Understanding the Chi-Square Distribution and Chi-Square Test

Chi-Square Distribution

The chi-square distribution is a fundamental continuous probability distribution in statistics. It's derived from the sum of squares of independent standard normal random variables. The distribution is defined by a single parameter called degrees of freedom (df).

Key properties:

1. Always non-negative
2. Right-skewed, especially for low degrees of freedom
3. As df increases, it approaches a normal distribution

The probability density function (PDF) of the chi-square distribution is:

$f(x; k) = \frac{1}{2^{k/2}\Gamma(k/2)} x^{k/2-1}e^{-x/2}$

Where k is the degrees of freedom and Γ is the gamma function.

Applications:

1. Estimating population variance
2. Constructing confidence intervals for population variance
3. Hypothesis testing, particularly in chi-square tests

The shape of the distribution varies with degrees of freedom:

• Low df (1-2): Highly right-skewed
• Moderate df (5-10): Moderately right-skewed
• High df (>30): Approximately normal

Expected value: E(X) = k Variance: Var(X) = 2k

The chi-square distribution is related to other distributions:

• Square of a standard normal variable follows χ²(1)
• Sum of k independent χ²(1) variables follows χ²(k)
• Relationship with F-distribution and t-distribution

Critical values of the chi-square distribution are often used in hypothesis testing and can be found in statistical tables or calculated using software.

Understanding the chi-square distribution is crucial for various statistical analyses, including goodness-of-fit tests, tests of independence, and analysis of variance (ANOVA).

Chi-Square Test

The chi-square test is a statistical hypothesis test used to determine if there is a significant association between categorical variables or if a sample comes from a population with a specific distribution.

Types of chi-square tests:

1. Goodness-of-fit test: Compares observed frequencies to expected frequencies based on a hypothesized distribution.
2. Test of independence: Examines the relationship between two categorical variables in a contingency table.

The chi-square statistic is calculated as:

$\chi^2 = \sum_{i=1}^{n} \frac{(O_i - E_i)^2}{E_i}$

Where $O_i$ is the observed frequency and $E_i$ is the expected frequency.

Key assumptions:

1. Random sampling
2. Independence of observations
3. Mutually exclusive and exhaustive categories
4. Large expected frequencies (typically > 5 in 80% of cells)
5. Categorical data
6. Sufficient sample size

Steps in conducting a chi-square test:

1. State null and alternative hypotheses
2. Choose significance level (α)
3. Calculate expected frequencies
4. Compute chi-square statistic
5. Determine degrees of freedom
6. Find critical value or p-value
7. Make decision and interpret results

Example (Test of Independence): H0: Variable A and Variable B are independent H1: Variable A and Variable B are not independent

A\B	B1	B2	Total
A1	30	20	50
A2	40	10	50
Total	70	30	100

Calculate χ² statistic, df = (rows-1)(columns-1) = 1 Compare to critical value or find p-value Interpret results based on significance level

Limitations:

• Sensitive to sample size
• Doesn't provide information about strength of association
• Affected by small expected frequencies

The chi-square test is widely used in various fields, including social sciences, biology, and market research, to analyze categorical data and make inferences about populations.