One-way ANOVA tests the effect of one independent variable on a dependent variable, while two-way ANOVA tests the effect of two independent variables on a dependent variable. Additionally, two-way ANOVA allows for the examination of interactions between the independent variables.

Residual analysis is important because it helps us to validate the assumptions of the regression model, such as linearity, independence, normality, and equal variance (homoscedasticity). This is crucial for the reliability and validity of the regression model.

The Total Probability Rule provides a way to compute the probability of an event from the probabilities of that event occurring within disjoint subsets of the sample space. It essentially allows you to break down the probability of complex events into simpler or more basic component events.

In a Chi-square test for goodness of fit, the degrees of freedom are calculated as the number of categories minus one. This reflects the number of values in the final calculation that are free to vary.

The choice of bin size in a histogram can greatly affect the resulting visualization. If the bins are too large, important features of the data may be obscured. If the bins are too small, the histogram may appear too 'noisy' and it may be difficult to interpret underlying patterns. Thus, the choice of bin size can indeed change the perceived shape of the histogram.

Heteroscedasticity can be resolved by transforming the dependent variable, typically using a logarithmic transformation. This often stabilizes the variance of the residuals across different levels of the predictors.

A Poisson distribution is used when we are counting the number of times an event happens over a fixed interval of time or space, and the events are independent and occur at a constant average rate. It's often used to model random events such as calls to a call center or arrivals at a website.

Qualitative data can be transformed into quantitative data for analysis by coding the responses. This is a process where categories or themes identified in the qualitative data are assigned numerical codes. These numerical codes can then be used in statistical analyses. For instance, if you have data on types of pets (dogs, cats, etc.), you can assign a numerical code (1 for dogs, 2 for cats, etc.) to transform this qualitative data into quantitative data.

When we say that a distribution is skewed, we mean that the distribution is not symmetric about its mean. In a skewed distribution, the data points are not evenly distributed around the mean, with more data on one side of the mean than the other.

If the p-value in a Chi-square test is smaller than the significance level, we reject the null hypothesis in favor of the alternative hypothesis. This suggests that there is a significant association between the variables.