Student /-values

The precision limits r and R are given by equations (4.3a) and (4.3b), respectively, where tv>a is the Student /-value for v degrees of freedom and a corresponds to the stated probability, sr is the repeatability standard deviation and Sr is the reproducibility standard deviation calculated from (v + 1) results ... 

A confidence interval is calculated from t x s/+fn (see Section 6.1.3). To obtain a standard uncertainty we need to calculate sl Jn. We therefore need to know the appropriate Student /-value (see Appendix, p. 253). However, statements of this type are generally given without specifying the degrees of freedom. Under these circumstances, if it can be assumed that the producer of the material carried out a reasonable number of measurements to determine the stated value, it is acceptable to use the value of t for infinite degrees of freedom, which is 1.96 at the 95% confidence level. If the degrees of freedom are known, then the appropriate /-value can be obtained from statistical tables. In this example, the standard uncertainty is 3/1.96 = 1.53 mg D1. 

To illustrate the difference between these two calculations, look at figure 2.7, which shows the difference between the 95% point on the normal distribution (z = 1.96) and the corresponding Student /-value for different degrees of freedom. The /-value depends on the degrees of freedom, and asymptotically approaches the value of z as the degrees of freedom tends to infinity. The important point is to note how quickly this happens. For a sample of three measurements there are two degrees of freedom and /o.o5,2 is 4.3, more than twice z0.025-Therefore a confidence interval based on a standard deviation of three results will be twice that if the population standard deviation, a is... 

Figure 2.7 Plot of the Student /-value for calculation of a 95% confidence interval with increasing degrees of freedom. The corresponding z-value from the normal distribution is shown (z0 025 = 1-96).

An unnecessary complication, which was possibly once introduced to make life easier, is the distinction between one- and two-tailed Student /-values (tails are also used in other statistics). Two-tailed probabilities are spread over the two ends of the distribution with half the given probability in each tail, and are denoted by putting a double prime (") after the probability value. One-tailed probabilities are shown as a single prime ( ) and refer to just one tail of the distribution. For example, for a 95% confidence interval and 10 degrees of freedom, 0.025, 10 is equal to o.o5",io> as can be seen from figure 2.9. Annoyingly, in Excel the z values obtained from the normal distribution are always one tailed (=—NORMSINV(p)1) but the Student /-values... 

What is the probability associated with a two-tailed Student /-value of 2.23 with 10 degrees of freedom ... 

It would be a lot easier if we could ignore Student /-values and just assume that the s calculated from our data is a. The value of z from the normal distribution for the 95% confidence interval is, as we have learned, 1.96. For 30 degrees of freedom, the error in using z and not t (t0 05">30 = 2.04) is about 4%. Hence the answer is that it depends on the error you can tolerate, but you should usually consider the Student /-distribution for less than about 30 data. 

The difficulty with using equation 2.14 or 2.15 is that we need to know n to calculate the degrees of freedom, to give the Student /-value. It is possible to iterate equation 2.14 or 2.15 with an initial guess at n to give t which is then put back in n and so on. After the experiments are performed it may be necessary to recalculate n to take into account the new value for the standard deviation. In practice, we are usually only interested in ball-park figures, for example 5% with anywhere between 4 and 6% being acceptable, and hence the process is not too tedious. 

Confidence intervals on the slope and intercept are determined by multiplying the standard deviations by a two-tailed Student /-value (recall from chapter 2 that two tails refers to both halves of the distribution) at an appropriate probability and degrees of freedom of the regression. These can be used when the slope or intercept is needed for further calculations, for example in determining the activation energy from the slope of an Arrhenius plot (log (7c) against I/7 ). 

A student reported the age of an ice layer at 70 m as 425 years. The accepted value is 427 years. What is the percent error of the students value ... 

Instead of removing a specific value, an analogy to a trim mean might be employed. Say 10% of the most extreme absolute residual values. Cook s distance values, or deleted Studentized values are simply removed this is 5% of the extreme positive residuals and 5% of the negative ones. This sort of determination helps prevent distorting the data for one s gain. 

Fifth, alternative assessments need to be implemented to measure all the possible outcomes of working in the laboratory. Perhaps the laboratory does not change students understanding of individual concepts so much as changing the way in which students connect those concepts or the ways in which students value those concepts. Perhaps the laboratory changes motivation and other affective domain constmcts. Again, qualitative research techniques are best able to investigate questions of this nature. 

Formative and summative assessments are used during the course to evaluate the achievement of the learning objectives. Example assessment methods and results are discussed in the paper. Also discussed are the results of recent teaching evaluations, which show that the students value this approach to course design and organization. 

Kem, E. L. and Carpenter, J. R. (1984) Enhancement of student values, interests and attitudes in Earth Science through a field oriented approach Journal of Geological Education, 32 pp. 299—305. 

To investigate how the students valued their LARAM experience, since the first year of the School, a questionnaire was set up with reference to both the didactic and logistic aspects of the School and handed out to the students at the end of each year s course. Figure 7 shows the results of questionnaires... 

Students experiences of program culture are much different. Student values reflect adaptations students make to successfully navigate engineering departments. Students value ... 

On the other hand, Chilean citizens have a value profile more similar to American citizens than to the immigrant citizens/residents group. Chilean citizens, like U.S. citizens, also highly rate values related to personal interest the difference is that U.S. citizens value more Achievement while Chilean citizens value more Hedonism. On the collective interest side, both groups give lower rates, but Chilean students value more UniversaUsm while U.S. citizens value more Conformity (see Fig. 12.7). 

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