Hypothesis rejecting, falsely

Clinical trials are carried out to show that the null hypothesis is false. The p value is the probability of having an effect by chance if the null hypothesis were actually true. The null hypothesis is rejected in favor of the alternative hypothesis when the p value is less than a. 

If we can demonstrate to our satisfaction that the null hypothesis is false, then we can reject that hypothesis and accept the alternative hypothesis that Po 0. 

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 ... 

Interpretation While good batches of the quality produced (= 99,81% purity) have a probability of being rejected (false negative) less than 5% of the time, even if no replicates are performed, false positives are a problem an effective purity of /r = 98.5% will be taxed acceptable in 12.7% of all cases because the found Xmean is 99% or more. Incidentally, plotting 100 (1 -p) versus n creates the so-called power-curve, see file POWER.xls and program HYPOTHESIS.exe. 

Now, suppose all null hypotheses are true, so that Mq = m, then m — Wq = 0 and therefore 5=0 and testing each hypothesis at level a will give E [V jm = a. In general, if there are some false null hypotheses, then the expected value of V will be less but the value of m will be unchanged so that the E [V jm < a. Thus the PCER is controlled at or below a by this strategy. On the other hand, the family-wise error rate (FWER) is simply the probability that at least one hypothesis is falsely rejected, and this is P (V > 1). This... 

The significance level represents the probability that the null hypothesis is falsely rejected. 

The second type of error occurs when the null hypothesis is retained even though it is false and should be rejected. This is known as a type 2 error, and its probability of occurrence is [3. Unfortunately, in most cases [3 cannot be easily calculated or estimated. 

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 / ). 

This type of error equates to box B and is variously described as a type I error, a false-positive error or the a error. A type I error in a study result would lead to the incorrect rejection of the null hypothesis. 

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. 

First the hypotheses must be chosen. There are two (1) the null hypothesis denoted by H sub zero which Is assumed true until rejected, and (2) the alternative hypothesis denoted by H sub one or sub A for alternative which Is assumed false until the null hypothesis Is rejected. The logic of the test requires that the hypotheses be "mutually exclusive" and "jointly exhaustive." "Mutually exclusive" means that one and only one of the hypotheses can be true "jointly exhaustive" means that one or the other of the hypotheses must be true. Both cannot be false. The null hypothesis Is to reflect the status quo, which means that failure to reject It Is only continuation of a present loss. For the agricultural station, failure to Improve the status quo means that the old brand of seed, pesticide, or fertilizer Is used when. In fact, a new and better brand Is available. This Is a status quo loss of productivity (e.g. 

Type I error (alpha error) An incorrect decision resulting from rejecting the null hypothesis when the null hypothesis is true. A false positive decision. 

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. 

Tor instance, suppose the statistical criterion for testing a null hypothesis is p < 0.05. If p < 0.05, the researcher does not reject the hypothesis, because to accept the hypothesis would be to take a greater risk of treating as true a proposition that might turn out to be false. Because scientists have been historically more averse to false positives than to false negatives, they have been willing to reject hypotheses rather than take the stance of not accepting them. 

Suppose now that we repeat the calculation of the power for other specific values of Hj and plot them as shown in Fig. 1.8. Inspection of Fig. 1.8 shows that the power would vary from a value of a where H, ii=48 to a value of 1.0 where Hj °°. Such calculations yield the power function curve, as shown in Fig. 1.8. As expected, the further is removed from p0=48, the higher is the probability of rejecting the false hypothesis H0. Inspection of Fig. 1.7 shows that (3 decreases as a increases, so that we could obtain a higher power at the sacrifice of the level of significance. A higher power at the same a is possible if a large sample size is used. 

In this part, we have considered the fundamentals of statistical tests and have seen that no test is free from possible error. We can reduce the probability of rejecting a true hypothesis only by running a greater risk of accepting a false hypothesis. We note that a larger sample size reduces the probability of error. 

Based on the sample data, we may reject the null hypothesis when in fact it is true, and consequently accept the alternative hypothesis. By failing to recognize a true state and rejecting it in favor of a false state, we will make a decision error called a false rejection decision error. It is also called a false positive error, or in statistical terms, Type I decision error. The measure of the size of this error or the probability is named alpha (a). The probability of making a correct decision (accepting the null hypothesis when it is true) is then equal to 1—a. For environmental projects, a is usually selected in the range of 0.05-0.20. 

Suppose that the true mean concentration p is llOmg/kg, but the sample mean concentration is 90mg/kg. In this case, the null hypothesis is true (H0 llOmg/kg > lOOmg/kg). However, basing our decision on the sample data, we reject it in favor of the alternative hypothesis (Ha 90mg/kg< lOOmg/kg) and make a false rejection decision error. 

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