The statement "With the same data set and the same \mathbf{P}(Type I error) but using a different test statistic, \mathbf{P}(Type II error) can differ" is true.
The statement "We should look for \alpha such that \mathbf{P} (Type I error) + \mathbf{P} (Type II error) is minimized" is also true.
3. Which of the following statement(s) is/are true?
Once we set \mathbf{P}(Type I error), \mathbf{P}(Type II error) is automatically set, does not depend on the data, and cannot be computed.
Once we set \mathbf{P}(Type I error), \mathbf{P}(Type II error) depends on both the data and the test statistic, and can be computed.
Once we set \mathbf{P}(Type I error), \mathbf{P}(Type II error) depends on the test statistic but not the data, and can be computed.
Once we set \mathbf{P}(Type I error), \mathbf{P}(Type II error) depends on the data and the test statistic, and can sometimes be computed.
With the same data set and the same \mathbf{P}(Type I error) but using a different test statistic, \mathbf{P}(Type II error) can differ.
We should look for \alpha such that \mathbf{P} (Type I error) + \mathbf{P} (Type II error) is minimized.
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