Final answer:
The variable 'k' in chi-square tests typically represents the degrees of freedom which define the shape of the chi-square distribution and are essential for determining the critical values during hypothesis testing.
Step-by-step explanation:
In the context of chi-square tests, the variable k often stands for the degrees of freedom associated with the chi-square distribution. The chi-square random variable (x²) is the sum of the squares of k independent standard normal variables (Z²), which means x² = (Z₁) ² + (Z₂) ² + ... + (Zₖ) ². In chi-square tests, degrees of freedom are crucial as they help to define the shape of the chi-square curve, which is non-symmetrical and skewed to the right, and for each degree of freedom, there is a unique chi-square curve.
The term degrees of freedom (df) is key in understanding the chi-square test's result interpretation, as it affects the critical values we compare with our test statistics to make decisions regarding the null hypothesis.
Considering statistical tests, such as those for variances or independencies, the value of k plays additional roles. For instance, in the test of two variances or an F-test with k categories, k represents the number of different data cells or categories used. Furthermore, if k is used to denote a percentile, as in k = the 90th percentile, it refers to a specific point on the distribution curve, such that a certain percentage of data falls below it. This clarifies that k can represent critical values in statistical significance testing as well.