Question

Are the event “Political Affiliation” and “views on tariff” independent evets? Use statistical evidence to justify your answer.

	Opinion
Political Affiliation	F	O	U	Total
Democrat (D)	0.12	0.09	0.07	0.28
Republican (R)	0.16	0.12	0.14	0.42
Independent (I)	0.04	0.03	0.06	0.13
Green (G)	0.08	0.06	0.03	0.17
Total	0.4	0.3	0.3	1

Answer 1

We know that two variables X and Y are statistically independent if

P [X=i, Y=j] = P[X=i]. P[Y=j] for all i and j

Here if we consider Political Affiliation as X variable and Views on tariff as Y variable, and assume that i takes 4 values of Political Affiliation as D, R I and G

Also, consider j takes values of “Views on Tariff” as F, O and U

Here from the given table, we are clearly given with the joint probability distribution of X and Y.

Also the last column gives the marginal distribution of X and last row gives the marginal distribution of Y

Hence, marginal distribution of X is given as:

Value of X: Political Affiliation	P[X=i]; i= D, R, I,G
Democrat(D)	0.28
Republican(R)	0.42
Independent (I)	0.13
Green (G)	0.17
Total	1.0

Also, marginal distribution of Y is given as:

Value of Y: Views On Tariff	P[Y=j] ; j=F,O,U
F	0.4
O	0.3
U	0.3
Total	1.0

From the given table in the question, we get

P [X=D, Y=F] = 0.12

But P [X=D] P[ Y=F] = 0.28 x 0.4 = 0.112 which is not equal to 0.12

Hence , P [X=D, Y=F] =0.12 ≠ 0.112 = P [X=D] P[ Y=F]

Hence P [X=i, Y=j] = P[X=i]. P[Y=j] for all i and j -> this is a wrong statement.

Hence X and Y are not statistically independent.

Hence , “Political Affiliation” and “Views On Tariff” are not independent events.