RI S5 Hypothesis Testing_Add Prac_Solns

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 6 ______________________________________________________ Additional Practice Questions for Chapter S5: Hypothesis Testing Page 1 of 16 Additional Practice Questions for Chapter S5: Hypothesis Testing (Solutions) 1 9758/2017/02/Q7 The production manager of a food manufacturing company wishes to take a random sample of a certain type of biscuit bar from the thousands produced one day at his factory, for quality control purposes. He wishes to check that the mean mass of the bars is 32 grams, as stated on the packets. (i) State what it means for a sample to be random in this context. [1] The masses, x grams of a random sample of 40 biscuit bars are summarised as follows.     240 32 7.7 32 11.05n x x      (ii) Calculate unbiased estimates of the population mean and variance of the mass of biscuit bars. [2] (iii) Test at 1% level of significance, the claim that the mean mass of biscuit bars is 32 grams. You should state your hypotheses and define any symbols you use. [5] (iv) Explain why there is no need for the production manager to know anything about the population distribution of the masses of the biscuit bars. [2] Solution (i) For a sample to be random, every biscuit bar has equal chance of being selected into the sample, and the selection of one biscuit bar is independent of the selection of another biscuit bar. (ii) Let 32y x  Then 7.7 0.192540y    and 32 31.8075x y   (note that this is an exact answer) Therefore an unbiased estimate of the population mean is 31.8075 grams.    2 2 2 2 2 7.71 1 11.05 0.245331 39 40y x ys y sn n                     Therefore an unbiased estimate of the population variance is 0.245 grams2 (3.sf) (iii) Let  denote the population mean mass (in grams) of a biscuit bar 0H : 32 1H : 32 Perform a 2-tail test at 1% significance level. Under H0, since 40n is large 0.24533~ N 32,40X     approximately by Central Limit Theorem.

Raffles Institution H2 Mathematics 2025 Year 6 ______________________________________________________________________________________________ ______________________________________________________ Additional Practice Questions for Chapter S5: Hypothesis Testing Page 2 of 16 Using a z-test, p-value 2P 31.8075 0.0140 0.01X    We do not reject 0H and conclude that there is insufficient evidence at 1% level of significance, that the mean mass of a biscuit bar is not 32 grams. (iv) No assumption about the population is needed, as sample size of 40 is large and hence X follows a normal distribution approximately by Central Limit Theorem. 2 SRJC/2006/02/Q31OR(a) To investigate the length of time spent by students in the school canteen during lunch break, a random sample of 90 students was taken. The time, x minutes, spent by each student was measured and it was found that ( 40) 270x  and