Distribution of the sample means: Here we re-consider the GMAT scores from 2003 and take three different samples each of size 10. The average of these sample means (642.67) is very close to the population average of 643.48.
Rates of return for different groups of stocks: In this example, the sample mean is distributed as in this Excel file. Note that as the sample size n increases, the standard deviation of the sample mean reduces. (In this example, the population is NOT normallly distributed, but the distribution of the sample means resembles normal as the sample size increases.)
Note: If you are not comfortable visualizing stock returns, you can also assume that there is a lottery with four possible outcomes: There are 60 balls in a bag, and 15 of them are RED resulting in a win of $1, 15 are BLUE resulting in $2, 15 are YELLOW resulting in $3, and 15 are GREEN resulting in a $4 win.
We use Visual Statistics (Chapter 8: Properties of Estimators) to simulate the distribution of different sample means. Please download and install this on your computer. (Make sure you install using INSTRUCTOR COMPLETE option.)
Let's consider four different cases with normal and non-normal populations, and with small and large sample sizes: (A "large" sample size is usually at least 30.)
Population |
Number of samples |
Sample size, n |
Distribution of X-bar |
|---|---|---|---|
Normal |
1000 |
4 (small) |
|
Normal |
1000 |
49 (large) |
|
Non-normal |
1000 |
4 (small) |
|
Non-normal |
1000 |
49 (large) |
Here is a graphical summary of our results for the distribution of sample means for different shapes of populations and different sample sizes.
"New Coke'' : The Coca-Cola Company introduced the "New Coke" in April 1985 and stopped the production of the original coke. This turned out to be a poor decision as the sales plummeted. The company then decided to return to the original formula in June 1985. At that time the "Old Coke" became the "Coke Classic." Here's the sequence of events resulting in several name changes:
Here's a documentary 
telling the story of the disastrous launch of "New Coke" back in 1985. Let's pay attention to 1:40 and 2:50.
How would you conduct the market research to see if drinkers would switch to New Coke?
This problem requires calculating the probability Pr(Z < -4.08) which is not in any of the tables. So we use Megastat and get this result.