Assumptions and Testing Help
Introduction to Assumptions and Testing
A hypothesis is a supposition. Some hypotheses are based on experience, some on reason, and some on opinion. A hypothesis can be ''picked out of the blue'' to see what would happen (or be likely to happen) under certain circumstances. Often, the truth or falsity of a hypothesis can greatly affect the outcome of a statistical experiment.
It's An Assumption
A hypothesis is always an assumption. Often it takes the form of a prediction. Maybe it is correct; maybe not. Maybe it will prove true, maybe not. Maybe we will never know one way or the other.
When data is collected in a real-world experiment, there is always some error. This error can result from instrument hardware imperfections, limitations of the human senses in reading instrument displays, and sometimes plain carelessness. But in some experiments or situations, there is another source of potential error: large portions of vital data are missing. Maybe it is missing because it can't be obtained. Maybe it has been omitted intentionally by someone who hopes to influence the outcome of the experiment. In such a situation, we might have to make an ''educated guess'' concerning the missing data. This ''guess'' can take the form of a hypothesis.
Imagine a major hurricane named Emma churning in the North Atlantic. Suppose you live in Wilmington, Delaware, and your house is on the shore of the Delaware River. Does the hurricane pose a threat to you? If so, when should you expect the danger to be greatest? How serious will the situation be if the hurricane strikes? The answers to these questions are dependent variables. They depend on several factors. Some of the factors can be observed and forecast easily and with accuracy. Some of the factors are difficult to observe or forecast, or can be only roughly approximated.
The best the meteorologist and statistician can do in this kind of situation is formulate a graph that shows the probability that the hurricane will follow a path between two certain limits. An example of such a plot, showing the situation for the hypothetical Hurricane Emma, is shown in Fig. 6-1. The probability that the storm will strike between two limiting paths is indicated by the percentage numbers. This is why the values increase as the limiting paths get farther away, in either direction, from the predicted path, which lies midway between the two dashed lines marked 25%.
The weather experts use several models to make path predictions for Hurricane Emma. Each of these models uses data, obtained from instrument readings and satellite imagery, and processes the data using specialized programs on a supercomputer. The various models ''think'' in different ways, so they don't all agree. In addition to the computer programs, the hurricane experts use historical data, and also some of their own intuition, to come up with an official storm path forecast and an official storm intensity forecast for the next 24 hours, 2 days, 3 days, and 5 days.
Imagine that, in our hypothetical scenario, all the computer models agree on one thing: Emma, which is a category-5 hurricane (the most violent possible), is going to stay at this level of intensity for the next several days. The weather experts also agree that if Emma strikes land, the event will be remembered by the local residents for a long time. If it goes over Wilmington, the Delaware River will experience massive tidal flooding. You, who live on the riverfront, do not want to be there if and when that occurs. You also want to take every reasonable precaution to protect your property from damage in case the river rises.
Along with the data, there are hypotheses. Imagine that there is a large cyclonic weather system (called a low), with a long trailing cold front, moving from west to east across the USA. A continental low like this can pull a hurricane into or around itself. The hurricane tends to fall into, and then follow, the low, as if the hurricane were a rolling ball and the front were a trough. (This is where the expression ''trough'' comes from in weather jargon.) But this only happens when a hurricane wanders close enough to get caught up in the wind circulation of the low. If this occurs with Emma, the hurricane will likely be deflected away from the coast, or else make landfall further north than expected. Will the low, currently over the western United States, affect Emma? It's too early to know. So we formulate hypotheses:
- The low crossing North America will move fast, and will interact with Emma before the hurricane reaches land, causing the hurricane to follow a more northerly path than that implied by Fig. 6-1.
- The low crossing North America will stall or dissipate, or will move slowly, and Emma will follow a path near, or to the south of, the one implied by Fig. 6-1.
Imagine that we enter the first hypothesis into the various computer models along with the known data. In effect, we treat the hypothesis as if it were factual data. The computer models create forecasts for us. In the case of the first hypothesis, we might get a forecast map that looks like Fig. 6-2A. Then we enter the second hypothesis into the computer programs. The result might be a forecast map that looks like Fig. 6-2B.
We can make the hypothesis a variable. We can assign various forward speeds to the low-pressure system and the associated front, enter several values into the computer models, and get maps for each value.
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