R

When we have a normally-distributed data, parameters \(\mu\) and \(\sigma\) from our PDF can completely explain the behaviour seen in our sample. With \(\mu\) represents the central tendency and \(\sigma\) the spread, we can directly compare similarly distributed samples. Often, we need to confirm how much our average value differs from other observations. In doing so, we are facing a mean difference problem in our venture of statistics. This lecture will help us proving mean differences in one-sample and two-sample problems.

The limitation when using T-Test is its inability to directly compare multiple group at once. Often times, we are interested to see whether our groups of interest present with at least on differing average value. To alleviate this issue, we can assign a generalized form of a T-Test. We will do so by analyzing between and within group variances. This analysis resulted in the sum of square with two degree of freedoms, one coming from the number of groups and another from the calculation of withing group variability.

A parametric test requires us to assume or hypothesize parameters in a population. Often, a small sample size or a highly-skewed distribution does not resemble a normal distribution. In such a case, it becomes impertinent to assume normality in our data. Even though parametric tests are quite robust against non-normal data to a certain degree, it still requires a large number of sample. With larger \(n\) and homogeneous intergroup variance, the parametric test may have a sufficient power to correctly reject the \(H_0\). However, if we cannot satisfy the required assumption, we need to drop our hypothesized claim of population parameters. In other words, we are employing a non-parametric test to measure observed differences.

Even though parametric tests on multiple groups have a favourable statistical power in non-normal data, it requires a large enough sample to correctly reject the \(H_0\). Moreover, parametric tests only applies on numeric data, as it compares the mean between assigned groups. In cases of having an ordinal data or data with a low number of sample, non-parametric tests may provide a better inference.

So far, we have solved hypotheses testings for condition where we have numeric values as our dependent variable and categoric data as our independent variables. We have yet to see the solution if we have numeric variables for both of the dependent and independent variables. After learning the descriptive statistics, we understand that we can observe the spread in our data by measuring the variance, i.e. the dispersion of each data element relative to the mean value. Similarly, we can understand the dispersion of two numeric variables by accounting the covariance.