T two-tailed

P(T < t) one-tail t Critical one-tail P(T < t) two-tail t Critical two-tail... 

Student t, two tailed value (95% for 9 degrees of freedom in this example)... 

Mean Standard derivation t P (T[Pg.295]

We used a two-tailed test. Upon rereading the problem, we realize that this was pure FeO whose iron content was 77.60% so that p = 77.60 and the confidence interval does not include the known value. Since the FeO was a standard, a one-tailed test should have been used since only random values would be expected to exceed 77.60%. Now the Student t value of 2.13 (for —to05) should have been used, and now the confidence interval becomes 77.11 0.23. A systematic error is presumed to exist. 

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. 

The abbreviated table on the next page, which gives critical values of z for both one-tailed and two-tailed tests at various levels of significance, will be found useful for purposes of reference. Critical values of z for other levels of significance are found by the use of Table 2.26b. For a small number of samples we replace z, obtained from above or from Table 2.26b, by t from Table 2.27, and we replace cr by ... 

The confidence limits for the slope are given by fc where the r-value is taken at the desired confidence level and (A — 2) degrees of freedom. Similarly, the confidence limits for the intercept are given by a ts. The closeness of x to X is answered in terms of a confidence interval for that extends from an upper confidence (UCL) to a lower confidence (LCL) level. Let us choose 95% for the confidence interval. Then, remembering that this is a two-tailed test (UCL and LCL), we obtain from a table of Student s t distribution the critical value of L (U975) the appropriate number of degrees of freedom. 

The t-values in this table are for a two-tailed test. For a one-tailed test, the a values for each column are half of the stated value, column for a one-tailed test is for the 95% confidence level, a = 0.05. For example, the first... 

Above values refer to a single tail outside the indicated limit of t. For example, for 95 percent of the area to be between —t and-t-t in a two-tailed t distribution, use the values for tooss or 2.5 percent for each tail. 

Ot = significance level, usually set at. 10,. 05, or. 01 t = tabled t value corresponding to the significance level Ot. For a two-tailed test, each corresponding tail would have an area of Ot/2, and for a one-tailed test, one tail area would be equal to Ot. If O" is known, then z would be used rather than the t. t = (x- il )/ s/Vn) = sample value of the test statistic. 

The critical values or value of t would be defined by the tabled value of t with (n — I) df corresponding to a tail area of Ot. For a two-tailed test, each tail area would be Ot/2, and for a one-tailed test there would be an upper-tail or a lower-tail area of Ot corresponding to forms 2 and 3 respectively. 

Had the calculated value for t been less than 1.80 then there would have been no significance in the results and no apparent bias in the laboratory procedure, as the tables would have indicated a probability of greater than 1 in 10 of obtaining that value. It should be pointed out that these values refer to what is known as a double-sided, or two-tailed, distribution because it concerns probabilities of values both less and greater than the mean. In some calculations an analyst may only be interested in one of these two cases, and under these conditions the -test becomes single-tailed so that the probability from the tables is halved. 

Figure 17. Order parameter profile for the two tails of DMPC bilayers. The chain closest to the head group is named sn and the other one sn2. Segments closest to the glycerol backbone are numbered t = 1, and the chain ends are t = 16. The lines are drawn to guide the eye...

The table on this slide is an example for sneh t-valnes. In the first two rows are the eonfidenee levels for one-tailed (IT) and two-tailed (2T) questions. 

We get the critical value from the table of t-values. We have a two-tailed question, we want to have 95% confidence level, and with 10 measurements we have 9 degrees of freedom. Therefore we get from the table a critical value of 2.262. 

The example we jnst looked at is called one-sample t-test. It compares the mean of analytical resnlts with a stated valne. This is a typical analytical question. The problem may be two-tailed as in onr example, where it doesn t matter, if the analytical valne is biased to the one or the other direction. Or the question conld be one-tailed, e.g. if we want to know whether the copper content analysed in an alloy is below the specificatioa... 

If we have a one-tailed question we have to use other t-values. In the table shown in this shde we have to select the level of confidence from the first row for one-tailed and from the second row for two-tailed questions. 

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