The t-test

Each of these confidence intervals (the calculated interval and the critical interval) can be expressed in terms of b, s, and some value of t (see Equation 6.5). Because the same values of bg and are used for the construction of these intervals, the 

In short, if then Hq is rejected at the given level of confidence and the 

Let us now consider a slightly different question. Rather than inquiring about the significance of the specific parameter Pq, we might ask instead, Does the model y, = 0 + Ti, provide an adequate fit to the experimental data, or does this model show a significant lack of fit  

We begin by examining more closely the sum of squares of residuals between the measured response, yi and the predicted response, (y, = 0 for all i of this model), which is given by 

If the model = 0 + r, does describe the true behavior of the system, we would expect replicate experiments to have a mean value of zero (y,- = 0) the sum of squares due to purely experimental uncertainty would be expected to be 

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The t test can be applied to differences between pairs of observations. Perhaps only a single pair can be performed at one time, or possibly one wishes to compare two methods using samples of differing analytical content. It is still necessary that the two methods possess the same inherent standard deviation. An average difference d calculated, and individual deviations from d are used to evaluate the variance of the differences. 

The t test is also used to judge whether a given lot of material conforms to a particular specification. If both plus and minus departures from the known value are to be guarded against, a two-tailed test is involved. If departures in only one direction are undesirable, then the 10% level values for t are appropriate for the 5% level in one direction. Similarly, the 2% level should be used to obtain the 1% level to test the departure from the known value in one direction only these constitute a one-tailed test. More on this subject will be in the next section. 

A modified form of the t-test for comparing several sets of data. 

Thus, the calculated value of F (1.87) is less than the tabulated value therefore the methods have comparable precisions (standard deviations) and so the t-test can be used with confidence. 

Does the found mean Xmean correspond to expectations The expected value E(x) written as /r (Greek mu), is either a theoretical value, or an experimental average underpinned by so many measurements that one is very certain of its numerical value. The question can be answered by the t-test explained in Section 1.5.2. A rough assessment is obtained by checking to see whether and Xmean are separated by more than 2 Sx or not if the difference Ax is larger, x ean is probably not a good estimate for /t. 

The same is true if another situation is considered if in a batch process a sample is taken before and after the operation under scrutiny, say, impurity elimination by recrystallization, and both samples are subjected to the same test method, the results from, say, 10 batch processes can be analyzed pairwise. If the investigated operation has a strictly additive effect on the measured parameter, this will be seen in the t-test in all other cases both the difference Axmean and the standard deviation will be affected. 

Extension of the t-Test to More Than Two Series of Measurements... 

Both cases are amenable to the same test, the distinction being a matter of the number of degrees of freedom/. The F-test is used in connection with the t-test. (See program TTEST.)... 

It is easy to prove that there are differences between two given means using the t-test (Case bi or c). 

For a given deviation z = (x - ju)/a, CP = f z) calculates the cumulative probability of finding a deviation as large purely by chance this corresponds to the t-test for very large numbers of degrees of freedom. 

The dry weights (104 C, 48 hr) of ten plants from each treatment group were taken at the termination of each experiment in order to compare growth effects with plant water status. Dry weight data were analyzed using analysis of variance (ANOVA) and Duncan s multiple-range test. Diffusive resistance and water potential were evaluated using the t-test. Each of these and subsequent experiments was replicated. 

Note — Values each day are the mean of six seedlings for resistance and two for water potential. Resistance data and the 6-day mean for water potential were analyzed with the t-test. 

The relation of measured results to given values, e.g., critical levels, legally fixed values, regulatory limits, maximum acceptable values, is of continual relevance in analytical chemistry. In the analytical reality, the problematic nature of detection leads to the test statistics, strictly speaking to the t-test (Currie [1995, 1997] Ehrlich and Danzer [2006]). By means of that, it is tested, if the determined analytical result is significantly different from the average blank of the critical value, respectively. 

The simplest such type of design is the sequential design, simplest if for no other reason than that the type of design it replaces is one of the simplest designs itself. This design is the simple test for comparison of means, using the Z-test or the t-test as the test statistic we have discussed these in our previous column series and book Statistics in Spectroscopy (now in its second edition [1]). 

The test to determine whether the bias is significant incorporates the Student s /-test. The method for calculating the t-test statistic is shown in equation 38-10 using MathCad symbolic notation. Equations 38-8 and 38-9 are used to calculate the standard deviation of the differences between the sums of X and Y for both analytical methods A and B, whereas equation 38-10 is used to calculate the standard deviation of the mean. The /-table statistic for comparison of the test statistic is given in equations 38-11 and 38-12. The F-statistic and f-statistic tables can be found in standard statistical texts such as references [1-3]. The null hypothesis (H0) states that there is no systematic difference between the two methods, whereas the alternate hypothesis (Hf) states that there is a significant systematic difference between the methods. It can be seen from these results that the bias is significant between these two methods and that METHOD B has results biased by 0.084 above the results obtained by METHOD A. The estimated bias is given by the Mean Difference calculation. 

Having established that the standard deviations of two sets of data agree at a reasonable confidence level it is possible to proceed to a comparison of the mean results derived from the two sets, using the t-test in one of its forms. As in the previous case, the factor is calculated from the experimental set of results and compared with the table of critical values (Table 2.3). If /jX ) exceeds the critical value for the appropriate number of degrees of freedom, the difference between the means is said to be significant. When there is an accepted value for the result based on extensive previous analysis t is computed from equation (2.9)... 

The formula for the t test described in Procedure 1.3 compares the mean of replicate analyses of only one sample but it may be preferable to compare the accuracy over the analytical range of the method. To do this a paired f test may be used in which samples with different concentrations are analysed using both methods and the difference between each pair of results is compared. A simplified example is given in Procedure 1.4. 

Significant differences between the means of the two populations ( < 0.01) as determined by the t>test. 

According to the t-tests, there was a significant difference between peat alone and all the other litters, and also between straw and straw and peat combined. 

The statistical evaluation leads to a limit value, i.e., a critical effect, and all effects. Ex, that are in absolute value larger than or equal to the limit value are considered significant. The limit value is usually based on the t-test statistic given in the following equation " " ... 

Analysis of the LLGs gives 4 nodes which pass the t-test. They are listed in table 8. One has a negative LLG and so is discarded. 

Although there are many useful statistical tools, there are two that have particular relevance to chemometrics the t-test and the f-test [22,23]. The t-test is used to determine whether a single value is statistically different from the rest of the values in a series. Given a series of values, and the number of values in the series, the f-value for a specific value is given by the following equation ... 

The t value is the number of standard deviations that the single value differs from the mean value. This t value is then compared to the critical t value obtained from a t-table, given a desired statistical confidence (i.e., 90%, 95%, or 99% confidence) and the number of degrees of freedom (typically iV-1), to assess whether the value is statistically different from the other values in the series. In chemometrics, the t test can be useful for evaluating outliers in data sets. 

The f-test is similar to the t-test, but is used to determine whether two different standard deviations are statistically different. In the context of chemometrics, the f-test is often used to compare distributions in regression model errors in order to assess whether one model is significantly different than another. The f-statistic is simply the ratio of the squares of two standard deviations obtained from two different distributions ... 

Thereby we have to consider that the outlier test assumes the chosen approach for the regression function to be correct. First we should have a look on the plot of the residual analysis, because from there we can recognise potential outliers. We calculate the regression both with and without the potential outlier. Then we can apply either the F-test or the t-test... 

CRITICAL t TEST FOR COMPARING AVERAGES. Whenever two sets of analytical determinations have different arithmetic averages, but comparable standard deviations, one may apply the t test to assess the statistical significance of the difference in these averages. 

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