Student’s t Test
The t test is ideal when we have a single constraint, that is, q = 1. The F test, discussed in the next section, must be used if q > 1.
Since (J is normal as shown above, we have
(12.4.7) Q’0 ~ N[c, a2Q’ (X’X^Q] under the null hypothesis (that is, if Q’3 = c). Note that here Q’ is a row vector and c is a scalar. Therefore,
(12.4.8) т z ~ N(0, 1).
The random variables defined in (12.4.2) and (12.4.8) are independent because of Theorem 12.4.2. Hence, by Definition 2 of the Appendix, we have
QP ~ c
Student’s t with T — К degrees of freedom, where a is the square root of
the unbiased estimator of a defined in equation (12.2.29). Note that the denominator in (12.4.9) is an estimate of the standard deviation of the numerator. The null hypothesis Q’P = c can be tested by the statistic
(12.4.9) . We use a one-tail or two-tail test, depending on the alternative hypothesis.
The following are some of the values of Q and c that frequently occur in practice:
The ith element of Q is unity and all other elements are zero. Then the null hypothesis is simply (3, = c.
The zth and 7th elements of Q are 1 and — 1, respectively, and c = 0. Then the null hypothesis becomes (3, = (37.
Q is a А-vector of ones. Then the null hypothesis becomes Х,:=1рг = c.