Statistical Inference: A Short Course

13.3 The Chi-Square Distribution

Having rationalized the test statistic for conducting a multinomial goodness-of-fit test (Eq. 13.4), we next examine the properties of the chi-square distribution and its attendant probabilities. The chi-square distribution is a continuous distribution that represents the sampling distribution of a sum of squares of independent standard normal variables. That is, if the observations X₁, . . ., X_n constitute a random sample of size n taken from a normal population with mean μ and standard deviation σ, then the Z_i = (X_i − μ)/σ, i = 1, . . ., n, are independent N(0,1) random variables and

Looking to the properties of the chi-square distribution:

1. The mean and standard deviation of a chi-square random variable X are E(X) = v and , respectively, where v denotes degrees of freedom.

2. The chi-square distribution is positively skewed and it has a peak that is sharper than that of a normal distribution.

3. Selected quantiles of the chi-square distribution can be determined from the chi-square table (Table A.3) for various values of the degrees of freedom parameter v. For various cumulative probabilities 1 − α (Fig. 13.1), the quantile satisfies

or, alternatively, ...