Test of Independence 2×2 Problems

Problem 1: Test of Independence

The following table shows the relationship between Gender and Preference for Online Shopping.

Test whether gender and preference for online shopping are independent at 5% level of significance.

Step 1: State the Hypotheses

Null Hypothesis (H₀): Gender and preference for online shopping are independent.

Alternative Hypothesis (H₁): Gender and preference for online shopping are not independent.

Step 2: Calculate Expected Frequencies

Expected Frequency (E) = (Row Total × Column Total) / Grand Total

• E₁₁ = (60 × 70) / 100 = 42
• E₁₂ = (60 × 30) / 100 = 18
• E₂₁ = (40 × 70) / 100 = 28
• E₂₂ = (40 × 30) / 100 = 12

Step 3: Prepare Chi-square Table

χ² = 0.095 + 0.222 + 0.143 + 0.333 = 0.793

Step 4: Degrees of Freedom

df = (r − 1)(c − 1)
df = (2 − 1)(2 − 1) = 1

Step 5: Table Value

At 5% level of significance and 1 df,
χ² (table) = 3.84

Step 6: Decision

Calculated χ² = 0.793
Table χ² = 3.84

Since calculated χ² < table χ², accept H₀.

Problem 2: Test of Independence

A survey examined the relationship between Advertisement Exposure and Purchase Decision.

Test the independence at 5% level of significance.

Step 1: Hypotheses

H₀: Advertisement exposure and purchase decision are independent.
H₁: They are associated.

Step 2: Expected Frequencies

• E₁₁ = (80 × 70) / 140 = 40
• E₁₂ = (80 × 70) / 140 = 40
• E₂₁ = (60 × 70) / 140 = 30
• E₂₂ = (60 × 70) / 140 = 30

Step 3: Chi-square Calculation

χ² = 11.66

Step 4: Degrees of Freedom

df = (2 − 1)(2 − 1) = 1

Step 5: Table Value

χ² (0.05, 1 df) = 3.84

Step 6: Decision

Calculated χ² > Table χ²
Reject H₀

Important Exam Notes

• Test of independence is applied only to qualitative data
• Expected frequency should preferably be ≥ 5
• Degrees of freedom for 2×2 table is always 1
• Used in market research, consumer behavior, HR studies