Consider test of \(H_{0}: \mu=0\) against \(H_{a}: \mu eq 0\) at nominal size 0.05 when the dgp is \(y \sim \mathcal{N}[\mu, 100]\), so the standard deviation is 10 , and the sample size is \(N=10\). The test statistic is the usual \(t\)-test statistic \(t=\widehat{\mu} / \sqrt{s / 10}\), where \(s^{2}=\) (1/9) \(\sum_{i}\left(y_{i}-\bar{y}\right)^{2}\). Perform 1,000 simulations to answer the following.

(a) Obtain the actual size of the \(t\)-test if the correct finite-sample critical values \(\pm t_{.025}(8)= \pm 2.306\) are used. Is there size distortion?

(b) Obtain the actual size of the \(t\)-test if the asymptotic approximation critical values \(\pm z_{.025}= \pm 1.960\) are used. Is there size distortion?

(c) Obtain the power of the \(t\)-test against the alternative \(H_{a}: \mu=1\), when the critical values \(\pm t_{.025}(8)= \pm 2.306\) are used. Is the test powerful against this particular alternative?