Stats Solutions

(Steps Shown) Suppose that Y_i denotes the stock returns for the i -th firm before the subprime mortgage crisis in 2008 , and X_i denotes the stock returns for

Question: Suppose that \(Y_{i}\) denotes the stock returns for the \(i\) -th firm before the subprime mortgage crisis in 2008 , and \(X_{i}\) denotes the stock returns for the \(i\) -th firm after the subprime mortgage crisis. The researcher has a random sample of 1000 firms. Assume the following:

\[\begin{aligned} Y_{1}-X_{1} & \sim N\left(0, \sigma^{2}\right) \text { under the null hypothesis } \\ Y_{1}-X_{1} & \sim N\left(a, \sigma^{2}\right) \text { under the alternative hypothesis for some } a>0 \end{aligned}\]

where we assume that the researcher knows the variance \(\sigma^{2}=36\). Suppose that

\[\begin{aligned} \bar{X}_{n} &=100 \text { and } \\ \bar{Y}_{n} &=200 \end{aligned}\]

where we recall \(n=1000\).

  1. Construct a test statistic that tests:
    \[\begin{aligned} H_{0} &: \mathbf{E} Y_{1}=\mathbf{E} X_{1} \text { against } \\ H_{1} &: \mathbf{E} Y_{1}>\mathbf{E} X_{1} \end{aligned}\]
  2. Find a critical value that gives Type I error of the test exactly equal to 0.05.
  3. See if the researcher would reject the null hypothesis at \(5 \%\) significance level based on the test defined through (a) and (b).
  4. Compute the \(p\) -value of the test statistic. Would the researcher reject the null hypothesis at \(1 \%\) ?
  5. Compute the power of the test when \(\alpha=0.07\).

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