Chi-Square Test Practice Practice Questions

The chi-square (χ²) test is a statistical test used to determine whether there is a significant association between categorical variables or whether the observed frequencies differ significantly from the expected frequencies.

Example 1: A study investigates the relationship between eye color (blue, brown, green) and hair color (blonde, brunette, redhead). The following data is collected:

Step 1: Hypotheses

H₀: Eye color and hair color are independent
H₁: They are associated

Step 2: Expected Frequencies

Using E = \frac{(\text{Row Total} \times \text{Column Total})}{\text{Grand Total}}

Blue: (40.91, 40.91, 18.18) {color: blonde,brunette,redhead }
Brown: (32.73, 32.73, 14.55)
Green: (16.36, 16.36, 7.27)

Step 3: Chi-Square Calculation

\chi^2 = \sum \frac{(O - E)^2}{E} \approx 12.67

Step 4: Degrees of Freedom

df = (3 − 1)(3 − 1) = 4

Step 5: Decision

Critical value (α = 0.05, df = 4) = 9.488

Since 12.67 > 9.488 → Reject H₀

There is a significant association between eye color and hair color

Example 2: 100 flips of a coin are performed. The coin is fair, with an equal chance of heads and tails, according to the null hypothesis. 55 heads and 45 tails are the observed findings.

Step 1: Hypotheses

H₀: Coin is fair
H₁: Coin is not fair

Step 2: Expected Values

Heads = 50, Tails = 50

Step 3: Chi-Square Calculation

\chi^2 = \frac{(55-50)^2}{50} + \frac{(45-50)^2}{50} = 1

Step 4: Degrees of Freedom

df = 1

Step 5: Decision

Critical value (α = 0.05) = 3.84

Since 1 < 3.84 → Fail to reject H₀

The coin is likely fair

Practice Problems

Q1. Market Research on Beverages

A company conducts a survey to determine whether there's a relationship between age groups and preferred beverages. The data collected is as follows:

Use a chi-square test to determine if there is an association between age groups and preferred beverages.

Q2. Student Performance

A teacher wants to find out if there is a relationship between study habits and grades. The data collected is as follows:

Perform a chi-square test to determine if study habits and grades are associated.

Q3. Gender and Major

A university wants to see if there is an association between gender and chosen major. The data collected is:

Conduct a chi-square test to examine if gender and chosen major are related.

Q4. Voting Preferences

A political analyst wants to know if there is a relationship between gender and voting preference. The data is:

Test the hypothesis that gender and voting preference are independent.