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Assumptions and Testing Help (page 2)

By — McGraw-Hill Professional
Updated on Aug 26, 2011

Null Hypothesis

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).

Assumptions and Testing

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'').

Alternative Hypothesis

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.

Assumptions and Testing

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.

Ice-Cream Lovers

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.

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