Linguistics 483 Notes

A. C. Brett acbrett@uvic.ca Department of Linguistics University of Victoria Clearihue C139

The t-Statistic

The t-statistic is often used to determine, on the basis of a random sample x₁, x₂, ... , x_n of n independent observations of a variable X, whether the mean, m_X, of the population of values on X differs from a standard or expected value, m_e.

Typically, the mean, m_X, and variance, s_X², of the population are unknown. The mean, m_X, and variance, s_X², of the random sample of observations of X are therefore used to estimate them.

A value for the t-statistic is then obtained as follows:

          m_X - m_e
t = —————
        [s_X²/n] ^1/2

• m_X = (1/n) S_iⁿ x_i, with S_iⁿ denoting the summation over i from i = 1 to i = n, is the sample mean;



• s_X² = (1/(n - 1)) S_iⁿ (x_i - m_X )² is the sample variance;



• m_e is a standard or expected value for the population mean, m_X; and,



• [s_X²/n] ^1/2 is the standard error of the mean, with the exponent 1/2 denoting the square root operation.

Table 1. Null hypotheses, alternative hypotheses, and decision criteria for 1- and 2-tailed t-tests of the difference between the mean, mX, of the population of values of a variable X and a standard or expected value, me, on the basis of a random sample of n independent observations of X. Test Hypotheses Decision Criterion Null, H0 Alternative, HA 2-Tailed mX - me = 0      (mX = me) mX - me ≠ 0      (mX ≠ me) | t | ≥ ta/2; n 1-Tailed mX - me ≤ 0      (mX ≤ me) mX - me > 0      (mX > me) t ≥ ta; n 1-Tailed mX - me ≥ 0      (mX ≥ me) mX - me < 0      (mX < me) t ≤ - ta; n

Table 1. shows the three possible null hypotheses, H₀, regarding the difference between the population mean, m_X, and a standard or expected value, m_e. Also shown are the criteria on the basis of which the decision is made to reject H₀ and claim the corresponding alternative, H_A.

Two-Tailed Test

If the difference between the sample estimate, m_X, of the population mean, m_X, and the value m_e is sufficiently different from zero, relative to the standard error of the mean, [s_X²/n] ^1/2, then one can reject H₀ and claim the alternative, H_A. In a two-tailed test, however, m_X can be less than m_e, which yields a negative t-value, or m_X can be greater m_X, which yields a positive t.

It is this fact that leads to the test being called a two-tailed test. The two tails are those regions of the t-distribution that extend to the left and right, and which decrease toward the horizontal axis. The tail that extends to the right toward the more positive values of t is called the upper tail; the tail to the left toward the more negative values is called the lower tail. The t-value obtained in the test can be either in the upper or in the lower tail of the distribution. Thus, if the calculated t is sufficiently positive (it is in the upper tail), or sufficiently negative (it is in the lower tail), one can reject H₀, and claim H_A.

In practice, when applying a two-tailed test, one examines only the magnitude or absolute value of t, denoted as | t |. The decision to reject H₀, and claim H_A, is then based upon the magnitude obtained for t relative to a selected critical value of t.

A critical t-value, denoted as t_{a/2; n}, depends on the chosen significance level, a, and the number of degrees of freedom, n. (The division of a by 2 is explained below in the section on critical values.) If | t | is greater than or equal to t_{a/2; n}, one can claim H_A.

One-Tailed Test

The corresponding alternative hypothesis in one of the tests is that the population mean, m_X, is greater than the value m_e. In the other test, the alternative is that m_X is less than m_e. Thus, unlike the two-tailed test wherein one is concerned only with whether m_X and m_e are different, in a one-tailed test one is interested in whether m_X is either greater than, or less than m_e.

If the value calculated for t using a sample mean, m_X, and variance, s_X², is greater than or equal to the upper (positive tail) critical t-value, t_{a; n}, then one can reject the H₀ that m_X ≤ m_e and claim H_A that m_X > m_e with the chance of error (in rejecting a true H₀) being less than the selected significance level, a.

If the value calculated for t is less than or equal to the lower (negative tail) critical t-value, - t_{a; n}, then one can reject the H₀ that m_X ≥ m_e and claim H_A that m_X < m_e with the chance of error (in rejecting a true H₀) being less than a.

Critical Values

An upper critical value is used in a one-tailed test of the null hypothesis that the mean, m_X, of the population of the values of a variable X is less than, or equal to a value m_e against the alternative that m_X is greater than m_e.

Since the t-distribution is symmetrical about zero, the upper critical value, t_{a; n}, can also be used to test the hypothesis that m_X is greater than, or equal to m_e against the alternative that m_X is less than m_e. In this one-tailed test, the t-value obtained is compared with - t_{a; n}.

An upper critical value is also used for two-tailed tests of the equality of m_X and m_e against the alternative that they differ. In such tests, one is not concerned with whether m_X is greater or less than m_e; one cares only that they be sufficiently different. One claims the alternative if the t-value is either sufficiently negative or sufficiently positive.

In a two-tailed test, one compares the magnitude or absolute value of the t-value with the upper critical value t_{a/2; n}. The probability of obtaining a t-value greater than or equal to t_{a/2; n} is a/2, and the probability of obtaining a t-value less than or equal to -t_{a/2; n} is also a/2. Hence, the probability of either one or the other of these events is a/2 + a/2 = a, the chosen significance level.

Tables of the critical values of the t-distribution are available from a number of sources. One such source is the web site of the Information Technology Laboratory of the US National Institute of Standards and Technology. A table of the upper critical t-value can be found at http://www.itl.nist.gov/div898/handbook/eda/section3/eda3672.htm.

Degrees of Freedom

Central Limit Theorem

Central Limit Theorem: The distribution of the means of random samples of size n from a population with mean m and variance s² approaches a normal distribution with mean m and variance s²/n as n increases, regardless of the population distribution, provided only that m and s² are finite, and s² is not zero.

For small samples from a normally distributed population, the t-distribution is symmetric, like a normal distribution, but it is platykurtic; that is, it is flattened relative to the normal: the peak height is lower than the highest points of a normal distribution, while the tails are higher and more spread out than those of the normal.

The shape of the t-distribution for smaller sample sizes indicates that larger differences between a sample mean m_X and an expected value m_e are more likely than when the sample size is large and the t-distribution approaches the normal. This result is consonant with our intuition that small samples will yield a less reliable estimate of a population mean than will larger samples.

The Relation of the t-Distribution to the Normal and c²-Distributions

                    N-dist.
t-dist.  =  ————
                    c-dist.

• If the difference, m_X - m_e, has a normal distribution, and
• If the standard error of the mean, [s_X²/n] ^1/2, has a c-distribution,

• The ratio of  m_X - m_e  to  [s_X²/n] ^1/2  has a t-distribution.

The requirements that m_X - m_e be normally distributed and that [s_X²/n] ^1/2 be c distributed can be satisfied for small sample sizes, n, if the following condition is met:

• The population of values of the variable X is normally and independently distributed.

Distribution of m_X - m_e . For small samples from a normally distributed population, the mean, m_X, of the sample is normally distributed because the sum of normally distributed values is normally distributed, and a normally distributed value multiplied by a constant, such as 1/n, is normally distributed. The subtraction of a constant, such as m_e, from a normally distributed value, such as the sample mean, yields a normally distributed value. Hence, the difference m_X - m_e, is normally distributed.

Distribution of [s_X²/n] ^1/2 . The difference between normally distributed values is normally distributed so that, for each observation in a sample from a normally distributed population, the difference x_i - m_X is normally distributed. Multiplication of a normally distributed values by a constant, such as 1/(n - 1) ^1/2, yields a normally distributed value. Hence, (x_i - m_X)/ (n - 1)^1/2 is normally distributed for each value x_i in the sample. As was observed in the note on Contingency Tables and the c ² Distribution, the sum of squared normally distributed values has a c² distribution. Thus, the sample variance, s_X², has a c² distribution with, in this case, n - 1 degrees of freedom. Taking the square root of a c² distribution yields a c.

The t-Statistic as a Signal to Noise Ratio

       signal
t = ———
        noise

• The signal consists of the difference m_X - m_e between a sample mean and a standard or expected value; and,



• The noise depends upon the variability, scatter, or dispersion of the values comprising the sample, and is measured using the standard error of the mean, [s_X²/n] ^1/2.

The signal must be sufficiently large relative to the noise in order that it be detected. For largish samples, with n about 60, the signal must be about twice the magnitude of the noise (because the critical t-value for a two-tailed test with a = 0.05 is 2.000). Thus, if the signal = 1.0, the noise cannot be larger than 0.5. If the noise were larger than 0.5, but the signal magnitude or strength remains unchanged, then the signal cannot be detected above the noise. For example, if the noise were 10% greater, that is noise = 0.55, but the signal strength is still 1.0, the signal to noise ratio become 1.82, and the signal is no longer detectable above the noise (because 1.82 is less than the critical value of 2.00).

Historical Note

A brief biography of Gosset is available at http://william-sealey-gosset.brainsip.com/ which includes the link http://www.york.ac.uk/depts/maths/histstat/student.pdf to Gosset's original paper: "Student" (1908). The probable error of a mean, Biometrika, Vol. 6, No. 1. (Mar.), pp. 1-25.

It is the platykurtic (flattened) shape of the t-distribution that was of interest to Gosset and which motivated him to devise the t-statistic. He was concerned that use of the normal distribution with small samples would lead to a conclusion that a sample mean m_X differed from a standard value m_e when, in fact, the observed difference was to be expected for a small sample size n.

One might imagine that Gossett would be testing the alcohol concentration of a batch of beer. Typically, he might make only five measurements of the alcohol level so that his sample size, n = 5. Suppose he calculated a statistic such as t; that is, he took the difference between the mean of his sample and the standard alcohol level, and he divided this by the standard error of the mean to obtain a value of 2.5. But, instead of using the critical values of t (because they didn't exist) to test whether his sample mean was significantly different from the standard, he used the critical values of a normal distribution. He would have applied a two-tailed test because an alcohol level either significantly greater than, or significantly less than the standard value would be undesirable. Therefore, if he employed a significance level, a = 0.05, he would have compared his statistic value of 2.5 with the 0.025 (= a/2 = 0.05/2) upper critical value of the normal distribution. This critical value is 1.960, which is less than 2.5, so that Gossett would have rejected the hypothesis that his sample mean was equal to the standard. Consequently, he would have had to take remedial measures with the batch of beer, and in the worst case, he would have discarded it.

Remedial measures to correct the alcohol level of the batch would have been costly, and perhaps unnecessary. After devising the t-statistic, Gossett would have looked up the 0.025 (= a/2 = 0.05/2) upper critical value of the t-distribution for n = 4 degrees of freedom and obtained 2.776. Comparing this with his statistic value of 2.5 would have led him to the conclusion that he could not reject the null hypothesis that his sample mean was equal to the standard alcohol level. Note that his failure to reject the null hypothesis does not mean that he could actually have claimed that his sample mean was equal to the standard. This result meant only that the evidence was insufficient to reject the null hypothesis. Consequently, there was no need to take costly remedial measures to correct the alcohol level of the batch of beer.