Null rejection

Next, an equation for a test statistic is written, and the test statistic s critical value is found from an appropriate table. This critical value defines the breakpoint between values of the test statistic for which the null hypothesis will be retained or rejected. The test statistic is calculated from the data, compared with the critical value, and the null hypothesis is either rejected or retained. Finally, the result of the significance test is used to answer the original question. 

A statement that the difference between two values is too great to be explained by indeterminate error accepted if the significance test shows that null hypothesis should be rejected (Ha). 

Examples of (a) two-tailed, (b) and (c) one-tailed, significance tests. The shaded areas in each curve represent the values for which the null hypothesis is rejected. 

Significance test in which the null hypothesis is rejected for values at either end of the normal distribution. 

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 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. 

Relationship between confidence intervals and results of a significance test, (a) The shaded area under the normal distribution curves shows the apparent confidence intervals for the sample based on fexp. The solid bars in (b) and (c) show the actual confidence intervals that can be explained by indeterminate error using the critical value of (a,v). In part (b) the null hypothesis is rejected and the alternative hypothesis is accepted. In part (c) the null hypothesis is retained. 

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. 

If evidence for a determinate error is found, as in Example 4.16, its source should be identified and corrected before analyzing additional samples. Failing to reject the null hypothesis, however, does not imply that the method is accurate, but only indicates that there is insufficient evidence to prove the method inaccurate at the stated confidence level. 

Regardless of whether equation 4.19 or 4.20 is used to calculate fexp, the null hypothesis is rejected if fexp is greater than f(a, v), and retained if fexp is less than or equal to f(a, v). 

Since Fgxp is larger than the critical value of 7.15 for F(0.05, 5, 5), the null hypothesis is rejected and the alternative hypothesis that the variances are significantly different is accepted. As a result, a pooled standard deviation cannot be calculated. 

The value of fexp is then compared with a critical value, f(a, v), which is determined by the chosen significance level, a, the degrees of freedom for the sample, V, and whether the significance test is one-tailed or two-tailed. For paired data, the degrees of freedom is - 1. If fexp is greater than f(a, v), then the null hypothesis is rejected and the alternative hypothesis is accepted. If fexp is less than or equal to f(a, v), then the null hypothesis is retained, and a significant difference has not been demonstrated at the stated significance level. This is known as the paired f-test. 

Because (fexp)AB is greater than f(0.05, 18), we reject the null hypothesis and accept the alternative hypothesis that the results for analyst B are significantly greater than those for analyst A. Working in the same fashion, it is easy to show that... 

Ohm s law the statement that the current moving through a circuit is proportional to the applied potential and inversely proportional to the circuit s resistance (E = iR). (p. 463) on-column injection the direct injection of thermally unstable samples onto a capillary column, (p. 568) one-taUed significance test significance test in which the null hypothesis is rejected for values at only one end of the normal distribution, (p. 84)... 

If the null hypothesis is assumed to be true, say, in the case of a two-sided test, form 1, then the distribution of the test statistic t is known. Given a random sample, one can predict how far its sample value of t might be expected to deviate from zero (the midvalue of t) by chance alone. If the sample value oft does, in fact, deviate too far from zero, then this is defined to be sufficient evidence to refute the assumption of the null hypothesis. It is consequently rejected, and the converse or alternative hypothesis is accepted. 

Consider the hypothesis Ii = [Lo- If, iri fact, the hypothesis is correct, i.e., Ii = [Lo (under the condition Of = o ), then the sampling distribution of x — x is predictable through the t distribution. The obseiwed sample values then can be compared with the corresponding t distribution. If the sample values are reasonably close (as reflectedthrough the Ot level), that is, X andxg are not Too different from each other on the basis of the t distribution, the null hypothesis would be accepted. Conversely, if they deviate from each other too much and the deviation is therefore not ascribable to chance, the conjecture would be questioned and the null hypothesis rejected. 

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. 

This test does not require reconciliation before it is applied. However, should the null hypothesis be rejected, it only indicates that a gross error might be present. It does not isolate which of the measurements (or constraints) are in error. Consequently, gross-error isolation must be done subsequently. 

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. 

If the found G exceeds the tabulated critical value x (p. nt- 1), the null hypothesis Hq must be rejected in favor of Fix, the standard deviations do not all belong to the same population, that is, there is at least one that is larger than the rest. The correction factor F is necessary because Eq. (1.47) overestimates... 

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. 

This homogeneity value will become positive when the null hypothesis is not rejected by Fisher s E-test (F < Ei a)Vl)V2) and will be closer to 1 the more homogeneous the material is. If inhomogeneity is statistically proved by the test statistic F > Fi a>VuV2, the homogeneity value becomes negative. In the limiting case F = hom(A) becomes zero. 

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 / ). 

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