The main difference between a histogram and a bar plot is the type of data they represent. A histogram is used for continuous data, where the bins represent ranges of data, while a bar plot is used for categorical data to compare the frequency or count of different categories.

The error term in a simple linear regression model is the difference between the observed and predicted values. It captures the variability in the dependent variable that is not explained by the independent variable in the model.

If the residuals are not randomly distributed (e.g., if they form a pattern), it suggests that the linear regression model is not a good fit for the data. This could be because the relationship between the variables is not linear, or because the data exhibits heteroscedasticity (unequal variances of errors), among other reasons.

The Chi-square test for goodness of fit is used with categorical data. It compares the observed frequencies in each category with the frequencies we would expect to see if the data followed the theoretical distribution.

In the Mann-Whitney U test, the null hypothesis is that there is no significant difference between the groups. More specifically, it states that the probability that a randomly selected value from the first group is greater than a randomly selected value from the second group is equal to 0.5.

Increasing the sample size decreases the width of the confidence interval. The larger the sample size, the more information you have, and thus the less uncertainty (which translates into a smaller standard error and narrower confidence interval).

If A and B are independent events, the probability of both occurring is P(A ∩ B) which is equal to P(A) * P(B). This is the fundamental characteristic of independent events in probability.

The power of a test is the ability of the test to detect an effect if it truly exists. It's the probability that the test correctly rejects a false null hypothesis. High power is desirable because it means the test is less likely to make a Type II error (false negative). When designing a study, it's important to choose a sample size and significance level that will provide enough power to detect an effect if one exists.

In the context of statistics, a 'population' refers to the entire set of individuals or items that we are interested in studying. This could range from all the people living in a country, to all the particles in a chemical solution, to all the sales transactions in a store, and so forth. When we collect data from every individual or item in the population, it's called a census. The information gained from a population is reliable and accurate because it accounts for everyone or everything we want to study.

An event in the context of probability refers to a possible outcome or combination of outcomes from a random experiment. For instance, if we toss a coin, the possible outcomes are 'Heads' and 'Tails'. Each of these outcomes is an event.