Understanding Test Statistics in Hypothesis Testing

In hypothesis testing, constructing test statistics is a crucial and often challenging step. By leveraging known theorems, we can simplify complex problems by transforming them into familiar statistics. Here, I’ll explain some common derivations involving these transformations.

Transformations Between Distributions

Exponential and Chi-Square Distributions

The relationship between the exponential distribution and the chi-square distribution is foundational in statistics. Suppose we have a random variable that follows an exponential distribution with parameter :

The exponential distribution can be transformed into a chi-square distribution. If we scale by , we get a chi-square distribution with two degrees of freedom:

This transformation simplifies the analysis by allowing us to use properties of the chi-square distribution.

Gamma and Chi-Square Distributions

The gamma distribution is a generalization of the exponential distribution. If a random variable follows a gamma distribution with shape parameter and rate parameter :

when and , is equivalent to a chi-square distribution with degrees of freedom:

This equivalence is useful for deriving properties and performing tests involving the gamma distribution.

Sample Variance and Chi-Square Distribution

When dealing with normally distributed samples, the relationship between sample variance and the chi-square distribution is particularly important. For sample of size from a normal distribution :

The sample variance is defined as:

where is the sample mean. The statistic follows a chi-square distribution with degrees of freedom:

This result is fundamental in constructing confidence intervals and performing hypothesis tests about the population variance.

Conclusion

By understanding and applying these transformations, we can simplify the process of constructing test statistics.