Making Predictions

Suppose that you decide to sell commemorative T‐shirts at your town's centennial picnic. You know that you can make a tidy profit, but only if you can sell most of your supply of shirts because your supplier will not buy any of them back. How many shirts can you reasonably plan on selling?

Your first question, of course, is this: How many people will be attending the picnic? Suppose you know that 100,000 tickets to the event have been sold. How many T‐shirts should you purchase from your supplier to sell on the day of the event? 10,000? 50,000? 100,000? 150,000? How many of each size—small, medium, large, extra‐large? And the important question: How many T‐shirts must you sell in order to make some profit for your time and effort and not be left with an inventory of thousands of unsold shirts?

Ideally, before you buy your inventory of T‐shirts, you need to have an accurate idea of just how many ticket holders will want to purchase centennial T‐shirts and which sizes they will want. But, obviously, you have neither the time nor the resources to ask all 100,000 people whether they plan to purchase commemorative T‐shirts. If, however, you could locate a small number of those ticket holders—for example, 100—and get an accurate count of how many of those 100 would purchase a T‐shirt, you would have a better idea of how many of the 100,000 ticket holders would be willing to buy one.

That is, of course, if the 100 ticket holders that you asked (called the sample) are not too different in their intentions to purchase T‐shirts from the total 100,000 ticket holders (called the population). If the sample is indeed representative (typical) of the population, you could expect about the same percentage of T‐shirt sales (and sizes) for the population as for the sample, all things being equal. So, if 50 of your sample of 100 people say they cannot wait to plunk down $10 for a centennial T‐shirt, it would be reasonable to expect that you would sell about 50,000 T‐shirts to your population of 100,000. (At just $1 profit per T‐shirt, that is $50,000!)

But before you start shopping to buy a yacht with your profits, remember that this prediction of total T‐shirt sales relies heavily upon the sample being representative (similar to the population), which may not necessarily be the case with your sample. You may have inadvertently selected a sample that has more expendable income or is a greater proportion of souvenir T‐shirt enthusiasts or who knows what else. Are you reasonably certain that the intentions of the sample of 100 ticket holders reflect the intentions of the 100,000 ticket holders? If not, you may quite possibly be stuck with tens of thousands of centennial T‐shirts and no profit to splurge on a yacht.

You can see why choosing a random sample is a critical part of the process of statistics. Even with careful sampling methods, our conclusions are still educated guesses. This is because one sample does not perfectly represent the population, and different samples may give different results.