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 1.9600135.84 648 (round down) T T TZ T T           On the upper tail side, the critical value is 915 2 648 1182   . Thus the critical region is 648T  or 1182T  . (OR 915 267T   .) 5 (a), (b) and (c) are not appropriate as the two samples cannot be paired up by dependence. (d) may not be appropriate as the spending is unlikely to follow normal distribution, e.g. female’s spending especially. 6 It is more appropriate to use a non-parametric test when the population distribution is unknown or when the data lacking exact numerical values. (i) In both tests, ‘average’ means median. To test at 10% significance level (two-tail): H0: The median prices at the two stores are the same. H1: The median prices at the two stores are different. Item 1 2 3 4 5 6 7 8 9 10 11 12 Store A 12.50 19.50 4.20 11.00 4.30 7.20 8.60 9.50 4.60 7.20 5.20 7.30 Store B 11.20 18.50 3.70 9.50 4.00 6.80 9.30 9.40 4.80 6.60 4.10 8.10 Sign - - - - - - + - + - - + Diff –1.3 –1.0 –0.5 –1.5 –0.3 –0.4 0.7 –0.1 0.2 –0.6 –1.1 0.8 Signed Rank –11 –9 –5 –12 –3 –4 7 –1 2 –6 –10 8 In a sign test, let K denote the number of ‘+’ among the 12 differences. Under H0,  ~ B 12,0.5K . There are 3 ‘+’ signs in the sample,  -value 2P 3 0.146 0.1p K    . www.KiasuExamPaper.com 1109

National Junior College Mathematics Department 2017 2016 – 2017 / H2 FMaths / Non-parametric Tests Tutorial Solutions Page 3 of 4 Therefore, we do not reject H 0 at the 10% significance level and conclude there is insufficient evidence to conclude that the median prices are different. In a Wilcoxon matched-pairs ranked sign test, 7 2 8 17P    is the smaller value. 17T  , is in the critical region is 17T  (by MF26). Therefore, we reject H 0 at the 10% significance level and conclude there is sufficient evidence to conclude that the median prices are different. (ii) A paired-sample t-test might be used. This test is unlikely valid as the differences between the prices of items in different stores do not necessarily follow normal. 7 Sample point 1 2 3 4 5 6 7 8 9 10 % water in 1989 20 15 26 19 19 17 24 29 19 23 % water in 1990 19 24 21 29 23 28 30 21 32 26 Difference d –1 9 –5 10 4 11 6 –8 13 3 Signed Rank –1 7 –4 8 3 9 5 –6 10 2 (i) To test at 5% significance level: H0: 0dm  H1: 0dm  1 4 6 11T Q     , is not in the critical region 10T  . Therefore, we do not reject H0 at 5% significance level and conclude there is insufficient evidence that the median difference is more than 0. (ii) To test at 5% significance level: H0: 0d  H1: 0d  Under H0,  0 ~ t 9 10 Dt S  . By GC, p-value = 0.0458 < 0.5. (OR the critical region is 0.95, 9 1.833t t  , 1.888t  is in the critical region.) Therefore, we reject H0 at 5% significance level and conclude there is sufficient evidence that the median difference is more than 0. Both observed test statistics are close to their critical value, indicating a insignificant difference between the two tests. 8 Let m denote the median reaction time in s. ‘Average’ refers to median in this context. To test at 5% significance level (two-tail): H0: 0.6m H1: 0.6m Let K denote the number of students whose reaction time exceeds 0.6 in a random sample of 10. Under H0,  ~ B 10,0.5K . Signs for the given samples: –, –, –, –, +, –, –, –, +, – www.KiasuExamPaper.com 1110

National Junior College Mathematics Department 2017 2016 – 2017 / H2 FMaths / Non-parametric Tests Tutorial Solutions Page 4 of 4  -value 2P 2 0.109 0.05p K    . Therefore, we do not reject H 0 at 5% significance level and conclude there is insufficient evidence that the median reaction time is not 0.6. Wilcoxon’s signed rank test is preferred as it takes more information, mag nitudes of the differences in this case, into consideration. Let d denote the increase in the reaction time after drinking. Difference d 0.04 –0.01 0.02 0.06 –0.05 0.07 0.09 0.08 –0.03 0.12 Signed Rank 4 –1 2 6 –5 7 9 8 –3 10 1 5 3 9Q    is smaller. 9T  is in the critical region 10T  . Therefore, we reject H0 at 5% significance level and conclude there is sufficient evidence that the median reaction time is increased by drinking. 9 (i) Tasters should not know which sauce is with or without preservative. The order of tasting, with and without preservatives, should be randomised or balanced. Tasters should be allocated randomly to samples. Tasters should clean their mouths after tasting a source. (ii) H0: There is no difference between the median taste ratings. H1: There is a difference between the median taste ratings. The signs of the differences (with – without) are 0, +, –, –, 0, +, –, –, –, –, –, +. Let K denote the number of + in a random sample of 10. Under H0,  ~ B 10,0.5K .  -value 2P 3 0.344 0.05p X    Therefore, we do not reject H0 at 5% significance level and conclude there is insufficient