V8 Sampling Distribution Theory

This is part three of our video series related to sampling distribution theory. I'm Renee Clark from the Swanson School of Engineering. In this video, we are going to discuss the following topics. We're first going to do a review of the definition of a sampling distribution, and then we're going to talk about the shape of the sampling distribution of the mean when the underlying parent population is normal, and why this is the case. Okay, so, first, let's quickly review what is a sampling distribution. Okay, a sampling distribution is the distribution of a statistic such as the sample average.

So, pictorially, if you were to take different random samples out of a population, okay, and calculate a statistic such as the average for each sample, okay, as shown here, those averages, or those statistics, would likely differ ,right? 16.0, 16.7, 16.8, etc. Okay, because those averages, or those statistics, are based on completely different random samples. Okay, and the items in each sample are very likely to differ. Okay, so these averages, or these statistics, are random variables. Okay, therefore we can plot them. They're going to have a distribution. So, if we were to plot these statistics or these averages, they might form a distribution… distribution such as the following. So, as you can see, these are distributions of the averages. Okay, based on different random samples that are taken out of the population for the random variable X, okay, this distribution of X bars is known as the sampling distribution of the mean. So, next we're going to talk about the shape of the sampling distribution of xbar. Okay, so, if your population, or your parent distribution, is normally distributed. Okay, so, another way to write that is if x is distributed normally as shown on the right. Okay, if your parent population has a normal distribution, then automatically the sampling distribution will be normally distributed. Or another way to write that is xbar will be normally distributed. Okay, so, if x distributed normally, xbar automatically distributed normally as shown on the right here. Okay, so, on the right is the distribution for xbar. Okay, this is your sampling distribution. Okay, this is your parent distribution.

Another way to say it- it's your population of X, okay? Okay, so, why is this the case? It’s due to the following theorem that states the following. Okay, this theorem states that if your X's are random variables having a normal distribution. So, another way to state that is if x is normally distributed or you… if you want to write Xi are normally distributed. Okay, then this random variable right here, which as you will notice is a linear combination of the X's, okay, then that variable also has a normal distribution. Okay, so, let's explain this just a bit further.

We know that xbar, or the sample average, is equal to the following, right? You add up your various x's and you divide by the number of x's that there are. Okay, you sum up your items, divide by the number of items. Okay, another way to write that is you can bring the N or 1 over n out in front of each term, right, and write it like this. Well, you will notice what I'm writing is a linear combination of the X's, or the xi with constant 1/ n in front of each term. Okay, xbar is therefore a linear combination of the X I’s with constant 1 / n as a coefficient of each term.

Okay, and again, going back up to our theorem, it says that, if your X's are normally distributed, then any combination of them- any linear combination of them- will be normally distributed. Okay, so again, right here, if your various X's are normally distributed, any linear combination of them will also be normally distributed. That is why X bar will be automatically normally distributed, or the sampling distribution will be automatically normally distributed if the parent population or the x's are normally distributed.

Thank you to the National Science… National Science Foundation under Grant 233 582 for supporting our work.

Thank you for watching.