Sampling Distribution of a Mean for Personal Income

Statistical Topic:
We have worked with the theoretical distributions for a population, the normal and standard normal but these are not the distributions we need to complete an inferential test for one-sample mean. This distribution is called the sampling distribution of xbar. Instead of personal income values for each state, this distribution consists of means of random samples.

Student Issue:
How does the sampling distribution of xbar compare with the population distribution.

Data Set:
Table 1. Per Capita Personal Income by State for 1997 contains U.S. Census Bureau estimates of personal income.

Goals of Data Analysis Lab:
Using the population data set, we can generate enough random samples to draw a sampling distribution for xbar. We can then compare this distribution of sample means to the distribution of income values. We can also compare the parameters for these two distributions.

Statistical Techniques:

Table 1 provides random samples of 20 incomes randomly selected from the population of incomes. Each student will compute two sample means from two sets of these random numbers as designated by the instructor. Place both of your sample means on the board for other classmates to use. How many possible random samples of size 20 could be generated from this population of state income data?
Draw a histogram of all of these sample means using the same x-axes as in Figure 1. This histogram is a small example of the sampling distribution for one sample mean. If we could we would like to have at least a 100 sample means to get a good picture of this distribution. Compare this distribution with Figure 1. Compare the middle, spread, shape and outliers for this histogram of sample means. How does this histogram of sample means compare with the histogram in Figure 1? Why are there differences?
What does the Central Limit Theorem has to do with this distribution? State this theorem.