The t-Distribution

FIGURE 3.2 Illustration of the f-distribution (a) for various degrees of freedom and (b) the area under the curve equal to 0.95 at 5 degrees of freedom. 

---

In applications sample sizes are usually small and O unknown. In these cases, the t distribution can be used where... 

The t distribution is also symmetric and centered at zero. It is said to be robust in the sense that even when the individual obseiwations x are not normally distributed, sample averages o x have distributions which tend toward normahty as n gets large. Even for small n of 5 through 10, the approximation is usually relatively accurate. 

Since the t distribution relies on the sample standard deviation. s, the resultant distribution will differ according to the sample size n. To designate this difference, the respec tive distributions are classified according to what are called the degrees of freedom and abbreviated as df. In simple problems, the df are just the sample size minus I. In more complicated applications the df can be different. In general, degrees of freedom are the number of quantities minus the number of constraints. For example, four numbers in a square which must have row and column sums equal to zero have only one df, i.e., four numbers minus three constraints (the fourth constraint is redundant). 

Consider the hypothesis Ii = [Lo- If, iri fact, the hypothesis is correct, i.e., Ii = [Lo (under the condition Of = o ), then the sampling distribution of x — x is predictable through the t distribution. The obseiwed sample values then can be compared with the corresponding t distribution. If the sample values are reasonably close (as reflectedthrough the Ot level), that is, X andxg are not Too different from each other on the basis of the t distribution, the null hypothesis would be accepted. Conversely, if they deviate from each other too much and the deviation is therefore not ascribable to chance, the conjecture would be questioned and the null hypothesis rejected. 

The population of differences is normally distributed with a mean [L ansample size is 10 or greater in most situations. 

The decision rule for each of the three forms would be to reject the null hypothesis if the sample value oft fell in that area of the t distribution defined by Ot, which is called the critical region. Other wise, the alternative hypothesis would be accepted for lack of contrary evidence. 

So how does one infer that two samples come from different populations when only small samples are available The key is the discovery of the t-distribution by Gosset in 1908 (publishing under the pseudonym of Student) and development of the concept by Fisher in 1926. This revolutionary concept enables the estimation of ct ( standard deviation of the population) from values of standard errors of the mean and thus to estimate... 

The t distribution allows the use of samples to make inferences about populations... 

The table gives the value of f . — the 100a percentage point of the t-distribution for v degrees of freedom. 

Figure 1.15. Student s f-distiibutions for 1 (bottom), 2, 5, and 100 (top) degrees of freedom /. The hatched area between the innermost marks is in all cases 80% of the total area under the respective curve. The other marks designate the points at which the area reaches 90, resp. 95% of the total area. This shows how the r-factor varies with /. The t-distribution for / = 100 already very closely matches the normal distribution. The normal distribution, which corresponds to t(f = o), does not depend on/.

Its mean is zero and its variance n/(n—2). The pth percentile of the t distribution with v degrees of freedom is noted tp v. The Student s t-distribution converges rapidly towards the normal distribution in practice, when v > 30, the two distributions become indistinguishable. 

A widely used a = 5 percent significance level produces a 95 percent confidence interval extending over t91 confidence interval for a standard normal distribution. Therefore, the normal approximation of the t-distribution is correct to 12 percent for m> 10 and to 4 percent for m> 30. 

Parallel to the case of a single random variable, the mean vector and covariance matrix of random variables involved in a measurement are usually unknown, suggesting the use of their sampling distributions instead. Let us assume that x is a vector of n normally distributed variables with mean n-column vector ft and covariance matrix L. A sample of m observations has a mean vector x and annxn covariance matrix S. The properties of the t-distribution are extended to n variables by stating that the scalar m(x—p)TS ( —p) is distributed as the Hotelling s-T2 distribution. The matrix S/m is simply the covariance matrix of the estimate x. There is no need to tabulate the T2 distribution since the statistic... 

Measure of reaction time Half-life of a reaction The 100(1 — a/2)% point of the t distribution with v degrees of freedom Transformed dependent variable defined by Eq. (125) Variance of the parameter estimate b,... 

From the above procedure, we can estimate the (1—a) confidence interval of the optimal gap. For a given t ]ti/2 where t is the critical value of the t-distribution with (n—1) degrees of freedom, the following can be estimated ... 

It turns out that the signal-to-noise ratio, under the assumption that the two treatment means are the same, has a predictable behaviour (we will say more about this in the next chapter) and the probabilities associated with values of this ratio are given by a particular distribution, the t-distribution. Figure 3.3 displays these probabilities for the example we are considering. Note that we have labelled this the t-distribution on 38 degree of freedom again we will say more about where the 38 comes from in the next chapter. 

The probabilities associated with the values that this signal-to-noise ratio can take when the treatments are the same are again given by the t shape in this case the appropriate t-distribution is tjj, the t-distribution on 31 degrees of freedom. Why tjj The appropriate t-distribution is indexed by the number of patients minus one. 

However, the T-distribution permits an extension of the plate theory, which is also usable in case of asymmetric peaks. The chromatogram (1 component) is considered to be the result of a pure time delay and a T-distribution response. The procedure implies the fitting of a function f(t) given in Eq. (15) to the chromatographic peak. The asymmetry of the peak determines the new plate number n, decreasing with increasing asymmetry. 

The t-statistic follows what is known as the Student s t-distribution, after the statistician William Sealy Gosset (1876-1937) who published under the pseudonym StudenT. The shape of the t-distribution is similar to that of the normal distribution, but forms a family of curves distinguished by a parameter known as the degrees of freedom. The 5% critical point in the t-distribution always exceeds the normal value of 1.96, but is nevertheless close to 2.0 for all but quite small values of degrees of freedom. 

---