Question

Consider the accompanying 2 × 3 table displaying the sample proportions that fell in the various combinations of categories (e.g., 12% of those in the sample were in the first category of both factors).

	1	2	3
1	0.12	0.20	0.27
2	0.08	0.11	0.22

(a) Suppose the sample consisted of n = 100 people. Use the chi-squared test for independence with significance level 0.10. State the appropriate hypotheses

.H₀: p_ij =

pi

pj

for every pair (i, j)
H_a: at least one p_ij ≠

pi

pj

H₀: p_ij ≠

pi

pj

for every pair (i, j)
H_a: at least one p_ij =

pi

pj

    

H₀: p_ij = p_i · p_j

for every pair (i, j)

H_a: at least one p_ij ≠ p_i · p_j

H₀: p_ij ≠ p_i · p_j

for every pair (i, j)

H_a: at least one p_ij = p_i · p_j

Calculate the test statistic. (Round your answer to two decimal places.)
χ² =  

What can be said about the P-value for the test?

P-value < 0.0050.005 < P-value < 0.01    0.01 < P-value < 0.0250.025 < P-value < 0.050.05 < P-value < 0.10P-value > 0.10

State the conclusion in the problem context.

Reject H₀. An individual's category with respect to factor 1 is not independent of the category with respect to factor 2.Fail to reject

H₀. An individual's category with respect to factor 1 is independent of the category with respect to factor 2.   

Reject H₀. An individual's category with respect to factor 1 is independent of the category with respect to factor 2.

Fail to reject H₀. An individual's category with respect to factor 1 is not independent of the category with respect to factor 2.

(b) Suppose the sample consisted of n = 1000 people. Use the chi-squared test for independence with significance level 0.10.
Calculate the test statistic. (Round your answer to two decimal places.)
χ² =  

What can be said about the P-value for the test?

P-value < 0.005

0.005 < P-value < 0.01   

0.01 < P-value < 0.025

0.025 < P-value < 0.05

0.05 < P-value < 0.10

P-value > 0.10

State the conclusion in the problem context.

Fail to reject H₀. An individual's category with respect to factor 1 is independent of the category with respect to factor 2.Reject

H₀. An individual's category with respect to factor 1 is independent of the category with respect to factor 2.   

Reject H₀. An individual's category with respect to factor 1 is not independent of the category with respect to factor 2.

Fail to reject H₀. An individual's category with respect to factor 1 is not independent of the category with respect to factor 2.

(c) What is the smallest sample size n for which these observed proportions would result in rejection of the independence hypothesis? (Round your answer up to the next whole number.)
n =  

Answer 1

Ho: pij =pi/pj

Ha: pij ≠ p_{i/pj for at least one pair}

Applying chi square test of independence:

Expected	Ei=row total*column total/grand total	1st	2nd	3rd	Total
	1st	11.800	18.290	28.910	59
	2nd	8.200	12.710	20.090	41
	total	20	31	49	100
chi square    χ2	=(Oi-Ei)2/Ei	1st	2nd	3rd	Total
	1st	0.0034	0.1599	0.1262	0.2895
	2nd	0.0049	0.2301	0.1816	0.4165
	total	0.0083	0.3899	0.3078	0.7060
test statistic X2 =		0.71

P-value > 0.10

Fail to reject

H₀. An individual's category with respect to factor 1 is independent of the category with respect to factor 2.   

b)

Applying chi square test of independence:

Expected	Ei=row total*column total/grand total	1st	2nd	3rd	Total
	1st	118.000	182.900	289.100	590
	2nd	82.000	127.100	200.900	410
	total	200	310	490	1000
chi square    χ2	=(Oi-Ei)2/Ei	1st	2nd	3rd	Total
	1st	0.0339	1.5987	1.2619	2.8945
	2nd	0.0488	2.3006	1.8159	4.1653
	total	0.0827	3.8994	3.0778	7.0598
test statistic X2 =		7.060

0.025 < P-value < 0.05

Reject H₀. An individual's category with respect to factor 1 is not independent of the category with respect to factor 2.
c(

n=653