Sampling Distributions 7 2

In this video, we'll be learning about the sampling distribution. But before we talk about sampling distributions, we need to know the difference between sample distribution and sampling distribution. So, let's go through an example. Suppose we have population of 10,000 people, and we know that the mean height of everybody in this population is 5'4. Recall that sample is small portion of the population that we examine and draw conclusions from. So, from one sample, the average height could be 5'3, but in different sample, the average height could be 5'7 or 5'4. Notice how the sample mean doesn't always have to be equal to the population mean. This is because samples are relatively smaller in size, and because of that, they have more variability, contain less information, and don't always accurately represent the population. So, what's the difference between sample distribution and sampling distribution? sample distribution involves taking singular sample from population and interpreting the data. On the other hand, sampling distribution is distribution of statistic made from multiple simple random samples drawn from specific population. So, if you were creating sampling distribution of the sample mean, the first step is to find your population of interest. Let's say we were interested in measuring height. Next, you would take random sample of size For this example, we'll choose an of five. Now, we'll measure the heights of each individual in the sample. Once that's complete, we will calculate the mean height x-bar for that sample. And then finally, we would create frequency distribution of the sample mean by plotting the value of x-bar for this sample. Remember that sampling distribution involves taking multiple samples. So, we would have to do this entire process for another sample. Therefore, we would take another simple random sample, calculate x-bar, and then plot that value onto the histogram. Take another random sample, calculate bar, and then plot that onto the histogram. If we repeatedly do this hundreds and thousands of times, we eventually end up with sampling distribution. Essentially, sampling distribution is just whole bunch of bars that are stacked on top of each other. And the interesting part is, if you have enough data, the sampling distribution will be normally distributed. This is due to the central limit theorem, but we will save the discussion for this topic in the next video. So now that you know what sampling distribution is, let's compare the difference between population distribution and sampling distribution. population distribution has mean of mu and standard deviation of sigma. And when random variable follows normal distribution with mean mu and standard deviation sigma, this can be represented by the following notation. And the standardization formula for this normally distributed population would be equal to an observation minus the population mean mu divided by the population standard deviation sigma. For the sampling distribution, this is created by taking multiple random samples from the original population, then calculating the bar for each sample, and then combining it into one graph. When we do this, we find that the population mean of all the bars is actually equal to mu. In other words, the mean of the sampling distribution is equal to the mean of the original population distribution that you sampled from. However, for the standard deviation, we can clearly see that we have something different between these two graphs. The sampling distribution has much smaller spread than the population distribution, which clearly has larger spread or larger standard deviation. The standard deviation for sampling distribution is always smaller than the population standard deviation. This is because the averages are less variable than individual observations, and we know that sampling distribution is made up of averages, whereas population distribution is made up of individual observations, which can have very wide range of measurements, and therefore more variability. We find that the standard deviation for sampling distribution is equal to the population standard deviation sigma divided by the square root of where is the size of your sample. The standard deviation of sampling distribution is also called the standard error. So, when random variable follows normal distribution with mean mu and standard deviation of sigma over the square root of we can write this as notation to formally state that. And finally, the standardization formula for sampling distribution is equal to an observation minus the population mean mu of all the bars, which is just equal to mu, divided by the standard deviation, which is equal to sigma divided by the square root of So, to quickly recap what we've talked about, population distribution is distribution that is created from measuring every single individual in the population. sample distribution is distribution that is created from measuring every single individual in the sample. And most importantly, sampling distribution involves repeatedly taking sample and calculating statistic for each individual sample, and then combining that information to create distribution. Now, you might be asking, what's the point of sampling distribution in the first place? And to answer that question, it all boils down to convenience and efficiency. For example, if we wanted to know the average height of all humans living on Earth, we would first have to measure and record the heights of every person on Earth. As of 2021, there are about 8 billion people living on Earth right now. That's lot of people. Measuring every single person would take lot of time, effort, and money. So, instead, we can create sampling distribution. And from this sampling distribution, it can give us an idea of what the value of the population mean mu is without having to measure every single individual on Earth. Another great thing about the sampling distribution is that it allows us to calculate the probability of getting certain outcome based on what our sample size is equal to. So, now that you know what sampling distribution is, let's do some practice questions. Suppose it is known that the heights of all Canadians follows normal distribution with mean of 160 cm and standard deviation of 7 cm. What is the probability that the average height of 10 random Canadians is less than 157 cm? Before solve this question, I'm going to make some illustrations to help you understand what is happening. According to the question, we have normal distribution with mu of 160 and sigma of 7. Mu is always located in the center of the distribution, so we will put 160 in the center. From here, we can create intervals that increase by the standard deviation. So, one standard deviation to the right of the mean is 167. Two standard deviations to the right of the mean is 174. And one standard deviation to the left of the mean is 153, and so on and so forth. This is the population distribution. For the sampling distribution of the sample mean, we know that is equal to 10 because the question says that this is the size of the sample we are dealing with. We also know that mu bar is always equal to the mean of the parent population, so it will also be equal to 160 cm. However, the standard deviation of the sampling distribution is equal to sigma divided by the square root of In this case, the standard error is equal to 7 divided by the square root of 10, giving us an answer of 2.21. This means we can create intervals that increase by the standard deviation. Now that you can visually see the difference between the population distribution and the sampling distribution, let's tackle the question. What is the probability that the average height of 10 random Canadians is less than 157 cm. This refers to the sampling distribution, not the population distribution. Therefore, this is the area we are concerned about. We are looking for the proportion of less than 157, and this is unknown until we standardize the distribution, so we will use the standardization formula. Mu is equal to 160, sigma is equal to 7, and is equal to 10. We can replace the entire denominator by the standard error, which we have already calculated to be 2.21. All we have to do now is plug 157 into the formula, and when we do, we get score of -1.36, which corresponds to our value of 157. Now, we can use the score table to determine how much area is associated with the score. According to the table, score of -1.36 corresponds to an area of 0.0869. Therefore, we can say that the proportion of less than -1.36 is equal to 0.0869. This value is in fact the same proportion of less than 157. As result, we can say that the probability that the average height of 10 random Canadians being less than 157 is equal to 0.0869, or 8.69%, and that is the answer. For the next example, I'll introduce part two to this question. What is the proportion of all people that have heights greater than 170 cm? For this question, it's important you understand that we will not be dealing with the sampling distribution anymore, and this is because the question asks for all people. Therefore, we are going to be working with population distribution instead. We are looking for the proportion of greater than we need to standardize the distribution, and we will do this by using the standardization formula. This time, we have to use the formula for the population distribution, rather than the formula for the sampling distribution. Plugging 170 and everything else into this formula gives us Z-score value of 1.43. From here, we can use the Z-score table to see how much area is associated with the Z-score. Z-score value of 1.43 is associated with an area of 0.9236. However, remember that the Z-score table only gives us areas to the left of Z-score value. According to the table, Z-score of 1.43 has an area of 0.9236 to the left of it. However, we want the amount of area associated to the right of the Z-score value. Remember that the total area of normal distribution is equal to 100% or 1. Therefore, we can do 1 - 0.9236, giving us an area of 0.0764 to the right of this Z-score value. As result, the proportion of greater than 1.43 is equal to 0.0764. This also means that the proportion of greater than 170 is equal to 0.0764. Therefore, we can say that the proportion of all people that have heights greater than 170 cm is equal to 0.0764 or 7.64%. If you found this video helpful, consider supporting us on Patreon to help us make more videos. You can also visit our website at simplelearningpro.com to get access to mini study guides and practice questions. Thanks for watching.