1) Suppose that the school decides to get some new vending machines for non-carbonated beverages. The student body is given a choice between Arizona Iced Tea and Snapple, and the FBLA is doing a survey to determine which brand of drink people prefer. The Randomania software can help you to simulate such a survey and see how many people you need to ask in order to get a good picture of what the population really thinks.
From the Options menu, choose 4) Polling Simulations. Imagine a hat with 1500 slips of paper (one for each person in the school). On each slip of paper will be written the drink preference for that person. The computer simulation will allow you to set up that "hat" and then draw samples from it. Object 1 will be "Arizona." Let's say that 675 people in the school prefer that drink, so your hat would need to have 675 slips of paper saying "Arizona." Object 2, then, is "Snapple," with 825 slips of paper. When you get to Object 3, just hit enter and the computer will set up the hat with the slips of paper you requested.
2) Setting up the hat gives us a model of the preferences of the entire population of the school. The FBLA students wouldn't know how the hat is filled, however. They are going to sample from the hat to try to figure out what the preferences are. Ask the computer for a sample size of 100. The computer will then reach into the hat and pull out 100 slips of paper. Each slip has an equal chance of being picked, so we refer to this as a "random" sample.
Notice that in the true population, Snapple wins over Arizona. Does it also come out that way in your sample? Or could you maybe just happen to pick a lot of Arizona drinkers and give a distorted picture of how things really are? Just how far off is your sample?
3) Repeat the experiment several times, maintaining that sample size of 100. The computer will refill the hat and draw out a different clump of 100 slips each time. See if you get a sample that shows Arizona winning, when in fact it is pretty far behind in the true population.
4) If the percentages in your sample are more than three percentage points away from the true values, that result is considered to be outside the margin of acceptable error. Of course, your sample can't be perfect, but we'd like it to be fairly close to the true situation. Anything off by more than three percent should be rejected. Run the same experiment ten times and keep track of how often the results are too far off to be acceptable.
5) You can also get the computer to run that same experiment a whole bunch of times. Hit m for multiple experiments. Then ask it to run the experiment 1000 times. The computer will display a bar graph to show how many of the trials came out with a result within the margin of error (valid) and how many were outside of the margin of error. How often is the experiment outside the margin of error? Half the time? A third of the time? A tenth of the time?
6) Actually, we won't be satisfied with a sample size if it produces misleading results half the time. It needs to be within the margin of error at least 95% of the time. That means it could be misleading 1 out of 20 times, on average, but no more. This is called a 95% confidence level.
7) You can see that the sample size of 100 is clearly inadequate. We can't be at all confident that it's a good representation. Try increasing the sample size to 200 and see how that helps.
8) Keep increasing the sample size until you feel confident that it will give valid results at least 95% of the time.
9) Of course, you could get a delightfully accurate sample just by asking every single person in the population. As a pollster, why might you want to avoid having a really big sample?
10) Now consider a larger population-perhaps a small town of 15000 people. Keep the proportions the same for Arizona and Snapple. Design a new experiment to investigate what sample size you would need for that new population.
11) Next think about sampling the entire country of 300 million people, still with the same proportion of Snapple lovers. How big would the sample need to be then to maintain your 95% confidence level?
12) Try changing the margin of error (both higher and lower) and see what effect that has on the sample size needed to maintain 95% confidence.
13) 95% confidence sounds really good until you think about the fact that the sample will be misleading 1 out of 20 times. Repeat your experiments, but this time seek a 99% confidence level. How does that affect necessary sample size?
14) All of these experiments were run with a population that was split 45%-55% between the two choices. Try changing that to a 50%-50% distribution and see if it changes your other conclusions. Then try 70%-30%.
15) Try similar experiments with three possible drinks instead of two.
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