1.4.4.md

Solution to Review Question

by Qiang Gao, updated at May 11, 2017

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Chapter 1 Finite-Sample Properties of OLS

Section 4 Hypothesis Testing under Normality

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Review Question 1.4.4 (One-tailed $$t$$-test)

The $$t$$-test described in the text is the tow-tailed $$t$$-test because the significance $$\alpha$$ is equally distributed between both tails of the $$t$$ distribution. Suppose the alternative is one-sided and written as $$ \mathrm{H}_1: \beta_k > \bar{\beta}_k $$. Consider the following modification of the decision rule of the $$t$$-test.

1. Same as above.
2. Find the critical value $$t_\alpha$$ such that the area in the $$t$$ distribution to the right of $$t_\alpha$$ is $$\alpha$$. Note the difference from the two-tailed test: the left tail is ignored and the area of $$\alpha$$ is assigned to the upper tail only.
3. Accept if $$t_k < t_\alpha$$; reject otherwise.

Show that the size (significance level) of this one-tailed $$t$$-test is $$\alpha$$.

Solution

The size of a test is the probability of falsely rejection. When $$\mathrm{H}0$$ is true, the test statistic $$t_k$$ has a $$ t(n-K) $$ distribution. If we reject whenever $$t_k \ge t\alpha$$, then the probability is $$\alpha$$ by construction. So the size of this one-tailed $$t$$-test is $$\alpha$$.

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