Alternate hypothesis

In attempting to reach decisions, it is useful to make assumptions or guesses about the populations involved. Such assumptions, which may or may not be true, are called statistical hypotheses and in general are statements about the probability distributions of the populations. A common procedure is to set up a null hypothesis, denoted by which states that there is no significant difference between two sets of data or that a variable exerts no significant effect. Any hypothesis which differs from a null hypothesis is called an alternative hypothesis, denoted by Tfj. 

A one-tailed test is required since the alternative hypothesis states that the population parameter is equal to or less than the hypothesized value. 

If the significance test is conducted at the 95% confidence level (a = 0.05), then the null hypothesis will be retained if a 95% confidence interval around X contains p,. If the alternative hypothesis is... 

The alternative hypothesis also can be stated in one of two additional ways... 

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. 

In a study involving paired data the difference, d[, between the paired values for each sample is calculated. The average difference, d, and standard deviation of the differences, are then calculated. The null hypothesis is that d is 0, and that there is no difference in the results for the two data sets. The alternative hypothesis is that the results for the two sets of data are significantly different, and, therefore, d is not equal to 0. 

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. 

On occasion, a data set appears to be skewed by the presence of one or more data points that are not consistent with the remaining data points. Such values are called outliers. The most commonly used significance test for identifying outliers is Dixon s Q-test. The null hypothesis is that the apparent outlier is taken from the same population as the remaining data. The alternative hypothesis is that the outlier comes from a different population, and, therefore, should be excluded from consideration. 

Individual comparisons using Fisher s least significant difference test are based on the following null hypothesis and one-tailed alternative hypothesis... 

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. 

Hq = assumption or null hypothesis regarding the population proportion Hi = alternative hypothesis... 

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... 

Since it is conceivable that some slight change in a process might lead to a different content, a mental note is made of this by stating that if the new result differs from the old one, the alternate hypothesis H applies. The difference between Hq and might be due to pa pb and/or a a ob-The first possibility is explored in the following section, the second one will be dealt with in Section 1.7.1. 

Interpretation If the alternate hypothesis had been stated as //i Xmean is different from /r, a two-sided test is applied with 2.5% probability being provided for each possibility Xmean smaller than p" resp. Xmean larger than /i . Because 1.92 is smaller than 2.45, the test criterion is not exceeded, so Hi is rejected. On the other hand, if it was known beforehand that Xmean can only be smaller than p, the one-sided test is conducted under the alternate hypothesis H Xn,ean smaller than p in this case the result is elose, with 1.92 almost exceeding 1.94. 

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.

Table 1.28. Interpretation of a Null/Alternative Hypothesis Situation...

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