Question. 1 A tourist company
Question. 1A tourist company organizes various types of sight tours, offeringsome discount rates for certain typesof customers, among them children (under 12) and seniors (over 65years of age). The company wantsto estimate the total number of seniors in its tours on the basisof a random sample of site tours recordedin their books. Each tour record shows the number of seniors, thenumber of children, and some otherdata. There were 4,000 tours organized during the last year, listedas they appeared in calendar time. Therecords show seasonal variations in numbers and types of people onthe tours. Most of the tours wereorganized in summer and least in winter.(a) Describe briefly how you would select an SRS of 100 tours fromthe last year, using a TRN.(b) The company’s manager prefers a simpler method, using one-in-40systematic sample of tours.Describe briefly how you would select this sample, using aTRN.(c) Can the sample obtained in (b) be treated as an SRS for thepurpose of estimation? Explain why, orwhy not.
(d) The sample in (a) was selected and the average number ofseniors per tour of 20 seniors wasobtained from the sample, with the sampling standard deviation of 5seniors. Estimate the total numberof seniors taking tours last year, and place a bound on the errorof estimation.(e) The company also wants to estimate the percentage of seniorsout of all tourists on the tours lastyear, (i) What information should be collected from every tour inthe sample for the company to be ableto estimate this percentage, and to place a bound on the error ofestimation? (ii) Exactly what summaryvalues should be calculated from the sample to be able to completethe required tasks? Give appropriateformulas.
(f) It is estimated that the total number of tours next yearwill be 10% higher than this year. How largeshould next year’s SRS sample be so that the estimated total numberof seniors will be within ± 2,000 ofthe true total, with probability 95%? [hint: use appropriateinformation from this year sample](g) What sampling design would you suggest that you think wouldproduce better results than one in (a)or (b)? Explain your choice in some details, considering actualconditions of the problem, not just ingeneral.
Answer:
A tourist company organizes various types of sight tours,offering some discount rates for certain typesof customers, among them children (under 12) and seniors (over 65years of age). The company wantsto estimate the total number of seniors in itstours on the basis of a random sample of site toursrecordedin their books. Each tour record shows the number of seniors, thenumber of children, and some otherdata. There were 4,000 tours organized during the lastyear, listed as they appeared in calendar time. Therecords show seasonal variations in numbers andtypes of people on the tours. Most of the tours wereorganised in summer with least in winter.
a.) To select a simple random sample of 100tours, firstly we have to number all the 4000 tours from 1 to 4000serially. Then starting from any point on the TRN, we startchoosing 4 digit numbers column wise.At any stage if the randomnumber is less than 4000 then we choose the tour corresponding tothat number.If the random number is above 4000, (say 4500),then thefinal random number shall be the number obtained as remainder ondividing the initial random number by 4000(ie 4500 divided by 4000gives 500 and we choose the 500th tour). Based on our choice,we maydo the sampling with or without replacement accordingly.We stop theprocess when the sample size of 100 is reached.
b.) If the manager intends to perform asystematic sampling of one-in-forty, then one in every 40 elementis chosen in the sample. For the starting point , we draw a 2 digitrandom number,if it lies under 40 then we keep it otherwise theremainder on divison by 40 is kept.This random number willcorrespond to the first unit of the sample.Thereafter,we chooseelements in the sample at intervals of 40.For example,if ourstarting point is 37,successive elements will be 37+40,37+80,37+120and so on.Hence a systematic sample will be obtained.
c.)Sample in b can be treated as a simplerandom sample because such a sample would give adquaterepresentation of all the tours taking place throughout theyear.SRS may tend to cluster the sample to a specific season.
d.) Average number of seniors estimated in atour was 20.
For simple randomsampling, population total of seniors toured previous year isestimated as 4000*20=80000.
The variance in theestimation is 40002 *20
e.) To estimate the percentageof senior in the tours, corresponding to every tour in the samplewe shall need the total number of people in the tour and the numberof seniors in it.
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