Hypothesis, null

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. 

From Table 2.28 with four degrees of freedom for A and five degrees of freedom for B, the value of F would exceed 5.19 five percent of the time. Therefore, the null hypothesis is valid, and comparable skills are exhibited by the two analysts. 

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%. 

The difference between retaining a null hypothesis and proving the null hypothesis is important. To appreciate this point, let us return to our example on determining the mass of a penny. After looking at the data in Table 4.12, you might pose the following null and alternative hypotheses... 

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. 

Consider the situation when the accuracy of a new analytical method is evaluated by analyzing a standard reference material with a known )J,. A sample of the standard is analyzed, and the sample s mean is determined. The null hypothesis is that the sample s mean is equal to p. 

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. 

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. 

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 critical value for F(0.05, 6, 4) is 9.197. Since Fexp is less than F(0.05, 6, 4), the null hypothesis is retained. There is no evidence at the chosen significance level to suggest that the difference in precisions is significant. 

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). 

The critical value for f(0.05, 10), from Appendix IB, is 2.23. Since fexp is less than f(0.05, 10) the null hypothesis is retained, and there is no evidence that the two sets of pennies are significantly different at the chosen significance level. 

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. 

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... 

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... 

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