Question

Question. 1
A tourist company organizes various types of sight tours, offering some discount rates for certain types
of customers, among them children (under 12) and seniors (over 65 years of age). The company wants
to estimate the total number of seniors in its tours on the basis of a random sample of site tours recorded
in their books. Each tour record shows the number of seniors, the number of children, and some other
data. There were 4,000 tours organized during the last year, listed as they appeared in calendar time. The
records show seasonal variations in numbers and types of people on the tours. Most of the tours were
organized in summer and least in winter.
(a) Describe briefly how you would select an SRS of 100 tours from the last year, using a TRN.
(b) The company's manager prefers a simpler method, using one-in-40 systematic sample of tours.
Describe briefly how you would select this sample, using a TRN.
(c) Can the sample obtained in (b) be treated as an SRS for the purpose of estimation? Explain why, or
why not.

(d) The sample in (a) was selected and the average number of seniors per tour of 20 seniors was
obtained from the sample, with the sampling standard deviation of 5 seniors. Estimate the total number
of seniors taking tours last year, and place a bound on the error of estimation.
(e) The company also wants to estimate the percentage of seniors out of all tourists on the tours last
year, (i) What information should be collected from every tour in the sample for the company to be able
to estimate this percentage, and to place a bound on the error of estimation? (ii) Exactly what summary
values should be calculated from the sample to be able to complete the required tasks? Give appropriate
formulas.

(f) It is estimated that the total number of tours next year will be 10% higher than this year. How large
should next year's SRS sample be so that the estimated total number of seniors will be within ± 2,000 of
the true total, with probability 95%? [hint: use appropriate information from this year sample]
(g) What sampling design would you suggest that you think would produce better results than one in (a)
or (b)? Explain your choice in some details, considering actual conditions of the problem, not just in
general.

Answer 1

A tourist company organizes various types of sight tours, offering some discount rates for certain types
of customers, among them children (under 12) and seniors (over 65 years of age). The company wants
to estimate the total number of seniors in its tours on the basis of a random sample of site tours recorded
in their books. Each tour record shows the number of seniors, the number of children, and some other
data. There were 4,000 tours organized during the last year, listed as they appeared in calendar time. The
records show seasonal variations in numbers and types of people on the tours. Most of the tours were organised in summer with least in winter.

a.) To select a simple random sample of 100 tours, firstly we have to number all the 4000 tours from 1 to 4000 serially. Then starting from any point on the TRN, we start choosing 4 digit numbers column wise.At any stage if the random number is less than 4000 then we choose the tour corresponding to that number.If the random number is above 4000, (say 4500),then the final random number shall be the number obtained as remainder on dividing the initial random number by 4000(ie 4500 divided by 4000 gives 500 and we choose the 500th tour). Based on our choice,we may do the sampling with or without replacement accordingly.We stop the process when the sample size of 100 is reached.

b.) If the manager intends to perform a systematic sampling of one-in-forty, then one in every 40 element is chosen in the sample. For the starting point , we draw a 2 digit random number,if it lies under 40 then we keep it otherwise the remainder on divison by 40 is kept.This random number will correspond to the first unit of the sample.Thereafter,we choose elements in the sample at intervals of 40.For example,if our starting point is 37,successive elements will be 37+40,37+80,37+120 and so on.Hence a systematic sample will be obtained.

c.)Sample in b can be treated as a simple random sample because such a sample would give adquate representation of all the tours taking place throughout the year.SRS may tend to cluster the sample to a specific season.

d.) Average number of seniors estimated in a tour was 20.

        For simple random sampling, population total of seniors toured previous year is estimated as 4000*20=80000.

        The variance in the estimation is 4000² *20

e.) To estimate the percentage of senior in the tours, corresponding to every tour in the sample we shall need the total number of people in the tour and the number of seniors in it.