Quantile Regression

Appendix BApplications of the t‐Test Statistic for Testing Alternative Hypotheses

This appendix presents applications of the t‐test statistic for directly testing two‐sided and one‐sided hypotheses using the observed t‐statistic and its probability presented in the corresponding statistical results. For an illustration, see the statistical results in Figure B.1 and its model parameters presented in Table B.1, which are taken from the results presented in Chapter 2, in Figure 2.11 and Table 2.4. Based on Figure B.1 and Table B.1, we will define and test several alternative hypotheses.

 

B.1 Testing a Two‐Sided Hypothesis

A two‐sided statistical hypothesis can be presented as follows:

(B.1)

with the p‐value of each hypothesis equals to Prob. in the output.

Hence, at the α level of significance, the null hypothesis H₀ is rejected if the p‐value = Prob. < α. And it is accepted if the p‐value ≥ α.

For instance, for k = 2, at the 5% level of significance, the null hypothesis is rejected based on the t‐statistic of t₀ = −2.523 with df = (300 − 6) and p‐value = Prob./2 = 0.0122 < 0.05. Hence, we can conclude that the medians of Y1 in the two levels G = 1 and G = 2 have a significant difference, conditional for the level of H = 2.

B.2 Testing a Right‐Sided Hypothesis

A right‐sided statistical hypothesis can be presented as follows:

(B.2)

and the conclusion of the testing of each ...