The requirements for this assignment were to compare and contrast the “…standard normal and the student t, or simply the t distribution. If we use the standard normal when Ïƒ [sigma, standard deviation] for the population is known and use the t distribution when Ïƒ is not known, explain any differences in the two distributions when n [sample size] <” 30? Explain why, using an example if necessary, there is not a difference between z and t when n > 30.” This assignment was the third written response for my Statistics II class (Quantitative Tools for Management) during the Spring 2007 semester at the University of Massachusetts at Amherst’s online program; I earned a 5/5 for the below response:
In general, as the sample size decreases in a t distribution, it becomes more and more similar to a z distribution. As the sample size increases, the standard deviation decreases in both z and t distributions. In essence, a larger sample size leads to a more accurate end result. The exponential decline in differences between the t and z distribution is an outcome of the central limit theorem. As the standard deviation decreases (due to the increased reliability of an increasingly larger sample size), the differences between the two distributions are so accurate that they both approximate a standard normal distribution.
In the real world, we see this principle in practice all the time, as I’ve mostly been a windows user, more time with Mac OS X will decrease the amount of times I use control-c for copy instead of apple-c; [although WebCT threw me for a loop since it still uses control-c as does OpenOffice.org]. It’s the basic concept behind learning, the more information you take in the smarter you’ll be until you’re very nearly perfect (time plays a factor as you can’t learn everything in a life time) and a very pervasive concept in business; the more cars Honda makes in the exact same manner, the fewer mistakes there’ll be. If their new plant made a car that wouldn’t start in their first batch of five cars, the 20% failure rate is not very indicative of Honda quality. You would expect that as they made a million of the same model they’re failure rate would be much, much lower. Each million cars produced should bring their failure rate closer to 0%. If it doesn’t then their is a flaw in the manufacturing process.