Question

Consider the following contingency table of observed frequencies. Complete parts a. through d. below.

Column Variable

Row Variable

C1

C2

C3

R1

7

6

9

R2

11

9

8

a. Identify the null and alternative hypotheses for a​ chi-square test of independence with based on the information in the table. This test will have a significance level of

α=0.05.

Choose the correct answer below.

A.

H0​:

The variables

R1​,

R2​,

C1​,

C2​,

and

C3

are independent.

H1​:

At least one of the variables is not independent.

B.

H0​:

The row and column variables are independent of one another.

H1​:

The row and column variables are not independent of one another.Your answer is correct.

C.

H0​:

The row and column variables are not independent of one another.

H1​:

The row and column variables are independent of one another.

D.

H0​:

The variables

R1​,

R2​,

C1​,

C2​,

and

C3

are independent.

H1​:

None of the variables are independent.

​b, Calculate the expected frequencies for each cell in the contingency table.

		Column Variable
Row Variable		C1	C2	C3
R1		nothing	nothing	nothing
R2		nothing	nothing	nothing

​(Round to two decimal places as​ needed.)

Answer 1

The contingency table of observed frequencies is given as follows:

	C1	C2	C3	Row Total
R1	7	6	9	22
R2	11	9	8	28
Column Total	18	15	17	Grand Total = 50

(a) Chi-square test is a test of independence between two variables. The null and alternative hypothesis for chi-square test is given by:

H0: There is no association between the row and column variables.

H1: The exists an association between the row and column variables.

Thus, for the given test the hypothesis is given by:

H0​: The row and column variables are independent of one another.

H1​: The row and column variables are not independent of one another.

Answer: Option (B)

(b) The expected frequencies are calculated using the following formula:

Expected Frequency = (Row Total * Column Total) / Grand Total

The contingency table of expected frequencies is calculated as follows:

	C1	C2	C3	Row Total
R1	22*18/50 = 7.92	22*15/50 = 6.60	22*17/50 = 7.48	22
R2	28*18/50 = 10.08	28*15/50 = 8.40	28*17/50 = 9.52	28
Column Total	18	15	17	Grand Total = 50