The purpose of using MLE (Maximum Likelihood Estimation) in solving Chi Square problems is to find the most likely values for the parameters of a statistical model, based on the observed data. In other words, MLE allows us to estimate the parameters that best fit the data, and therefore, make more accurate conclusions about the population being studied.
The expected value of s^2 (E(s^2)) is calculated using the formula E(s^2) = (n-1) * (σ^2), where n is the sample size and σ^2 is the variance of the population. This formula is derived from the MLE method, which aims to find the parameter values that maximize the likelihood of observing the sample data.
E(s^2) and Var(s^2) both represent the expected value of s^2, but they have different interpretations. E(s^2) is the expected value of the sample variance, while Var(s^2) is the variance of the expected value of the sample variance. In other words, E(s^2) is the average value of s^2 we would expect to obtain from multiple samples, while Var(s^2) measures the variability of this expected value.
When using MLE to solve Chi Square problems, we assume that the data follows a Chi Square distribution, the sample is random and independent, and the sample size is large enough for the Central Limit Theorem to hold. Additionally, we assume that the data is continuous and that the parameters being estimated are fixed and not random.
No, MLE is specifically used for solving Chi Square problems that involve estimating the variance of a population. It cannot be used for other types of Chi Square problems, such as testing for independence or goodness of fit. MLE is most commonly used in regression analysis and other statistical models that involve estimating parameters.