Question: If the variance exceeds 2, there will be quality problems. Suppose that the manufacturer tests the hypotheses \[{{H}_{0}}:{{\sigma }^{2}}\le 2\] vs. \[{{H}_{1}}:{{\sigma }^{2}}>2\] and the rejection region is the set of all realizations \[{{x}_{1}},\ldots ,{{x}_{10}}\] for which the sample variance \[{{s}^{2}}>3.75977\].

a) What is the significance level of the test?

b) What is the power of the test if \[{{\sigma }^{2}}=6\] ?

c) Suppose the realized measurements are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Is \[{{H}_{0}}:{{\sigma }^{2}}\le 2\] accepted or rejected? What is the p-value of the test?

d) Briefly discuss the issues (pros and cons) involved in interchanging the two hypotheses – i.e., for testing \[{{H}_{0}}:{{\sigma }^{2}}\ge 2\] vs. \[{{H}_{1}}:{{\sigma }^{2}}<2\], instead of \[{{H}_{0}}:{{\sigma }^{2}}\le 2\] vs. \[{{H}_{1}}:{{\sigma }^{2}}>2\]. [You are not required to do the interchanged test - just discuss it.]