# Testing the Difference Between Two Means, Two Proportions, and Two Variances

In statistics, the two-sample test is a statistical test used to compare two populations after one of them has undergone a transformation. This test is known as the t test when comparing populations based on a normal distribution, which is the most common case. In other cases, the name of the test is the Kruskalâ€“Wallis test. The two-sample test compares two populations. Population 1 may be a sample of a larger population or a population that was randomly selected from a larger population. In either case, population 1 is referred to as the sample of interest. The population of interest may be a population of individuals or a population of measurements. In the case of a population of individuals, the sample of individuals is called the sample of individuals. In this case, the population of interest is the population of individuals. In the case of a population of measurements, the sample of measurements is called the sample of measurements. In this case, the population of interest is the population of measurements. The second population is referred to as the second sample. In the case where the second population is a sample of the first population, it is called the second sample of individuals or measurements. In this case, the second population is also the sample of interest. In the case where the second population is a random sample of the first population, it is called the second sample of measurements. In this case, the second population is the sample of interest. Both samples may be populations of measurements. In the case of a population of individuals, the second population is a sample of measurements of the individuals from the population. In a population of measurements, the second population is a sample of measurements of the measurements from the population. The population of interest may or may not be a population of values. The population of measurements may or may not be a population of values. In the case of a population of individuals, the population of measurements is the population of measurements for the individuals from the population. In a population of measurements, the population of measurements is the population of measurements for the measurements from the population. The two samples of interest are compared. The two samples of measurements are compared. In the case of a population of individuals, the two samples of measurements are compared. In the case of a population of measurements, the two samples of measurements are compared. The two means of the two populations are compared. The two variances of the two populations are compared. In the case of a population of individuals, the two variances of the two populations are compared. In the case of a population of measurements, the two variances of the two populations are compared. The two proportions of the two populations are compared. In the case of a population of measurements, the two proportions of the two populations are compared. If the two samples of interest are samples of a population, ?, ?, and the significance level of the test is ?, ?, then the two-sample test is a type I error when the null hypothesis is true and a type II error when the null hypothesis is not true. This is known as the alpha level of the test. The p value of the test is the probability that the observed difference in the two means is equal to or less than the smallest possible difference, given that the null hypothesis is true. This is known as the confidence interval of the test. If the two samples of interest are not samples of a population, but are instead samples of measurements from a population, then the two-sample test is a type II error when the null hypothesis is true and a type I error when the null hypothesis is not true. This is known as the beta level of the test. The p value of the test is the probability that the observed difference in the two means is equal to or greater than the largest possible difference, given that the null hypothesis is true. This is known as the confidence interval of the test. The interpretation of the two-sample test is as follows: If the two populations are different, the two-sample test rejects the null hypothesis that the two populations are the same (are equal). In this case, the common name for the test is the t or "two-sample t test" (see also t-distribution). If the two samples or measures are different, the two-sample test rejects the null hypothesis that the two samples or measures are the same (are equal). In this case, the common name for the test is the d or "two-sample d test" (see also standard deviation). Rejecting the null hypothesis that the two populations are equal is a type I error. Rejecting the null that the two samples are equal is a type II error. The p value of the two-sample test can be used to determine the probability of obtaining a difference of equal or larger magnitude as the observed difference in