Suppose (Y_{i} sim) i.i.d. (Nleft(mu_{Y}, sigma_{Y}^{2} ight)) for (i=1, ldots, n). With (sigma_{Y}^{2}) known, the (t)-statistic for

Suppose \(Y_{i} \sim\) i.i.d. \(N\left(\mu_{Y}, \sigma_{Y}^{2}\right)\) for \(i=1, \ldots, n\). With \(\sigma_{Y}^{2}\) known, the \(t\)-statistic for testing \(H_{0}: \mu_{Y}=0\) vs. \(H_{1}: \mu_{Y}>0\) is \(t=(\bar{Y}-0) / \operatorname{SE}(\bar{Y})\), where \(S E(\bar{Y})=\sigma_{Y} / \sqrt{n}\). Suppose \(\sigma_{Y}=10\) and \(n=100\), so that \(S E(\bar{Y})=1\). Using a test with a size of \(5 \%\), the null hypothesis is rejected if \(t>1.64\).

a. Suppose \(\mu_{Y}=0\), so the null hypothesis is true. What is the probability that the null hypothesis is rejected?

b. Suppose \(\mu_{Y}=2\), so the alternative hypothesis is true. What is the probability that the null hypothesis is rejected?

c. Suppose that in \(90 \%\) of cases the data are drawn from a population where the null is true \(\left(\mu_{Y}=0\right)\) and in \(10 \%\) of cases the data come from a population where the alternative is true and \(\mu_{Y}=2\). Your data came from either the first or the second population, but you don't know which.

i. You compute the \(t\)-statistic. What is the probability that \(t>1.64-\) that is, that you reject the null hypothesis?

ii. Suppose you reject the null hypothesis; that is, \(t>1\).64. What is the probability that the sample data were drawn from the \(\mu_{Y}=0\) population?

d. It is hard to discover a new effective drug. Suppose \(90 \%\) of new drugs are ineffective and only \(10 \%\) are effective. Let \(Y\) denote the drop in the level of a specific blood toxin for a patient taking a new drug. If the drug is ineffective, \(\mu_{Y}=0\) and \(\sigma_{Y}=10\); if the drug is effective, \(\mu_{Y}=2\) and \(\sigma_{Y}=10\).

i. A new drug is tested on a random sample of \(n=100\) patients, data are collected, and the resulting \(t\)-statistic is found to be greater than 1.64. What is the probability that the drug is ineffective (i.e., what is the false positive rate for the test using \(t>1.64)\) ?

ii. Suppose the one-sided test uses instead the \(0.5 \%\) significance level. What is the probability that the drug is ineffective (i.e., what is the false positive rate)?