RI T2W9_Hypo+Testing_%28Soln%29

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 6 _______________________________ Y6 H2 Math Term 2 Revision W9 Hypothesis Testing Page 1 of 4 Term 2 W9 Revision`: Hypothesis Testing Questions 1 DHS JC2 Mid-Year CT 9758/2018/02/Q1 The government of a particular country reported that the mean monthly water consumption of households is 317.9 m . This is followed by a nationwide water conservation campaign to encourage households to reduce their water consumption. Subsequently after the campaign, the water consumption 3 mx of a random sample of 50 households is recorded and their data are summarised as follows. ( 15) 103,x  2( 15) 599.x  (i) Find unbiased estimates of the population mean and variance. [2] (ii) Test at the 5% significance level whether the campaign is effective. State, giving a reason, whether any assumption about the distribution of the monthly water consumption of households is needed in order for the test to be valid. [4] (iii) Explain the meaning of ‘5% significance level’ in the context of the question. [1] Solution : (i) An unbiased estimate for the population mean is ( 15) 10315 15 17.0650 50 xx      An unbiased estimate for the population variance is   2 2 2 2( 15)1 ( 15)50 1 50 1 10359949 50 7.8943 (to 5 s.f.) = 7.89 (to 3 s.f.) xs x                (ii) Let X represent the water consumption (in cubic metres) of a household with population mean . To test H0:  = 17.9 vs H1:  <17.9 Perform 1-tail test at 5% significance level

Raffles Institution H2 Mathematics 2025 Year 6 _________________________________________________________________________________________ _______________________________ Y6 H2 Math Term 2 Revision W9 Hypothesis Testing Page 2 of 4 Under H0, 7.8943~ N , 51 07.9X     approximately, by Central Limit Theorem, since n = 50 is large. From the sample, 6.17.0x Using z-test, p-value = P 17.06X = 0.0173 (3 s.f.) Since p-value <0.05, we reject H0 and conclude that there is sufficient evidence at 5% significance level that the campaign was effective. No assumptions on the distribution of X are required as the sample size is large, by Central Limit Theorem, the distribution of the sample mean (X) is approximately normal. (iii) The 5% significance level means that there is a probability of 0.05 that we conclude that the mean water consumption of households has decreased, when it is actually unchanged at 317.9 m . 2 RI Prelim 9758/2020/02/Q7 In a factory, machines pack sugar into bags of 1 kg each on average, with variance 2 kg2. The manufacturer is concerned that the machines are putting too much sugar into the bags and decides to carry out a hypothesis test. A random sample of 8 bags are selected and their total mass is 8.4 kg. (i) Stating a necessary assumption, carry out a test of the manufacturer’s concern at the 5% sig