Question

Suppose we have a test statistic T_ n such that T_ n \sim |Z| where Z \sim N(0,1). In particular, for this problem we know the distribution of T_ n for any fixed n and not just asymptotically. You design the test

\psi _ n = \mathbf{1}(T_ n \geq q_{\eta /2})

where q_\eta is the 1-\eta quantile of a standard Gaussian (i.e., if Z \sim N(0,1), then P(Z > q_{\eta }) = \eta). If \psi = 1, we will reject H_0, and if \psi = 0, we will fail to reject H_0.

With this set-up, you observe a data set and compute T_ n. Consider the following figure:


On which side, to the left or to the right, of T_ n should the value q_{\eta /2} be such that \psi _ n rejects on our data set?

Answer 1

What is the largest value of q_{\eta /2} such that \psi _ n rejects on our data set?

Answer 2

What is the smallest value of \eta so that \psi _ n rejects on our data set? (Note that this is the p-value for our data set.)



\eta =2\times (\text {the area under the curve to the right of A})

\eta =2\times (\text {the area under the curve to the right of B})

\eta =2\times (\text {the area under the curve to the right of C})
unanswered

Answer 3

Now you observe a new data set and compute a new value of the test statistic, which we denote by T_ n'. Suppose that T_ n' < T_ n, i.e., the test statistic has a smaller value than from before.

Will the new p-value be larger or smaller than the p-value from the previous data set considered in this problem?