Alternative hypothesis acceptance

The null hypothesis test for this problem is stated as follows are two correlation coefficients rx and r2 statistically the same (i.e., rx = r2)l The alternative hypothesis is then rj r2. If the absolute value of the test statistic Z(n) is greater than the absolute value of the z-statistic, then the null hypothesis is rejected and the alternative hypothesis accepted - there is a significant difference between rx and r2. If the absolute value of Z(n) is less than the z-statistic, then the null hypothesis is accepted and the alternative hypothesis is rejected, thus there is not a significant difference between rx and r2. Let us look at a standard example again (equation 60-22). 

The calculated value of analysis of variance is F=1343.6 for the null hypothesis HqiP O. However, since the tabular value is F1 g 0 95=5.32 the null hypothesis is rejected and the alternative hypothesis accepted that the regression coefficient p, with 95% confidence level is statistically significant. 

The calculated probability of 0.011 is substantially less than 0.05. and the null hypothesis is rejected and the alternative hypothesis accepted. Hypothesis testing may be applied to any statistical parameter (/-distribution, / -distribution, etc.) for which a sampling distribution may be calculated or otherwise evaluated. 

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. 

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... 

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. 

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. 

There is a substantial weight of evidence for the cytoskeleton being responsible for the force production and control of cell locomotion. This view has not yet been accepted unanimously. However, an alternative hypothesis continues to be argued which states that membrane cycling is the motive force driving cell locomotion (Bretscher, 1987). One of the predictions of the membrane flow hypothesis is that there should be a discernible flow of lipid from the front to the rear of the cell. Lipid flow has proven very difficult to study, because of the lack of suitable methods to label single lipid molecules and the heterogenous behavior of membrane-associated proteins. The observation that particles were transported rearward when they bound... 

Figure 1.34. Alternative hypothesis and the power of a t-test. Alpha (a) is the probability of rejecting an event that belongs to the population associated with it is normally in the range 0.05. .. 0.01. Beta (/3) is the probability that an event that is effectively to be associated with H is accepted as belonging to the population associated with Hq. Note that the power of the test to discriminate between hypotheses increases with the distance between ha and hb- >-a is fixed either by theory or by previous measurements, while hb can be adjusted (shifted along the x-axis), for examples see H - H4, Section 4.1. Compare with program HYPOTHESIS.

With regard to an alternative hypothesis (Jouvet, 1984), there is no firm evidence to support the proposal that 5-HT released during W might act as a neurohormone and induce the synthesis and/or release of hypnogenic factors secondarily responsible for SWS and REMS occurrence. Therefore, based on neurochemical, electrophysiological, and neuropharmacological approaches, it is currently accepted that 5-HT functions to promote W and to inhibit REMS. 

Otherwise, the alternative hypothesis would be accepted for lack of contrary evidence. 

Two hypotheses are considered. Based on a random sample, the validity of the null hypothesis (Ho) is tested against the alternate hypothesis (Hi) in order to either reject or accept the first one. 

II the difference approach, which typically utilises 2-sided statistical tests (Hartmann et al., 1998), using either the null hypothesis (H0) or the alternative hypothesis (Hi). The evaluation of the method s bias (trueness) is determined by assessing the 95% confidence intervals (Cl) of the overall average bias compared to the 0% relative bias value (or 100% recovery). If the Cl brackets the 0% bias then the trueness that the method generates acceptable data is accepted, otherwise it is rejected. For precision measurements, if the Cl brackets the maximum RSDp at each concentration level of the validation standards then the method is acceptable. Typically, RSDn> is set at <3% (Bouabidi et al., 2010),... 

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. 

However, failure to disprove the null hypothesis does not mean we can reject the alternative hypothesis and accept the null hypothesis. This is a subtle but extremely important point in hypothesis testing, especially when hypothesis testing is used to identify factors in research and development projects (see Section 1.2 and Table 1.1). 

If the null hypothesis is rejected at the 95% level of confidence , why can we be at least 95% confident about accepting the alternative hypothesis ... 

Suppose, however, that the alternative hypothesis is H, bp > 0. Values of bp significantly less than zero would not satisfy the alternative hypothesis. If we did disprove the null hypothesis and bp were greater than zero, then we should be twice as confident about accepting the alternative hypotheses (or we should have only half the risk of being wrong). If is obtained from a regular two-tailed /-table specifying a risk a, then the level of confidence in the test is 100(1 - a/2)%. 

Table 21.3 Relationship between Type 1 (a) and Type 2 (P) errors and accepting the null hypothesis (HO) and rejecting the alternative hypothesis (HI)...

If the 95% confidence intervai for the mean difference (if) in the paired case contains zero, then one can accept the nuii hypothesis at the 0.05 level of significance. If zero is not included, then we reject the null hypothesis and accept the alternative hypothesis that the means are different. Simiiariy, in the unpaired case, if the 95% confidence interval for the difference of the means contains zero, then one can accept the null hypothesis if not the means are considered different at the 0.05 ievei of significance. 

The fact of the matter is that by accepting all of the safeguards that the skeptic rightiy demands and rejecting only the most absurd alternative hypothesis, we can confidently conclude that consciousness can be objective about itself. How else could consciousness be functionally adaptive and why else would it be evolutionarily conserved ... 

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