When the parameter value given in H0 falls outside of a 95% confidence interval, we should reject H0 at a 5% level in a two-tailed test based on the same sample.
If a sample is so extreme it is unlikely when Ho is true (close to 0.000). This means we have found evidence to support Ha.
If the null hypotheses are all true, α of the tests will yield statistically significant results just by random chance.
The smaller the p-value, the stronger the statistical evidence is against the null hypothesis and in favor of the alternative.
For instance, if the foregoing test is employed and if the sample size is 36, then the null hypothesis that the population mean is less than or equal to 1.5 will be rejected if ≥ 1.719 and will not be rejected if < 1.719. It is important to note that even when the estimate of μ—namely, the value of the sample mean —exceeds 1.5, the null hypothesis may still not be rejected. Indeed, when n = 36, a sample mean value of 1.7 will not result in rejection of the null hypothesis. This is true even though such a large value of the sample mean is certainly not evidence in support of the null hypothesis. Nevertheless, it is consistent with the null hypothesis in that if the population mean is 1.5, then there is a reasonable probability that the average of a sample of size 36 will be as large as 1.7. On the other hand, a value of the sample mean as large as 1.9 is so unlikely if the population mean is less than or equal to 1.5 that it will lead to rejection of this hypothesis.
Definition. The critical region, also called the rejection region, is that set of values of the test statistic for which the null hypothesis is rejected. The statistical test of the null hypothesis H 0 is completely specified once the test statistic and the critical region are specified.
Note that we never speak of rejecting the research hypothesis. The reason has to do with the favored status of the null hypothesis as default. Accepting the null hypothesis merely implies that you do not have enough evidence to decide against it. When we decide to “accept” a null hypothesis, H0, we should not necessarily believe that it is true, and should recognize that the research hypothesis H1 might well actually be true, but because the null hypothesis might be true (and has favored status) we will accept the null hypothesis. While accepting the null hypothesis as a reasonably possible scenario that could have generated the data, we nonetheless recognize that there are many other such believable scenarios close to the null hypothesis that also might have generated the data. For example, when we accept the null hypothesis that claims the population mean is $2,000, we have not usually ruled out the possibility that this mean is $2,001 or $1,999. For this reason, some statisticians prefer to say that we “fail to reject” the null hypothesis rather than simply say that we “accept” it.
There are two possible outcomes of a hypothesis test: either “accept the null hypothesis” or “reject the null hypothesis, accept the research hypothesis, and declare significance.” The result is defined to be statistically significant whenever you accept the research hypothesis because you have eliminated the null hypothesis as a reasonable possibility. By convention, the two possible outcomes are described as follows:
evidence to reject it. The size of the p -value can give a clue about the relationship of the null hypothesis to the data. A p -value near 0 or near 1 leaves little doubt as to the conclusion to be drawn from the study, but a p -value slightly exceeding α may suggest that further study is warranted. A listing of the actual p -value in a study result often adds information beyond just indicating a value greater or less than α. The next section further addresses this issue.
The error of rejecting the null hypothesis when it is true that can now be estimated using sample data is termed the p-value. If the p-value is smaller than α, we reject the null hypothesis; otherwise, we do not have enough
The rejection of the null hypothesis H 0 is a strong statement that H 0 does not appear to be consistent with the observed data. The result that H 0 is not rejected is a weak statement that should be interpreted to mean that H 0 is consistent with the data.