The Chi-square test for independence is used to test for a relationship or association between two categorical variables.

A two-way ANOVA involves two independent variables, each with any number of levels/groups. It allows simultaneous analysis of the effects of these variables.

A negative Spearman's rank correlation coefficient indicates an inverse association between two variables. That is, as one variable increases, the other tends to decrease.

A confidence interval for a proportion might be skewed when the proportion is very close to 0 or 1. In these cases, the distribution of sample proportions is not symmetrical, leading to skewed intervals.

When two or more data points have the same value, they are considered tied. The Kruskal-Wallis Test assigns them the average of the ranks that the tied values would have received had they been different.

Bayes' Theorem provides a mathematical framework for updating probabilities, which can be interpreted as degrees of belief, based on the evidence at hand. Thus, it assists in decision making under uncertainty by allowing for the incorporation of new information.

Inferential statistics is a branch of statistics that allows us to use data from a sample to infer or predict trends about the overall population. This technique is immensely useful as it's often impractical or impossible to collect data from an entire population. Inferential statistics makes use of various techniques such as probability, hypothesis testing, correlation, and regression to draw conclusions.

The key difference between a t-test and an ANOVA is the number of groups being compared. A t-test is used to compare the means of two groups, while ANOVA is used to compare the means of three or more groups.

The main difference between one-tailed and two-tailed tests is the directionality of the hypothesis. One-tailed tests look for an effect in a specific direction, while two-tailed tests look for an effect in either direction.

In hierarchical clustering, a dendrogram is used to visualize the hierarchy of clusters. It is a tree-like diagram that records the sequences of merges or splits.

The standard deviation is the square root of the variance. It measures the average distance between each data point and the mean. Like the variance, it expresses the dispersion of data around the mean, but unlike the variance, its units are the same as the original data, making it easier to interpret.

The primary purpose of PCA is to reduce the dimensionality of data while maintaining as much information as possible. It transforms the data into a new, lower-dimensional set of variables that are uncorrelated and that explain the maximum possible amount of variance in the data.