SA3202, Solution for Tutorial 3 1. # of Boys 0 1 2 3 4 Total # of Families 183 789 1250 875 246 3343 Expected 183.7 782.8 1251 888.7 236.7 3343 Probability .0549 .2342 .3742 .2658 .0708 1 H0: The number of boys follows a binomial distribution The estimated probability that a born child is a boy =.5159 T=.6270, df=5-1-1=3 ( 1 free parameter). 95% table=7.815. Do not reject H0. That is, it seems that the number of boys follow a binomial distribution. # of Goals 0 1 2 3 4 5 6+ Total Frequency 18 20 15 7 2 2 0 64 Expected 15.93 22.15 15.40 7.14 2.4823 .6904 .1983 64 Probability .2489 .3461 .2407 .1116 .03879 .01079 .0031 1 The estimated lambda=total goals scored/ total number of matches=1.39 After combining the last three categories, T=.6088, G=.59845, df=5-1-1=3, 95% table=7.815 Don’t reject H0. It seems that the number of goals scored follows a Poisson distribution. 5/7/2019 SA3202, Solution for Tutorial 3

SA3202, Solution for Tutorial 3 Interval [0,.3) [.3, .6) [.6,.9) [.9,1.2) [1.2, 1.5] Total Observed 3 2 3 6 11 25 Expected 5 5 5 5 5 25 For uniform distribution over [0,1.5], each interval has probability 1/5=.2 so that for each interval, the expected frequency is 25*.2=5. There is no any free parameter involved. T=10.8, G=9.73885, df=5-1-0=4, 95% table=9.488. Reject H0. Therefore, it is unlikely that the rupture follows a uniform distribution over [0,1.5]. Interval [10,20) [20,30) [30,40) [40, 50] Total Observed 6 6 5 3 20 Expected 3.17 6.827 6.827 3.17 20 Probability .1587 .3413 .3413 .1587 1 Since the hypothetical distribution is normal distribution, which has a range over the whole Real line, we should regard [10,20) as “20 and less”, [40,50] as “40 and over”, so that p1=P(X<20)=P(Z<(20-30)/10= -1)=P(Z>1)=.1587 p2=P(20<X<30)=P(-1<Z<0)=P(Z<0)-P(Z<-1)=.5-.1587=.3413. By Symmetry, p3=p2=.3413, p4=p1=.1587, no free parameter. T=3.1169, G=2.64435, df=4-1=3, 95%table=7.815. Do not reject H0. It seems that the N(30,100) fits the data well. (Other interval division schemes are acceptable too). 5/7/2019 SA3202, Solution for Tutorial 3