Assumptions and Testing Help (page 3)
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.
Suppose we observe the speed of the low-pressure weather system, and its associated cold front, that is crossing the continent. Imagine that we make use of the official weather forecasts to estimate the big system's speed over the next few days. Now suppose that we input this data into a hurricane-forecasting program, and come up with a forecast path for Emma. The program tells us that the mean path for Emma will take it across the mid-Atlantic coast of the USA. The computer even generates a graphic illustrating this path (Fig. 6-3).
We now have a hypothesis concerning the future path of Hurricane Emma. The actual outcome is independent of human control. If we decide to put the hypothesis illustrated by Fig. 6-3 to the test, it is called the null hypothesis. A null hypothesis is symbolized H0 (read ''H-null'' or ''H-nought'').
People who lie in the predicted path of Emma, as shown in Fig. 6-3, hope that H0 is wrong. The alternative hypothesis, symbolized H1, is the proposition that Emma will not follow the path near the one shown in Fig. 6-3. If someone asserts that Emma will go either north or south of the path assumed by H0, that person proposes a two-sided alternative. If someone else claims that Emma will travel north of the path assumed by H0, that person proposes a one-sided alternative. If yet another person proposes that Emma will go south of the path assumed by H0, it is also a one-sided alternative.
In a situation of this sort, it's possible for a whole crowd of people to come out and propose hypotheses: ''I think the storm will hit Washington, DC.'' ''No, I think it will hit New York City.'' ''I think it will hit somewhere between New York City and Boston, Massachusetts.'' ''You are all wrong. I think it will hit San Francisco, California.'' ''California! You're insane!'' The number of possible hypotheses in this scenario is limited only by the number of different opinions that can be obtained. Some are more plausible than others. We can take seriously the notion that the hurricane will strike somewhere between New York and Boston. Most people would reject the hypothesis that Emma will hit San Francisco, although the hypothesis that a person who says so is insane is subject to debate. Maybe she's only joking.
Some people, noting historical data showing that hurricanes almost never strike the mid-Atlantic coast of the USA in a manner such as that given by the computer model and illustrated in Fig. 6-3, claim that the storm will stay further south. Other people think the storm will travel north of the predicted path. There is good reason to believe either of these alternative hypotheses. In the past 100 years or so, the Carolinas and the Northeastern USA have taken direct hits from Atlantic hurricanes more often than has Delaware. There are many different computer programs in use by various government agencies, academic institutions, corporations, and think tanks. Each program produces a slightly different mean path prediction for Emma, given identical input of data. Alternative hypotheses abound. The null hypothesis H0 is a lonely proposition.
In order to determine whether or not H0 is correct, the experiment must be carried out. In this instance, that involves no active work on our part (other than getting prepared for the worst), we can only wait and see what happens. Emma will go where Nature steers her.
Here's another null/alternative hypothesis situation. Suppose we want to find out what proportion of ice-cream lovers in Canada prefer plain vanilla over all other flavors. Someone makes a claim that 25% of Canadian ice-cream connoisseurs go for vanilla, and 75% prefer some other flavor. This hypothesis is to be tested by conducting a massive survey. It is a null hypothesis, and is labeled H0.
The simple claim that H0 is wrong is the basic alternative hypothesis, H1. If an elderly woman claims that the proportion must be greater than 25%, she asserts a one-sided alternative. If a young boy claims that ''nobody in their right mind would like plain vanilla'' (so the proportion must be much lower), he also asserts a one-sided alternative. If a woman says she doesn't know whether or not the proportion is really 25%, but is almost certain that the proposition must be wrong one way or the other, then she asserts a two-sided alternative. The experiment in this case consists of carrying out the survey and tallying up the results.
In the ''USA hurricane scenario,'' H0 is almost certain to be rejected after the experiment has taken place. Even though Fig. 6-3 represents the mean path for Emma as determined by a computer program, the probability is low that Emma will actually follow right along this path. If you find this confusing, you can think of it in terms of a double negative. The computers are not saying that Emma is almost sure to follow the path shown in Fig. 6-3. They are telling us that the path shown is the least unlikely individual path according to their particular set of data and parameters. (When we talk about the probability that something will or will not occur, we mean the degree to which the forecasters believe it, based on historical and computer data. By clarifying this, we keep ourselves from committing the probability fallacy.)
Similarly, in the ''Canada ice-cream scenario,'' the probability is low that the proportion of vanilla lovers among Canadian ice-cream connoisseurs is exactly 25%. Even if we make this claim, we must be willing to accept that the experiment will almost surely produce results that are a little different, a little more or less than 25%. When we make H0 in this case, then we are asserting that all other exact proportion figures are less likely than 25%. (When we talk about the probability that something does or does not reflect reality, we mean the degree to which we believe it, based on experience, intuition, or plain guesswork. Again, we don't want to be guilty of the probability fallacy.)
Whenever someone makes a prediction or claim, someone else will refute it. In part, this is human nature. But logic also plays a role. Computer programs for hurricane forecasting get better with each passing year. Methods of conducting statistical surveys about all subjects, including people's ice-cream flavor preferences, also improve.
If a group of meteorologists comes up with a new computer program that says Hurricane Emma will pass over New York City instead of Wilmington, then the output of that program constitutes evidence against H0 in the ''USA hurricane scenario.'' If someone produces the results of a survey showing that only 17% of British ice-cream lovers prefer plain vanilla flavor and only 12% of USA ice-cream lovers prefer it, this might be considered evidence against H0 in the ''Canada ice-cream scenario.'' The gathering and presentation of data supporting or refuting a null hypothesis, and the conducting of experiments to figure out the true situation, is called statistical testing or hypothesis testing.
Species of Error
There are two major ways in which an error can be made when formulating hypotheses. One form of error involves rejecting or denying the potential truth of a null hypothesis, and then having the experiment end up demonstrating that the null hypothesis is true after all. This is sometimes called a type-1 error. The other species of error is the exact converse: accepting the null hypothesis and then having the experiment show that it is false. This is called a type-2 error.
How likely is either type of error in the ''USA hurricane scenario'' or the ''Canada ice-cream scenario''? These questions can be difficult to answer. It is hard enough to come up with good null hypotheses in the first place. Nevertheless, the chance for error is a good thing to know, because it tells us how seriously we ought to take a null hypothesis. The level of significance, symbolized by the lowercase Greek letter alpha (α), is the probability that H0 will turn out to be true after it has been rejected. This figure can be expressed as a ratio, in which case it is a number between 0 and 1, or as a percentage, in which case it is between 0% and 100%.
Practice problems for these concepts can be found at:
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