Today we reviewed weighted averages. Then we talked about how a probability model with numerical outcomes has a theoretical average which is the weighted average of the outcomes with the probabilities as the weights. This is also known as the expected value. We did the following workshop in class.
Today we talked some more about the Law of Large Numbers which says that the average of a sample (\(\bar{x}\)) tends to get close to the theoretical average (\(\mu\)) as the sample size N gets large.
We also talked about another idea called the Central Limit Theorem which says that when you take a large sample from any probability model with numerical outcomes, the distribution of the sample mean \(\bar{x}\) has the following three properties:
We didn’t do a workshop in class, but we did do the following exercises.
If you bet $1 on black in one game of roulette, \(\mu= \$0.947\) and \(\sigma = \$1.00\). What is the distribution of the average winnings if you play N=100 games of roulette? What if you play N=10,000 games? If you were to play N=10,000 games, what would the probability of losing money be?
The heights of men in the USA are normally distributed with mean \(\mu = 70\) inches and standard deviation \(\sigma=3\) inches. If I teach a class with \(N=20\) students, what is the theoretical mean and standard deviation of the average height \(\bar{x}\) for the whole class? What is the probability of getting a class of 20 students with an average height over 71 inches?
We talked about two-way tables today. We looked at how to find row and column proportions in a two-way table. We also made segmented bar graphs using Excel. And we defined the relative risk which is when you divide two corresponding row or column proportions to find out how many times larger one is than the other.
We did this workshop in class.