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Alternatively, the null hypothesis can postulate that the two samples are drawn from the same population, so that the variance and shape of the distributions are equal, as well as the means.

Formulation of the null hypothesis is a vital step in testing . Having formulated such a hypothesis, one can establish the probability of observing the obtained data or data more different from the prediction of the null hypothesis, if the null hypothesis is true. That probability is what is commonly called the "significance level" of the results.

Most statisticians believe that it is valid to state direction as a part of null hypothesis, or as part of a null hypothesis/alternative hypothesis pair (for example see ). The logic is quite simple: if the direction is omitted, then if the null hypothesis is not rejected it is quite confusing to interpret the conclusion. Say, the null is that the population mean = 10, and the one-tailed alternative: mean > 10. If the sample evidence obtained through x-bar equals -200 and the corresponding t-test statistic equals -50, what is the conclusion? Not enough evidence to reject the null hypothesis? Surely not! But we cannot accept the one-sided alternative in this case. Therefore, to overcome this ambiguity, it is better to include the direction of the effect if the test is one-sided. The statistical theory required to deal with the simple cases dealt with here, and more complicated ones, makes use of the concept of an .

Note that there are some who argue that the null hypothesis cannot be as general as indicated above: as Fisher, who first coined the term "null hypothesis" said, "the null hypothesis must be exact, that is free of vagueness and ambiguity, because it must supply the basis of the 'problem of distribution,' of which the test of significance is the solution."^{} Thus according to this view, the null hypothesis must be numerically exact — it must state that a particular quantity or difference is equal to a particular number. In classical science, it is most typically the statement that there is of a particular treatment; in observations, it is typically that there is between the value of a particular measured variable and that of a prediction. The usefulness of this viewpoint must be queried - one can note that the majority of null hypotheses test in practice do not meet this criterion of being "exact". For example, consider the usual test that two means are equal where the true values of the variances are unknown - exact values of the variances are not specified.

A test of a null hypothesis is useful because it sets a limit on the probability of observing a as or more extreme than that observed if the null hypothesis is true. In general it is much harder to be precise about the corresponding probability if the alternative hypothesis is true.

For example, one may want to compare the test scores of two random of men and women, and ask whether or not one population has a mean score different from the other. A null hypothesis would be that the mean score of the male population was the same as the mean score of the female population:

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In scientific and medical applications, the null hypothesis plays a major role in testing the significance of differences in treatment and groups. The assumption at the outset of the experiment is that no difference exists between the two groups (for the variable being compared): this is the null hypothesis in this instance. Other types of null hypothesis may be, for example, that:

That is, in scientific experimental design, we may predict that a particular factor will produce an effect on our dependent variable — this is our alternative hypothesis. We then consider how often we would expect to observe our experimental results, or results even more extreme, if we were to take many samples from a population where there was no effect (i.e. we test against our null hypothesis). If we find that this happens rarely (up to, say, 5% of the time), we can conclude that our results support our experimental prediction — we reject our null hypothesis.

In many statements of null hypotheses there is no appearance that these can have a "directionality", in that the statement says that values are identical. However, null hypotheses can and do have "direction" - in many of these instances statistical theory allows the formulation of the test procedure to be simplified so that the test is equivalent to testing for an exact identity. That is, if we formulate a one-tailed alternative hypothesis , the effective null hypothesis remains . It is not merely the opposite of the alternative hypothesis — that is, it is not However this does remain the true null hypothesis.

The reduction occurs because, in order to gauge support for the alternative hypothesis, classical hypothesis testing requires us to calculate how often we would have obtained results as or more extreme than our experimental observations. In order to do this, we need first to define the probability of rejecting the null hypothesis for each possibility included in the null hypothesis and second to ensure that these probabilities are all less than or equal to the quoted of the test. For any reasonable test procedure the largest of all these probabilities will occur on the boundary of the region _{T}, specifically for the cases included in _{0} only. Thus the test procedure can be defined (that is the can be defined) for testing the null hypothesis _{T} exactly as if the null hypothesis of interest was the reduced version _{0}.

If experimental observations contradict the prediction of the null hypothesis, it means that either the null hypothesis is false, or the event under observation occurs very improbably. This gives us high confidence in the falsehood of the null hypothesis, which can be improved in proportion to the number of trials conducted. However, accepting the alternative hypothesis only commits us to a difference in observed parameters; it does not prove that the theory or principles that predicted such a difference is true, since it is always possible that the difference could be due to additional factors not recognized by the theory.

For example, rejection of a null hypothesis that predicts that the rates of symptom relief in a sample of patients who received a and a sample who received a medicinal drug will be equal allows us to make a non-null statement (that the rates differed); it does not prove that the drug relieved the symptoms, though it gives us more confidence in that hypothesis.

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