Null hypothesis, testing

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 null hypothesis tested with the F-ratio is a general hypothesis stating that the true coefficients are all zero (note that b, is not included). The / "-ratio has an F-distribution with df= m and [Pg.126]

The null hypothesis tested by the Mantel-Haenszel method is as follows ... 

Another important consideration is the effect size. Because one is not only attempting to estimate the probability, but also the direction and magnitude of relationships, a direct index of the latter is very useful. Unfortunately, the tradition of null-hypothesis testing has tended to divert the focus of research away from the dimension of magnitude. In fact, any significance test represents the confluence of four mathematical components ... 

Table 1. Sample information, null hypothesis, test. statistic, and test interval for common tests on parameters of one or two measurement series...

Conversely, the null hypothesis tested for this effort may be stated as there is no difference observed between degradation rates for polystyrene SRMs weathered in un-accelerated exposures and ultra-accelerated exposures when compared (normalized) for UV radiant exposure on the jc-axis of the degradation curves. 

The question remains if there is a significant difference in the ability of the different probability distribution functions to describe the distillation data. It is generally not recommended to apply null hypothesis testing to information-theoretic ranking data to determine if the best model is significantly better than any of the lower ranked models (Burnham and Anderson, 1998). Model selection is best achieved through inspection of evidence ratios and residuals. A summary of the AIC and evidence ratios of the best 10 ranked functions are presented in Table 12.24. It can be... 

As applied in Example 12, the F test was one-tailed. The F test may also be applied as a two-tailed test in which the alternative to the null hypothesis is erj A cr. This doubles the probability that the null hypothesis is invalid and has the effect of changing the confidence level, in the above example, from 95% to 90%. 

To test the null hypothesis, you reach into your pocket, retrieve a penny, and determine its mass. If the mass of this penny is 2.512 g, then you have proved that the null hypothesis is incorrect. Finding that the mass of your penny is 3.162 g, however, does not prove that the null hypothesis is correct because the mass of the next penny you sample might fall outside the limits set by the null hypothesis. 

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. 

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... 

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. 

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 test statistic for evaluating the null hypothesis is called an f-test, and is given as either... 

The variance for the sample of ten tablets is 4.3. A two-tailed significance test is used since the measurement process is considered out of statistical control if the sample s variance is either too good or too poor. The null hypothesis and alternative hypotheses are... 

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. 

Significance tests, however, also are subject to type 2 errors in which the null hypothesis is falsely retained. Consider, for example, the situation shown in Figure 4.12b, where S is exactly equal to (Sa)dl. In this case the probability of a type 2 error is 50% since half of the signals arising from the sample s population fall below the detection limit. Thus, there is only a 50 50 probability that an analyte at the lUPAC detection limit will be detected. As defined, the lUPAC definition for the detection limit only indicates the smallest signal for which we can say, at a significance level of a, that an analyte is present in the sample. Failing to detect the analyte, however, does not imply that it is not present. 

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

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)... 

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