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3.6.1 Chi Square Distribution

3.6.1 Chi-Square Distribution

Key Concepts

  • Chi-Square Distribution: Derived from the square of a standard normal distribution, it is used extensively in hypothesis tests.
  • Degrees of Freedom: Refers to the number of independent values in the data after accounting for any parameters estimated.

Detailed Explanation

Definition and Formation

  • Standard Normal Squared: If Z is a standard normal random variable, Z^2 has a Chi-Square distribution with one degree of freedom.
  • Sum of Squares: More generally, if Z1, Z2, ..., Zn are independent standard normal random variables, then
    • X = Z_1^2 + Z_2^2 + ... + Z_n^2 has a Chi-Square distribution with n degrees of freedom.

Properties of Chi-Square Distribution

  • Skewness: The distribution is skewed right, with skewness decreasing as degrees of freedom increase.
  • Mean and Variance: The mean is equal to the degrees of freedom (n), and the variance is twice the degrees of freedom (2n).
  • Convergence to Normality: As the degrees of freedom increase, the distribution approximates a normal distribution due to the Central Limit Theorem.

Applications

  • Goodness of Fit Tests: Used to determine if observed data deviates significantly from expected data.
  • Independence in Contingency Tables: Tests whether two classifications are independent.
  • Variance Estimation: Used in the analysis of variance (ANOVA) for comparing sample variances.

Practical Application

Example Calculation:

  • Consider n=10 where each Zi is standard normal. Then X follows a Chi-Square distribution with 10 degrees of freedom.

Excel Computation:

  • Use CHISQ.DIST(x, k, TRUE) for the cumulative distribution function.
  • Use CHISQ.DIST.RT(x, k) for the right-tail probability.
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