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# Statistics and Forecast Help (page 2)

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By McGraw-Hill Professional
Updated on Sep 13, 2011

## Inference

The term inference refers to any process that is used to draw conclusions on the basis of data and hypotheses. In the simplest sense, inference is the application of reason, common sense, and logic. In statistics, inference requires the application of logic in specialized ways.

We have already seen two tools that can be used for statistical inference: confidence intervals and significance testing. Both of these tools give us numerical output. But ultimately, it is a matter of subjective judgment whether or not we should come to any particular conclusion based on such data. Sometimes a judgment is easy to make. Sometimes it is difficult. Sometimes inferences can be made and conclusions drawn with a ''cool head'' because nothing important depends on our decision. Sometimes there are emotional or life-and-death factors that can cloud our judgment. When our judgment is bad, we might make an inference when we should not, or else fail to make an inference when we should.

Consider again the ''USA hurricane scenario.'' If you live on the oceanfront and a hurricane is approaching, what should you do? Board up the windows? Go to a shelter? Find a friend who lives in a house that is better constructed than yours? Get in your car and flee? Statistics can help you decide what to do, but there are no numbers that can define an optimal course of action. No computer can tell you what to do.

In the ''Canada ice-cream scenario,'' suppose we conduct a survey by interviewing 12 people. Three of them (that's 25%) say that they prefer vanilla. Does this mean that H0, our null hypothesis, is correct? Most people would say no, because 12 people is not a big enough sample. But if we interview 12,000 people (taking care that the ages, ethnic backgrounds, and other factors present an unbiased cross-section of the Canadian population) and 2952 of them say they prefer vanilla, we can reasonably infer that H0 is valid, because 2952 is 24.6% of 12,000, and that is pretty close to 25%. If 1692 people say they prefer vanilla, we can infer that H0 is not valid, because 1692 is only 14.1% of 12,000, and that is nowhere near 25%.

How large a sample should we have in order to take the results of our survey seriously? That is a subjective decision. A dozen people is not enough, and 12,000 is plenty; few people will dispute that. But what about 120 people? Or 240? Or 480? The general rule in a situation like this is to get as large a sample as reasonably possible.

## Statistics and Forecast Practice Problems

#### Practice 1

Imagine that you are a man and that you live in a town of 1,000,000 people. Recently, you've been seeing a lot of women smoking. You start to suspect that there are more female smokers in your town than male smokers. You discuss this with a friend. Your friend says, ''You are wrong. The proportion of female to male smokers is 1:1.'' You say, ''Do you mean to tell me that the number of women smokers in this town is the same as the number of men smokers?'' Your friend says, ''Yes, or at least pretty close.'' You counter, ''There are far more women smokers than men smokers. I see it every evening. It seems that almost every woman I see has a cigarette in her mouth.'' Your friend has a quick retort: ''That's because you spend a lot of time at night clubs, where the number of women who smoke is out of proportion to the number of women smokers in the general population. Besides that, if I know you, you spend all your time looking at the women, so you haven't noticed whether the men are smoking or not.''

Suppose you and your friend decide to conduct an experiment. You intend to prove that there are more female smokers in your town than male smokers. Your friend offers the hypothesis that the number of male and female smokers is the same. What is a good null hypothesis here? What is the accompanying alternative hypothesis? How might we conduct a test to find out who is right?

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