. If it is found that only 140 of 400 voters selected atrandom favor the project, ﬁnd a 95% conﬁdence interval for the proportion of all voters in thiscommunity who favor the project.STAT1306/1603: Assignment 526. To determine which of the two types of seeds was better, a state agricultural station chose 9two-acre plots of land randomly within the state. Each plot was split in half, and a coin wastossed to determine in an unbiased way which half would be sown with seed A, and which halfwith seed B. The yields, in bushels, were recorded as follows:CountyPQRSTUVHKSeed A82681549378148897498Seed B896917391811509764117Which seed is better? To back up your answer, construct an appropriate 95% conﬁdence intervaland state the assumptions required.7. An industrial psychologist wishes to study the eﬀects of motivation on sales in a particular ﬁrm.Of 24 new salespersons, 12 are paid an hourly rate (Group A) and 12 are paid a commission(Group B). The 24 individuals are randomly assigned to the two groups. The following datarepresent the sales volume (in thousands of dollars) achieved during the ﬁrst month on the job.Hourly Rate256239222207228241212216236219225230Commission224254273285237277261228234225232245Assuming that the data come from independent normal populations with equal variance. Construct a 90% conﬁdence interval for the diﬀerence of the two population means. Is there evidencethat wage incentives through commission yield greater average sales volume?8. High blood pressure (hypertension) is a leading cause of strokes. Medical researchers are constantly seeking ways to treat patients suﬀering from this condition. A specialist in hypertensionclaims that regular aerobic exercise can reduce high blood pressure just as successfully as drugs,with none of the adverse side eﬀects. To test this claim, 30 patients who suﬀer from hypertensionwere chosen to participate in an experiment. For 60 days, 16 patients exercised three time perweek for one hour (Group 1); and the other 14 patients took the standard medication (Group 2).The percentage reduction in blood pressure was recorded for each individual, and the followingsummary statistics were obtained:x1 = 13.2;Â¯x2 = 9.51;Â¯s1 = 2.68;s2 = 3.61.Construct a 95% conﬁdence interval for the diﬀerence of the population mean percentage reduction in blood pressure between the two treatment groups.9. Telemarketers obtain names and telephone numbers from several sources. To determine whetherone particular source is better than a second, random samples of 400 and 420 names and numberswere obtained from sources A and B, respectively of which 52 and 46 from A and B made apurchase. Construct a 99% conﬁdence interval for pA − pB , the diﬀerence in the proportions ofcustomers making a purchase from the two sources.