F13 - Non-parametric Testing - Tutorial (Solutions)

National Junior College Mathematics Department 2017 2016 – 2017 / H2 FMaths / Non-parametric Tests Tutorial Solutions Page 1 of 4 National Junior College 2016 – 2017 H2 Further Mathematics Topic F12: Non-parametric Tests (Tutorial Solutions) Basic Mastery Questions 1 To test at 5% significance level (one-tail): H0: The pill does not affect the median blood pressure. H1: The pill lowers the median blood pressure. Before pill 120 136 160 98 115 110 180 190 138 128 After pill 118 122 143 105 98 98 180 175 105 112 The sign change - - - + - - 0 - - - Let X denote the number of ‘+’ signs in 9 volunteers. Under H0,  ~ B 9,0.5K .  -value P 1 0.0195 0.05p K     . Therefore, we reject H0 at the 5% significance level and conclude that the pill evidently lowers the median blood pressure. 2 To test at 5% significance level (two-tail): H0: The antitoxin does not change the median. H1: The antitoxin changes the median. Before 18.2 21.6 23.5 22.9 16.3 19.2 21.6 21.8 20.3 19.5 18.9 20.3 After 18.4 20.3 21.5 20.2 17.6 18.5 21.7 22.3 19.4 18.6 20.1 19.7 Diff 0.2 –1.3 –2.0 –2.7 1.3 –0.7 0.1 0.5 –0.9 –0.9 1.2 –0.6 Signed Rank 2 –9.5 –11 –12 9.5 –5 1 3 –6.5 –6.5 8 –4 2 9.5 1 3 8 23.5P      , 9.5 11 12 5 6.5 6.5 4 54.5Q        , 23.5T  , 12n . By MF26, the critical region is 13T  . 23.5T  is not in the critical region. Therefore, we do not reject H 0 at the 5% significance level and conclude there is insufficient evidence that the antitoxin changes the median. 3 To test at 10% significance level (two-tail): H0: The median life of a battery is 40 hours. H1: The median life of a battery is less than 40 hours. Let X denote the number of batteries in the sample 75 with a life less than 40 hours. Under H0,  ~ B 75,0.5X . (a)  -value P 32 0.124 0.1p X     . (b) Since 75n is large, both 37.5 5np nq   ,  ~ N 37.5,18.75X approximately           32.5 37.5-value P 32 P 32.5 P P 1.1547 18.75 1 P 1.155 1 0.8749 0.0010 0.1241 0.1 p X X Z Z Z                    In either approach, we do not reject H 0 at the 10% significance level and conclude there is insufficient evidence that the battery life is shorter than 40 hours. www.KiasuExamPaper.com 1108

National Junior College Mathematics Department 2017 2016 – 2017 / H2 FMaths / Non-parametric Tests Tutorial Solutions Page 2 of 4 4 (i)  ~ B 60,0.5K . Since n is large and 30 5np nq   ,  ~ N 30,15K approximately. Assuming critK is an integer,      crit crit crit2P 0.05 P 0.025 P 0.5 0.025K K K K K K           . Under normal distribution, crit 0.5 22.409K   , crit 21.9K  . Since we should not reject H0 when 22K  , critK should be round down to 21. Thus the critical region is 21K  or 39K  . (OR 30 9K   .) (ii) Under H0,        60 60 1 4 60 60 1 120 1 24 ~ N 0,1TZ     .  crit crit crit crit P 0.025 915P 0.025135.84 915 1.960