Statistics and Forecast Help (page 3)

By — McGraw-Hill Professional
Updated on Sep 13, 2011

Solution 1

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.

Practice 2

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.

Solution 2

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.

Practice 3

Name several different possible outcomes of the experiment described above, in which the null hypothesis is apparently verified.

Solution 3

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.

Practice 4

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.

Solution 4

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:

Hypotheses, Prediction, and Regression Practice Test

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