Lab 7 – Sampling Distribution.

 

Sampling distribution of xbar is like taking different samples, then after getting that data get their individual means and use those as the values to plot.

If you were to have sample size n=1, the shape center and variability will be the same as the graph of the population. For example, if the population graph had a left skewness to it, then the sample size will also have a similar skewness to it. This is because by getting sample size n=1, that is almost the same as getting the regular sample of the population. However, as the sample size increases, the sampling distribution will become more normal in shape, the center will be the same as the mean of the population and the standard deviation of the sample mean or variability will decrease.

During the lab, there were a few predictions that did not quite work. The reason for this was that I did not notice that changing the sample size very slightly would make such an impact on the graphs as to making them normal. Aside from this minor misconception, the predictions were not far off key. The predictions that were made were in the right direction but without the strong influence that the small change in the sampling size represented.

According to the Central Limit Theorem, the sampling distribution of sample means from any population with mean m , standard deviation s will have a normal shape, with mean mand standard deviation sigma/sqrt(n) for large values of n.

The Central Limit Theorem is only good when n is large enough to give you a normal shape. If you have n = 1, then usually, unless your population curve is normal, you will not get a normal shape. For example, in lab question #3, when the sample size is n=1, and the curve is skewed left, you can see that the graph is not normal. The Central Limit Theorem says that if n is a larger value, then the shape begins to look more normal till you can get to a point that the curvature is normal enough to be able to use the z score, or the 68-95-99.7 Rule.

In Monitoring Method 1, you cannot make the same calculations because z scores can only be used in the event that you have a normal curve. A curve with a sample size of n=1 and a population curve such as Joe’s would be generally the same. Because that is the case, the curve on the sample size of n= 1 is not normal. With this case, you cannot do similar calculations with z scores as you could with a sample size that is larger and will form a more normal curve.

Because of the information presented, I would recommend Monitoring Method 2. This is because there is a definite pattern that by increasing the sample size you decrease variability. With this in mind, the graph will look more narrowly centered, and also the proportions of getting above a 6.0 decrease, which is what you want when your job is on the line.

If his population had looked different, but had the same m value, then by increasing his sample size he would get the same general normal shape of the narrow mound with center m .

This is similar to the reasoning that stock brokers use when they select portfolios with several stocks instead of just one. By narrowing the range, your investments are safer, just as Joe’s job is safer in Monitoring Method 2. The less variability the graph has, the more certainty you have that it will only fall in between the narrow range. This means hat your chances of losing a lot is small, but this also means that your chance of gaining a lot is also small.

In looking at the graphs of question 8, The following applies to them concerning how those graphs may have been made.

  1. This graph looks like it has less variability, but it doesn’t look like its taking the form of a normal curve. This graph looks like it has a different population than the others.

  2. In this graph, the mounds are generally the same locations of the population but lower, which leads me to believe that this graph is a smaller number of samples with sample size of n=1.

  3. This graph is number of sample = 500 and sample size of n=4.

  4. This graph is number of sample = 500 and sample size of n=25.

  5. This graph looks like the population graph so this is probably sample size n=1. This could also be the population graph all together instead of it being a sampling distribution representation.