Question

if the mean=m,variance= v, observed value=o

we have four groups, g1 age 18-28,the observed resposibility of accident=127,m=0.809,v=0.848

for the same group the observed (not responsible) of accidents=192,m=1.22,v=1.849

group2 age 29-39 years ,o=155(responsible),m=0.5961,v=0.838

same group o=397 (not responsible),m=1.5269,v1.8336

group 3 age 40-50 o=127(responsible),m=0.635,v=0.841

same group o=277(not responsible),m=1.385,v=1.835

group 4 age >=51 ,o=54(responsible),m=0.446,v=0.849

same group o=54(not responsible),m=1.628,v=1.856

1-for the data shown which type of test we must use and why

2- using the test find the expected value for all responsible and not responsible accidents participations

3-if the test used is kai square find using (ovserved-expected)^2/expexted for all the data groups( resposible and not)

4- give an explanation or interpret the statistical test results for the significant values after finding p-value for the test

Answer 1

There are two distributions given one is responsibility of accidents under different age groups

the other one is non responsibility of accidents under the same age groups  

now our objective is to test whether responsibility of accidents under different age groups are uniformly distributed or not

similarly we have to test whether non responsibility of accidents under different age groups are uniformly distributed or not

1. Distribution of responsibility of accidents under different age groups

Age group	observed no.of responsibility of accidents Oi	Expected no. of responsibility of accidents Ei =Oi/n =463/4 =115.75	Oi-Ei	(Oi- Ei )2	(Oi- Ei )2/ Ei
18-28	127	115.75	11.25	126.562	1.0934
29-39	155	115.75	39,25	1540.5625	13.30
40-50	127	115.75	11.25	126.562	1.0934
>51	54	115.75	-61.75.	3813.062	32.942
Total	463	463	0		48.43

Kai square test of goodness of fit

For testing whether responsibility of accidents under different age groups is uniformily distributed or not we can apply kia square test of goodness of fit

set up null hypothesis H₀ ; Both observed and expected frequencies are not significantly different

i.e responsibility of accidents under different age groups are uniformly distributed  

under H_{0 the the value of kia square test statistic = 48.43 ( from last column of the above table)}

_{But the critical value of kia square at 5% level of significance for 3 degrees of freedom is 7.82}

_{we observe that the calculated value is greaterthan the critical value so we reject the null hypothesis}

_{i.e responsibility of accidents are not uniformly distributed between different age groups}

_{it means that age is influencing the responsibility of accidents i.e age and responsibility of accidents are not independent they are inter dependent}  

Distribution of non responsibility of accidents under different age groups

group	observed no.of Non responsibility of accidents Oi	Expected no. of Nonresponsibility of accidents Ei =Oi/n =920/4 =230	Oi-Ei	(Oi- Ei )2	(Oi- Ei )2/ Ei
18-28	192	230	-38	1444	6.2782
29-39	397	230	167	27889	121.256
40-50	277	230	47	2209	9.604
>51	54	230	-176	30976	134.678
Total	920	920	0		271.82

Kai square test of goodness of fit

For testing whether Non responsibility of accidents under different age groups is uniformily distributed or not we can apply kia square test of goodness of fit

set up null hypothesis H₀ ; Both observed and expected frequencies are not significantly different

i.enon responsibility of accidents under different age groups are uniformly distributed  

under H_{0 the the value of kia square test statistic = 271.82 ( from last column of the above table)}

_{But the critical value of kia square at 5% level of significance for 3 degrees of freedom is 7.82}

_{we observe that the calculated value is greaterthan the critical value so we reject the null hypothesis}

_{i.e Non responsibility of accidents are not uniformly distributed between different age groups}

_{it means that age is influencing the non responsibility of accidents i.e age and non responsibility of accidents are not independent they are inter dependent}