The principle of inclusion and exclusion is a counting principle used to calculate the probability of the union of multiple events. It's based on the idea that the union's probability should add the individual probabilities and subtract the probabilities of intersections to avoid double-counting.

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.

A confidence interval is a range of values, derived from the statistical analysis of the sample data, that is likely to contain an unknown population parameter.

According to the Central Limit Theorem, as the sample size increases, the distribution of the sample mean approaches a normal distribution more closely. This means the larger the sample size, the more accurately the sample mean will represent a normal distribution.

Non-parametric statistical methods do not require the data to follow a specific distribution, which is why they are often used when the assumptions of parametric tests are violated.

The peak of a distribution represents the mode of the data, that is, the value(s) that appear most frequently in the data set. In a perfectly symmetrical distribution, the mode, median, and mean coincide at the peak.

If we fail to reject the null hypothesis, this means that there is not enough evidence in the data to support the alternative hypothesis. We do not say the null hypothesis is true, because it is possible that a type II error (false negative) occurred.

In stratified sampling, the population is divided into non-overlapping groups, or strata, such as age groups, income levels, or gender. Then, a simple random sample is taken from each stratum. Stratified random sampling can provide more precise estimates if the strata are relevant to the characteristic of interest.

A low p-value in a t-test suggests that you can reject the null hypothesis. The p-value represents the probability that the results are due to random chance, so a lower p-value means the results are less likely to be due to chance.

Independence in probability theory refers to situations where the occurrence of one event does not affect the occurrence of another event. In other words, Events A and B are independent if the fact that A occurs does not affect the probability of B occurring.