Question

2. Central Statistics Office figures for 1980 show that the number of farms of various sizes in the four provinces of Ireland and across the state was as follows Province Farm Size Irish State (Acres) Leinster Munster Connaught Ulster 1-15 16,649 23,285 15,357 8,464 2,890 66,645 16,249 29,626 23,942 10,804 2,468 83,089 16,749 44,225 12,346 2,343 568 76,231 12,289 16,930 6,004 1,850 61,936 114,066 57,649 23,461 6,446 263,558 16-50 51-100 101-200 >200 All 520 37,593 (a) Calculate the mean, median and standard deviation of farm size in the Irish state. Comment on your results. (b) Convert the above table into a joint probability table. Choose one joint probability and one marginal probability and explain how to interpret these probabilities. (c) Give a definition of statistical independence. Using this definition, check if farm size is independent of province.

Answer 1

using ms-excel the mean, median and standard deviation is given as  

	Irish state
	61936
	114066
	57649
	23461
	6446
	263558
mean=	52712
medain=	57649
standard deviation=	41442.53

(b) the joint and marginal probability is given as

Joint probability is the probability that two events will occur simultaneously e.g. probability of farm size is between 16-50 and belongs to Munster is 0.1124. •

Marginal probability is the probability of the occurrence of the single event, probability of farm size between 51-100 is 0.2187 or probability of Linester is 0.2529.

farm size	Leinster	Munster	connaught	Ulster	Irish state
1-15	0.0632	0.0617	0.0635	0.0466	0.2350
16-50	0.0883	0.1124	0.1678	0.0642	0.4328
51-100	0.0583	0.0908	0.0468	0.0228	0.2187
101-200	0.0321	0.0410	0.0089	0.0070	0.0890
>200	0.0110	0.0094	0.0022	0.0020	0.0245
All	0.2529	0.3153	0.2892	0.1426	1.0000

(c)

here we use chi-square test and statistic chi-square=sum((O-E)2/E)=2142.5 with (r-1)(c-1)=(5-1)(4-1)=12 df

critical chi-square(0.05,12)=21.3 is less than calculated/observed chi-square=2142.5, so we reject the null hypothesis

H0: farm size and province are independent and conclude that farm size si not independent of province

Observed(O)
farm size	Leinster	Munster	connaught	Ulster	Irish state
1--15	16649	16249	16749	12289	61936
16-50	23285	29626	44225	16930	114066
51-100	15357	23942	12346	6004	57649
101-200	8464	10804	2343	1850	23461
>200	2890	2468	568	520	6446
All	66645	83089	76231	37593	263558
Expected (E)
farm size	Leinster	Munster	connaught	Ulster	Irish state
1--15	15661.5	19525.9	17914.2	8834.3	61936.0
16-50	28843.5	35960.3	32992.2	16270.0	114066.0
51-100	14577.5	18174.4	16674.3	8222.9	57649.0
101-200	5932.5	7396.3	6785.8	3346.4	23461.0
>200	1630.0	2032.2	1864.4	919.4	6446.0
All	66645.0	83089.0	76231.0	37593.0	263558.0
(O-E)
farm size	Leinster	Munster	connaught	Ulster	Irish state
1--15	987.5	-3276.9	-1165.2	3454.7	0.0
16-50	-5558.5	-6334.3	11232.8	660.0	0.0
51-100	779.5	5767.6	-4328.3	-2218.9	0.0
101-200	2531.5	3407.7	-4442.8	-1496.4	0.0
>200	1260.0	435.8	-1296.4	-399.4	0.0
	0.00	0.00	0.00	0.00	0.0
(O-E)2/E
farm size	Leinster	Munster	connaught	Ulster	Irish state
1--15	62.3	549.9	75.8	1350.9	2038.9
16-50	1071.2	1115.8	3824.4	26.8	6038.1
51-100	41.7	1830.4	1123.5	598.7	3594.3
101-200	1080.2	1570.0	2908.8	669.1	6228.2
>200	974.0	93.5	901.5	173.5	2142.5

.

E(16649)=row_total*column_total/Grand_total=61936*66645/263558=15661.5

and so on , the expected of each observed value is calculated.