Statistics and Forecast Help (page 3)
Introduction to Forecast
Let's revisit the ''USA hurricane scenario.'' The predicted path for Hurricane Emma shown in Fig. 6-3 is a hypothesis, not an absolute. The absolute truth will become known eventually, and a path for Emma will be drawn on a map with certainty, because it will represent history!
The "Purple Line"
If you live in a hurricane-prone region, perhaps you have logged on to the Internet to get an idea of whether a certain storm threatens you. You look on the tracking/forecast map, and see that the experts have drawn a ''purple line'' (also known as the ''line of doom'') going over your town on the map! Does this tell you that the forecasters think the storm is certain to hit you? No. It tells you that the ''purple line'' represents the mean predicted path based on computer models.
As time passes – that is, as the experiment plays itself out – you will get a better and better idea of how worried you ought to be. If you're the coolheaded scientific sort, you'll go to the Internet sites of the government weather agencies such as the National Hurricane Center, analyze the data for a while, and then make whatever preparations you think are wise. Perhaps you'll decide to take a short vacation to Nashville, Tennessee, and do your statistical analyses of Emma from there.
Confidence Intervals Revisited
Instead of drawing a single line on a map, indicating a predicted track for Hurricane Emma, it's better to draw path probability maps such as the ones in Figs. 6-1 and 6-2. These maps, in effect, show confidence intervals. As the storm draws closer to the mainland, the confidence intervals narrow. The forecasts are revised. The ''purple line'' – the mean path of the storm – might shift on the tracking map. (Then again, maybe it won't move at all.) Most hurricane Web sites have strike-probability maps that are more informative than the path-prediction maps.
Probability Depends on Range
Imagine that a couple of days pass, Emma has moved closer to the mainland, and the forecasters are still predicting a mean path that takes the center of Emma over Wilmington. The probability lines are more closely spaced than they were two days ago. We can generate a distribution curve that shows the relative danger at various points north and south of the predicted point of landfall (which is actually on the New Jersey coast east of Wilmington). Figure 6-4A is a path probability map, and Fig. 6-4B is an example of a statistical distribution showing the relative danger posed by Emma at various distances from the predicted point of landfall.
The vertical axis representing the landfall point (labeled 0 in Fig. 6-4B) does not depict the actual probability of strike. Strike probabilities can be ascertained only within various ranges – lengths of coastline – north and/or south of the predicted point of landfall. Examples are shown in Figs. 6-5A and B.
In the ''Canada ice-cream scenario,'' a similar situation exists. We can draw a distribution curve (Fig. 6-6) that shows the taste inclinations of people, based on the H0 that 25% of them like plain vanilla ice cream better than any other flavor. Given any range or ''margin of error'' that is a fixed number of percentage points wide, say ±2% either side of a particular point, H0 asserts that the greatest area under the curve will be obtained when that range is centered at 25%. Imagine that H0 happens to be true. If that is the case, then someone who says ''Our survey will show that 23% to 27% of the people prefer vanilla'' is more likely to be right than someone who says ''Our survey will show that 12% to 16% of the people prefer vanilla.'' In more general terms, let P be some percentage between 0% and 100%, and let x be a value much smaller than P. Then if someone says ''Our survey will show that P% ± x% of the people prefer vanilla,'' that person is most likely to be right if P% = 25%. That's where the distribution curve of Fig. 6-6 comes to a peak, and that's where the area under the curve, given any constant horizontal span, is the greatest.
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
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?
A reasonable null hypothesis, which your friend proposes, is the notion that the ratio of women to men smokers in your town is 1:1, that is, ''50-50.'' Then the alternative hypothesis, which you propose, is that there are considerably more women smokers than men smokers. To conduct a test to find out who is right, you'll have to choose an unbiased sample of the population of your town. The sample will have to consist of an equal number of men and women, and it should be as large as possible. You will have to ask all the subjects whether or not they smoke, and then assume that they're being honest with you.
Now imagine, for the sake of argument, that H0 is in fact true in the above-described scenario. (You don't know it and your friend doesn't know it, because you haven't conducted the survey yet.) You're about to conduct an experiment by taking a supposedly unbiased survey consisting of 100 people, 50 men and 50 women. Draw a simple graph showing the relative probabilities of the null hypothesis being verified, versus either one-sided alternative.
The curve is a normal distribution (Fig. 6-7). Of all the possible outcomes, the most likely is a 1:1 split, in which the same number of women as men say they smoke. This doesn't mean that this exact result is certain or even likely; it only means that it is the least unlikely of all the possible outcomes. It's reasonable to suppose that you might get a result of, say, 20 women asserting that they smoke while 22 men say they smoke. But you should be surprised if the survey comes back saying that 40 women say they smoke while only 10 men say so.
Fig. 6-7. Illustration for Practice 2.
Name several different possible outcomes of the experiment described above, in which the null hypothesis is apparently verified.
Recall that 50 men and 50 women are surveyed. If 20 men say they smoke and 20 women say they do, this suggests the null hypothesis is reasonable. The same goes for ratios such as 15:16 or 25:23.
Name two outcomes of the experiment described above, in which the null hypothesis is apparently verified, but in which the results should be highly suspect.
If none of the men and none of the women say they smoke, you ought to suspect that a lot of people are lying. Similarly, if all 50 men and all 50 women say they smoke, you should also expect deception. Even ratios of 2:3 or 48:47 would be suspect. (Results such as this might suggest that we conduct other experiments concerning the character of the people in this town.)
Practice problems for these concepts can be found at:
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