Student’s /-distribution

In the next several sections, the theoretical distributions and tests of significance will be examined beginning with Student s distribution or t test. If the data contained only random (or chance) errors, the cumulative estimates x and 5- would gradually approach the limits p and cr. The distribution of results would be normally distributed with mean p and standard deviation cr. Were the true mean of the infinite population known, it would also have some symmetrical type of distribution centered around p. However, it would be expected that the dispersion or spread of this dispersion about the mean would depend on the sample size. 

If a measurement is repeated only a few times, the estimate for the distribution variance calculated from this sample is uncertain and the tiornial distribution cannot be applied. In this case another distribution is used, f his distribution is Student s distribution or the /-distribution, and it has one more parameter the number of degrees of freedom, t>. The /-distribution takes into account, through the p parameter, the uncertainty of the variance. The values of the cumulative /-distribution function cannot be evaluated by elementary methods, and tabulated values or other calculation methods have to be used. 

Table 22.2 Values of the Student s bilateral confidence coefficient t (calculation of Student s distribution)...

The significance of the estimated coefficients was tested by using the Student s -distribution at the 0.05 level of significance. The multiple correlation coefficient (R) was derived from the total variability of measured properties and the residual variability of the calculated values. Variability was expressed as sums of squares. These calculations were performed for every property studied. 

Equation 2.17 is the mathematical expression of Student s conclusions. The new random variable is represented by the s anbol tn-i, and its distribution is called the t distribution or Student s distribution. The subscript n - 1 reminds us that the form of the distribution depends on the size of the sample. Actually there are several distributions, each one corresponding to a certain number of degrees of freedom involved in determining the value of s. 

It is important to stress, at this point, that one must use the number of degrees of freedom for calculating the standard deviation, not the average with Student s distribution. As we will see later, the average and the standard deviation can be obtained from different sets of observations. The value of n in is not necessarily the same value of n used to calculate the average, and which appears in the denominator of Eq. (2.17). 

Table A.2 contains t values for some right-tail areas of Student s distribution. The areas appear at the top of the table, in boldface. Like the standard normal distribution, the t distribution is symmetric about a zero mean, so we only need to know the tail values for one side.

Exercise 2.16. Repeat Exercise 2.14, this time using Student s distribution. 

Ruhng out crass and systematic errors, the titration values can vary only because of random errors. If so, the central limit theorem imphes that the average values of sets of titrations made on the same batch should follow Student s distribution. The average of the three titrations,... 

You are right a normal distribution, for the individual observations, or Student s -distribution, for the averages. When the process is under control, its variability is only due to random errors, and for this reason its responses should follow a normal distribution or a distribution closely related to it. This is the basic principle of quality control — again, another consequence of the central limit theorem. 

Since we are also assuming that the errors are normally distributed, we can use Student s distribution to test the significance of the value estimated for 6i. We follow the procedure discussed in Chapter 2 and define confidence intervals by means of... 

The confidence interval characterizes the range about the mean of a random variable, in which an observation can be expected with a given probability P or risk a = 1 — />. As a statistical factor, the t value from Student s distribution should be used in the case of a normal distribution (cf. Section 2.2). The confidence interval for the mean, x, is calculated for/ degrees of freedom by... 

The Z-distribution and Eq. 13 are applicable w hen both the population mean and the variance are known. When the variance must be estimated from a sample, the r -distribution and the proportion of sampling values that fall within the — and -f- limits as given above no longer apply. In such circumstances a distribution called Student s /-distribution is used, and / is a random variable defined by Eq. 17. 

The decision on statistical significance is based on the sampling distribution of the ratio of the difference between two means to the standard deviation of such differences as given by Student s /-distribution for the general case of unequal number of values for each mean. 

It is not always possible to evaluate the variances of two populations being compared. Under such circumstances a test program may be conducted by using the "paired sample" technique. As an e.xample, if the effect of a certain treatment on a particular polymer is being evaluated, a number of uniform polymer samples are prepared and divided into pairs. One sample of each pair is given the treatment and one is not. i.e., it is the control. Measurements are then eonducted on a suffieient number of pairs p (six or more) lor treated vs. untreated vvith some particular test. Inferences on the effect of the treatment t>r the dilfcrence in means of the two populations (treated vs. nontreaied) are based on the mean value of the dilTerenee between n paired values. The distribution of the ratio of this mean difference to the standard deviation of the.se differences follows a Student s /-distribution. 

PERCENTAGE POEMTS, STUDENT S -DISTRIBUTION This table gives values of t such that... 

As Fig. 5.4 shows, the Student s t distribution closely approximates the normal distribution as v, the number of degrees of freedom, increases. Equation (5.9) gives the distribution function for the Student s distribution curve shown in Fig. 5.4. Equation (5.9) is valid under the provisions of the following theorem. 

Theorem. If a variable u is normally distributed with mean zero and a variance of unity and a variable v has a distribution with u degrees of freedom (u and V must be independently distributed), then the variable t = u v/v has a Student s distribution with v degrees of freedom given by Eq. (5.9). 

The confidence limits for the mean of a small sample where x is normally distributed with mean and variance (with x and s as the sample estimates based on a r.s.s. n) can be determined using the Student s distribution. Let... 

Small-sample hypothesis testing follows the same pattern described in previous paragraphs except that a table of Student s distribution ordinates and areas would be used rather than a normal distribution table [2,3,5]. 

N number of. .. /. /q(v) X values of Student s distribution (quantiles) error probability (risk of first kind)... 

For type A evaluations, the degrees of freedom shall be calculated. In the simple case of m independent observations, the degrees of freedom equal m — 1. However, when only a small number of observations have been made, the use of the Student s -distribution is more appropriate than a Gaussian distribution (see textbooks of statistics). 

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