Rejection of null hypothesis

Mean Standard derivation t P (T[Pg.295]

Since significance tests are based on probabilities, their interpretation is naturally subject to error. As we have already seen, significance tests are carried out at a significance level, a, that defines the probability of rejecting a null hypothesis that is true. For example, when a significance test is conducted at a = 0.05, there is a 5% probability that the null hypothesis will be incorrectly rejected. This is known as a type 1 error, and its risk is always equivalent to a. Type 1 errors in two-tailed and one-tailed significance tests are represented by the shaded areas under the probability distribution curves in Figure 4.10. 

The critical value for f(0.05,4), as found in Appendix IB, is 2.78. Since fexp is greater than f(0.05, 4), we must reject the null hypothesis and accept the alternative hypothesis. At the 95% confidence level the difference between X and p, is significant and cannot be explained by indeterminate sources of error. There is evidence, therefore, that the results are affected by a determinate source of error. 

Therefore, the ratio of sample variances is no larger than one might expect to observe when in fact Cj = cf. There is not sufficient evidence to reject the null hypothesis that Of = <31. 

The decision rule for each of the three forms would be to reject the null hypothesis if the sample value oft fell in that area of the t distribution defined by Ot, which is called the critical region. Other wise, the alternative hypothesis would be accepted for lack of contrary evidence. 

In Fig. 30-25, representation of the fault detection monitoring activity, there appears to be two distinct time periods of unit operation with a transition period between the two. The mean parameter value and corresponding sample standard deviation can be calculated for each time. These means can be tested by setting the null hypothesis that the means are the same and performing the appropriate t-test. Rejecting the null hypothesis indicates that there may have been a shift in operation of the unit. Diagnosis (troubleshooting) is the next step. 

We begin with a simultaneous test of the null hypothesis that the single measurement xgj is drawn from the background population and that X and t provide the proper transformation to normality. Our simultaneous test, which is based on tj and r, rejects the null hypothesis if... 

The simultaneous test given by Equations 6 and 7 leads to a test appropriate for (X,t) unknown. The (X,T)-unknown test rejects the null hypothesis that xgj belongs to the background population if tj > c(r) for all (X,t). Since this test rejects the null hypothesis only if Equation 6 is satisfied for the true value of (X,t), this test has no greater probability of false detection than the simultaneous test. Thus, the (X,r)-unknown test is conservative in the sense that the probability of a false detection is less than a if the probability of false detection for the simultaneous test is a. 

To reject the null hypothesis erroneously although it is true (error of first kind, false-negative, risk a). 

Not to reject the null hypothesis by erroneously though the alternative hypothesis is true (error of second kind, false-positive, risk / ). 

The rule for accepting H0 is specified by selection of the a level as indicated in Fig. 3-65. For forms 2 and 3 the a area is defined to be in the upper or the lower tail respectively. The parameter a is the probability of rejecting the null hypothesis when it is actually true. 

If the null hypothesis is assumed to be true, say, in the case of a lower-tailed test, form 3, then the distribution of the test statistic t is known under the null hypothesis that limits p = Po- Given a random sample, one can predict how far its sample value of t might be expected to deviate from zero by chance alone when p = p> If the sample value of t is too small, as in the case of a negative value, then this would be defined as sufficient evidence to reject the null hypothesis. 

Since the sample t = 2.03 > critical t = 1.833, reject the null hypothesis. It has been demonstrated that the population of men from which the sample was drawn tend, as a whole, to have an increase in blood pressure after the stimulus has been given. The distribution of differences d seems to indicate that the degree of response varies by individuals. 

NOTE If Z(N) is greater than the absolute value of the z-statistic (Normal Curve one-tailed) we reject the null hypothesis and state that there is no significant difference in rl and r2 at the selected significance level. 

This leads to the term Power (1 - j3), which quantifies the ability of the study to find the true differences of various values of S. It is the probability of rejecting the null hypothesis when it is false or determining that the alternative hypothesis is true when indeed it is true. 

If the null hypothesis can be rejected on the basis of a 95% confidence test, then the risk of falsely rejecting the null hypothesis is at most 0.05, but might be much less. We don t know how much less it is unless we look up the critical value for, say. 

Similarly, if the null hypothesis cannot be rejected at the 95% level of confidence it does not mean that the quantity being tested is insignificant. Perhaps the null hypothesis could have been rejected at the 90% level of confidence. The quantity would still be rather significant, with a risk somewhere between 0.05 and 0.10 of having falsely rejected the null hypothesis. 

In other situations it is not necessary to decide before a test is made the risk one is willing to take. Such a situation is indicated by the subtly different question, What are my chances of being right if I reject this null hypothesis In this case, it is desirable to assign an exact level of confidence to the quantity being tested. Such a level of confidence would then designate the estimated level of risk associated with rejecting the null hypothesis. 

The relationship between the risk, a, of falsely rejecting the null hypothesis and the level of confidence, P, placed in the alternative hypothesis is P = 100(1 - a)%. If the null hypothesis is rejected at the 87% level of confidence, what is the risk that the null hypothesis was rejected falsely ... 

On the other hand, if we find tcntioai smaller than tobserved we can say we find significant bias at the 95% level or we reject the null hypothesis . But we cannot say there is bias. Since our calculations ate based on a 95% confidence level, there is a chance of about 5% that we are wrong. 

A test of the null h)rpothesis that the rates of infection are equal - Hq x jii/hnj = 1 gives a p-value of 0.894 using a chi-squared test. There is therefore no statistical evidence of a difference between the treatments and one is unable to reject the null hypothesis. However, the contrary statement is not true that therefore the treatments are the same. As Altman and Bland succinctly put it, absence of evidence is not evidence of absence. The individual estimated infection rates are jTi = 0.250 and = 0.231 that gives an estimated RR of 0.250/0.231 = 1.083 with an associated 95% confidence interval of 0.332-3.532. In other words, inoculation can potentially reduce the infection by a factor of three, or increase it by a factor of three with the implication that we are not justified in claiming that the treatments are equivalent. 

Power The probability of rejecting the null hypothesis in a statistical test when a particular alternative hypothesis happens to be true. 

There is a fair amount of language that we wrap around this process. We talk in terms of a test of significance. If p < 0.05, we declare statistical significance, and reject the null hypothesis at the 5 per cent level. We call 5 per cent the significance level, it is the level at which we declare statistical significance. If p > 0.05, then we say we have a non-significant difference and we are unable to reject the null hypothesis at the 5 per cent level. 

A significant p-value from this test would cause us to reject the null hypothesis, but the conclusion from this only tells us that there are some differences somewhere at least two of the ps are different. At that point we would want to look to identify where those differences lie and this would lead us to pairwise comparisons of the... 

With the p-value methodology we are rejecting the null hypothesis Hg in favour of the alternative hypothesis Hj, providing the two (one-sided) p-values are < 2.5 per cent. We have then established equivalence and we can talk in terms of the treatments being significantly equivalent. The terminology sounds almost contradictory, but is a correct statement. If either of the two p-values is above 2.5 per cent then the treatments are not significantly equivalent. 

We will focus our attention to the situation of non-inferiority. Within the testing framework the type I error in this case is as before, the false positive (rejecting the null hypothesis when it is true), which now translates into concluding noninferiority when the new treatment is in fact inferior. The type II error is the false negative (failing to reject the null hypothesis when it is false) and this translates into failing to conclude non-inferiority when the new treatment truly is non-inferior. The sample size calculations below relate to the evaluation of noninferiority when using either the confidence interval method or the alternative p-value approach recall these are mathematically the same. 

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