Sampling distributions are the foundations of statistical tests. In this video, learn about the sampling distribution at the basis of two-sample hypothesis testing.

- [Instructor] We turn our attention to the sampling distribution for independent samples. For testing the difference between two independent samples the possible hypothesis tests are One-Tailed test, where the null hypothesis is that the difference between the two parameters is less than or equal to zero, and the alternative hypothesis is that the difference between them is greater than zero. One-Tailed test in the other direction, where the null hypothesis is that the difference between the two parameters is greater than or equal to zero, and alternative hypothesis is that the difference between them is less than zero, or two-tailed, in which the null hypothesis is that the difference between the parameters is equal to zero and the alternative hypothesis is that the difference is not equal to zero. To test hypothesis you have to understand the appropriate sampling distributions. A sampling distribution is the distribution of all possible values of a statistic for a given sample size. The standard error is the standard deviation of a sampling distribution. Here's how the sampling distribution works for two independent samples. Select a sample from each of two populations and calculate their means. Calculate the difference between their means and repeat this many times for many samples. The set of all those differences is the sampling distribution. Elaborating on this, a set of these differences is called the Sampling Distribution of the Difference Between Means. Its standard deviation is called the Standard Error of the Difference Between Means. Bear in mind that we never actually create a sampling distribution when we do a study. We don't have to because the Central Limit Theorem tells us that if the samples are large, the sampling distribution of the difference between means is approximately a normal distribution, and that if the populations are normally distributed, the sampling distribution of the difference between means is a normal distribution even if the samples are small. It's also the case that the Mean of the Sampling Distribution of the Difference Between Means is the mean of the first population minus the mean of the second population, and that the Standard Error of the Difference Between Means is the square root of the variance of population over sample size one plus the variance of population two over sample size two, and this, according to the central limit theorem is what the Sampling Distribution of the Difference Between Means looks like. Notice the values along the x-axis. They're in terms of the standard error of the difference between means. Little more on the standard error. If the two population variances are equal, we just use the generic sigma squared term and by doing some algebra, the standard error of the difference between means is equal to the population standard deviation times the square root of one over sample size one plus one over sample size two. Now, in my experience, I've found that it's instructive to compare standard errors. The standard error of the mean for one sample after a little algebra, is the population standard deviation times the square root of one over the sample size. As we've just seen, the Standard Error of the Difference Between Means can be written as the population standard deviation times the square root of one over sample size one plus one over sample size two. Note the similarities and the differences. So to sum up, we looked at the sampling distribution of the difference between means. It follows the Central Limit Theorem. Its mean is equal to population mean one minus population mean two, and and the standard deviation and the standard error of the difference between means is the square root of the variance of population one over sample size one plus the variance of population two over sample size two.

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Video: Distributions for independent samples