How to Set Up a Hypothesis Test: Null versus Alternative

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If, for example, the null hypothesis says two population means are equal, the alternative says the means are unequal.

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How to Set Up a Hypothesis Test: Null versus Alternative

Which alternative hypothesis you choose in setting up your hypothesis test depends on what you’re interested in concluding, should you have enough evidence to refute the null hypothesis (the claim). The alternative hypothesis should be decided upon before collecting or looking at any data, so as not to influence the results.

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When we get the data we will calculate Z and then look it up in the Z table to see how unusual the obtained sample's mean is, if the null hypothesis Ho is true.

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The Z formula is:    Note that the population mean is 18 under the null hypothesis, and the standard error is 1, as we just calculated.
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Researchers often challenge claims about population parameters. You may hypothesize, for example, that the actual proportion of women who have varicose veins is lower than 0.25, based on your observations. Or you may hypothesize that due to the popularity of high heeled shoes, the proportion may be higher than 0.25. Or if you’re simply questioning whether the actual proportion is 0.25, your alternative hypothesis is: “No, it isn’t 0.25.”

Before actually conducting a hypothesis test, you have to put two possible hypotheses on the table — the null hypothesis is one of them. But, if the null hypothesis is rejected (that is, there was sufficient evidence against it), what’s your alternative going to be? Actually, three possibilities exist for the second (or alternative) hypothesis, denoted Ha. Here they are, along with their shorthand notations in the context of the pie example:

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Fisher, who first introduced the notion of significance tests in a formal systematic way, never considered alternative hypotheses.
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On the other hand, the null hypothesis is straightforward -- what is the probability that our treated and untreated samples are from the same population (that the treatment or predictor has no effect)? There is only one set of statistical probabilities -- calculation of chance effects. Instead of directly testing H, we test H. If we can reject H, (and factors are under control), we can accept H. To put it another way, the fate of the research hypothesis depends upon what happens to H.

In hypothesis tests, two errors are possible, .
Type I error: Supporting the when the null hypothesis is true.
Type II error: Not supporting the alternate hypothesis when the alternate hypothesis is true.

The Z formula is:    Note that the population mean is 460 under the null hypothesis, and the standard error is 15.6, as we just calculated.
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For example, if you want to test whether a company is correct in claiming its pie takes five minutes to make and it doesn’t matter whether the actual average time is more or less than that, you use the not-equal-to alternative. Your hypotheses for that test would be

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When you set up a hypothesis test to determine the validity of a statistical claim, you need to define both a null hypothesis and an alternative hypothesis.

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Finally, say you work for the company marketing the pie, and you think the pie can be made in less than five minutes (and could be marketed by the company as such). The less-than alternative is the one you want, and your two hypotheses would be

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In an example of a courtroom, let’s say that the null hypothesis is that a man is innocent and the alternate hypothesis is that he is guilty. if you convict an innocent man (Type I error), you support the alternate hypothesis (that he is guilty). A type II error would be letting a guilty man go free.

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For our example, we formally state: The alternative hypothesis (H1) is that prenatal exposure to alcohol has an effect on the birth weight for the population of lab rats.

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The (or hypotheses -- there may be more than one) is our working hypothesis -- our prediction, or what we expect to happen. It is also called the - because it is an alternative to the null hypothesis. Technically, the claim of the research hypothesis is that with respect to the outcome variable, our samples are from different populations (remember that refers to the group from which the sample is drawn). If we predict that math tutoring results in better performance, than we are predicting that after the treatment (tutoring), the treated sample truly is different from the untreated one (and therefore, from a different population).

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