RI S4 Sampling_Tutorial Qns

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 6 ___________________ Tutorial S4: Sampling Page 1 of 3 Tutorial S4: Sampling 1 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] 2 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] 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  ,   2 154 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 ] 4 The amount, x mg, of Vitamin B2 in a packet of cereals was measured. 10 packets of cereals were taken and the following data were obtained: 272, 285, 278, 293, 298, 283, 279, 281, 295, 271 Find unbiased estimates of the population mean and population variance of Vitamin B2 in packets of the cereal. [283.5, 86.7]

Raffles Institution H2 Mathematics 2025 Year 6 _____________________________________________________________________________________________ ________________ Tutorial S4: Sampling Page 2 of 3 5 A sample of 30 salesmen was surveyed and the ages of each of the salesman are given as follows: Age 21 22 23 24 25 26 27 28 Frequency 3 3 4 6 7 4 2 1 Calculate unbiased estimates for the population mean  and population variance 2. [24.2, 3.41] 6 Given that the random variables X and Y have distributions N(30,18) and N(20,16) respectively, and that X and Y denote the means of 15 independent observations of X and 8 independent observations of Y respectively. State the sampling distributions of (i) X Y , (ii) 5 3X Y , (iii) 4 2X Y , (iv) 1 2 15 1 2 8 23 X X X Y Y Y        . 7 A large number of samples of size