RI S4 Sampling_Tutorial Solns

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 6 ___________________ Tutorial S4: Sampling Page 1 of 10 Tutorial S4: Sampling 1 VJC Prelim 9740/2014/02/5 1st part There were 500 spectators seated at the grandstand of a stadium. A surveyor took the seat plan of the grandstand and threw a die on the seat plan 20 times and interviewed the spectators at the seats on which the die landed. Give a reason why this might not result in a random sample of spectators. [1] Solution The surveyor is likely to avoid throwing at the edge of the map. Hence not all the 500 seats have an equal chance of being selected. Alternative Solution After a seat is selected, the surveyor is likely to avoid the nearby seats. Hence the selections are not independent of one another. 2 9758/2019/02/Q6 In a certain country there are 100 professional football clubs, arranged in 4 divisions. There are 22 clubs in Division One, 24 in Division Two, 26 in Division Three and 28 in Division Four. (i) Alice wishes to find out about approaches to training by clubs in Division One, so she sends a questionnaire to the 22 clubs in Division One. Explain whether these 22 clubs form a sample or a population. [1] (ii) Dilip wishes to investigate the facilities for supporters at the football clubs. but does not want to obtain the detailed information necessary from all 100 clubs. Explain how he should carry out his investigation, and why he should do the investigation in this way. [2] (iii) Find the number of different possible samples of 20 clubs, with 5 clubs chosen from each division. [3] Solution (i) These 22 clubs form the population as they are ALL the clubs in Division One whose approaches to training she is interested to find out. (ii) How Assuming no special treatment with regard to facilities for supporters of different divisions, he could randomly pick a certain number of clubs out of the 100, say 10, to do a thorough investigation. Why This is to avoid bias and it would also be more cost effective and manageable. (iii) Number of different possible samples = 22 24 26 28 18 5 5 5 5 7.24 10C C C C    

Raffles Institution H2 Mathematics 2025 Year 6 _____________________________________________________________________________________________ ________________ Tutorial S4: Sampling Page 2 of 10 3 In each of the following cases, find unbiased estimates of the population mean and population variance of X. (a) sample size = 100, 160x , 2 265x  , (b) sample size = 150, 154 150x  ,   2154 11000,x  (c) sample size = 50, ( 10) 328x  , 2 16062x  , (d) sample size = 20, 1100x , sample variance = 10. [(a) 8 5 , 1 11 (b) 155, 10850 149 (c) 414 25, 8394 175 (d) 55, 200 19 ] Solution An unbiased estimate of population mean An unbiased estimate of population variance (a) 8 1.6100 5 xx   2 2 1 (160) 126599 100 11s       (b) Let 154y x  T