https://www.frontiersin.org/articles/10.3389/fnhum.2017.00390/full
Null Hypothesis
A null hypothesis is a statistical hypothesis that is tested for possible rejection under the assumption that it is true (usually that observations are the result of chance). The concept was introduced by R. A. Fisher.
The hypothesis contrary to the null hypothesis, usually that the observations are the result of a real effect, is known as the alternative hypothesis.
Null Hypothesis (H0)
A rejection of the null hypothesis H0 would then discredit the claim of the manufacturer.
From: Introductory Statistics (Fourth Edition), 2017
https://www.sciencedirect.com/topics/mathematics/null-hypothesis-h0
Truth, Possibility and Probability
In North-Holland Mathematics Studies, 1991
https://www.sciencedirect.com/topics/mathematics/null-hypothesis-h0
Linear Regression Models
Milan Meloun, Jiřà Militký, in Statistical Data Analysis, 2011
Problem 6.12 Simultaneous test of a composite hypothesis for a Lambert-Beer law model
For the data from Problem 6.10, test the composite null hypothesis H0: β2 = 0, β1 = 0.148 against HA: β2 ≠ 0, β1 ≠ 0.148. The false approach would be two separate tests of two null hypotheses, H0: β2 = 0 and H0: β1 = 0.148.
Solution: On substitution into Eq. (6.48), we obtain
Because T1, and T2 are less than the quantile of the Student t-distribution, t0.975(4) = 2.7764, both tests lead to a conclusion that H0: β2 = 0, β1 = 0.148 should be accepted. This conclusion is, however, false.
The more rigorous approach uses a simultaneous test of the composite hypothesis H0: β2 = 0 and β1 = 0.148.
The procedure starts with a calculation of RSC = 5.12 × 10 − 5 for estimates b1 = 0.1459 and b2 = 1.461 × 10− 4. Then, RSC1 = 5.3476 × 10− 4 for parameters β2,0 = 0 and β1,0 = 0.148 is calculated. From Eq. (6.50), the test criterion F1, is
Because the quantile of Fisher-Snedecor F-distribution is F0.95(2, 4) = 6.944, the null hypothesis H0: β2 = 0 and β1 = 0.148 cannot be accepted. The result of this F-test is not in agreement with conclusion of the previous t-tests. Figure 6.16 shows the 95% confidence ellipse of parameters β1 and β2, and the point β1,0 = 0.148 and β2,0 = 0 marked by a cross. This point lies outside the 95% confidence interval of the two parameters.
Figure 6.16. The 95% confidence interval for the parameters β1 and β2. The point β1 = 0, β2 = 0.148 is marked by a cross.
Conclusion: It may be concluded that a simultaneous test of the compositehypothesis cannot be replaced by tests of two separate hypotheses. Thus, testingof individual parameters in a vector β0 can lead to quite false conclusions.
Hypothesis Testing
Andrew F. Siegel, in Practical Business Statistics (Seventh Edition), 2016
Results, Decisions, and p-Values
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:
Results of a Hypothesis Test
| Either: | Accept the null hypothesis, H0, as a reasonable possibility. | A weak conclusion; not a significant result. |
| Or: | Reject the null hypothesis, H0, and accept the research hypothesis, H1 | A strong conclusion; a significant result. |
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.
It may help you to think of the hypotheses in terms of a criminal legal case. The null hypothesis is “innocent,” and the research hypothesis is “guilty.” Since our legal system is based on the principle of “innocent until proven guilty,” this assignment of hypotheses makes sense. Accepting the null hypothesis of innocence says that there was not enough evidence to convict; it does not prove that the person is truly innocent. On the other hand, rejecting the null hypothesis and accepting the research hypothesis of guilt says that there is enough evidence to rule out innocence as a possibility and to convincingly establish guilt. We do not have to rule out guilt in order to find someone innocent, but we do have to rule out innocence in order to find someone guilty.
While there is a vast variety of hypothesis tests covered here and in later chapters, depending on the type of data and the chosen model, and each test has its own particular detailed calculations (and its own important intuition) there is a useful, unifying fact: Every hypothesis test can produce a p-value that is interpreted in the same way:
Using the p-Value to Perform a Hypothesis Test
| If p > 0.05: | Accept the null hypothesis, H0, as a reasonable possibility. |
| If p < 0.05: | Reject the null hypothesis, H0, and accept the research hypothesis, H1 |
The p-value tells you how surprised you would be to learn that the null hypothesis had produced the data, with smaller p-values indicating more surprise and leading to rejection of H0 when p is less than the conventional 5% threshold. The p-value is computed (by statistical software) while assuming the null hypothesis is true, and tells the probability of observing your data (or data even farther from the null hypothesis). By convention, if the null hypothesis produces data like yours less than 5% of the time, this low probability is taken as evidence against the null hypothesis and leads to its rejection.2
Volume 4
M. Forina, ... P. Oliveri, in Comprehensive Chemometrics, 2009
4.04.4.3.3(i) Sensitivity and specificity
https://www.sciencedirect.com/topics/mathematics/null-hypothesis-h0
| Empty Cell | Null hypothesis H0 TRUE | Null hypothesis H0 FALSE |
|---|---|---|
| Statistical decision: Reject H0 | Type I error | Correct II decision |
| Statistical decision: Do not reject H0 | Correct I decision | Type II error |
https://www.sciencedirect.com/topics/mathematics/null-hypothesis-h0
No comments:
Post a Comment