Chi‐Squared - rFronteddu/general_wiki GitHub Wiki
Y~$χ^2$(k): "Random variable Y follows a chi-squared distribution with k degrees of freedom."
- This distribution is used for hypothesis testing and to compute confidence intervals (ex. testing goodness of fit of categorical values)
- E(X)=k
- Var(X)= 2k
- Has a table for CDF of known values of N,T
- asymmetric, skewed to the right.
- Fat tails
The relationship between chi and t is that if you take a random variable following a standard normal distribution (mean = 0, standard deviation = 1) and square it, the resulting distribution follows a chi-squared distribution with one degree of freedom, i.e., $χ^2$ 1.
Furthermore, when you have a random variable following a normal distribution and divide it by the square root of an independent chi-squared variable divided by its degrees of freedom, the resulting distribution follows a Student's t-distribution. Specifically, if X follows a standard normal distribution and Y follows a chi-squared distribution with ν degrees of freedom, then $T*\frac{X}{\sqrt{\frac{Y}{ν}}}$ follows a Student's t-distribution with ν degrees of freedom.
This relationship is often used in statistical hypothesis testing, particularly in situations where you need to compare means or test for differences between groups when working with sample data. The t-distribution is used to account for the variability in small samples, making it a valuable tool in inferential statistics.
Critical Value
For example, if you're conducting a chi-squared goodness-of-fit test with 4 degrees of freedom and a significance level of 0.05 (5%), you would look up the critical chi-squared value associated with a 5% significance level and 4 degrees of freedom.
The critical value is then compared to the test statistic (calculated from the sample data) to determine whether the null hypothesis is rejected. If the test statistic is greater than or equal to the critical value, you may reject the null hypothesis in favor of an alternative hypothesis.
The specific critical values can vary based on the chosen significance level and degrees of freedom, so it's essential to consult chi-squared distribution tables or statistical software to obtain the precise critical value for your particular hypothesis test.