Some of the components produced by a factory are defective. The management requires that no more than \(3 \%\) of the components produced are defective.
Niluki monitors the production process and takes a random sample of \(n\) components.
Write down the hypotheses Niluki should use in a test to assess whether or not the proportion of defective components is greater than 0.03
Niluki defines the random variable \(D _ { n }\) to represent the number of defective components in a sample of size \(n\). She considers two tests \(\mathbf { A }\) and \(\mathbf { B }\)
In test \(\mathbf { A }\), Niluki uses \(n = 100\) and if \(D _ { 100 } \geqslant 5\) she rejects \(H _ { 0 }\)
Find the size of test \(\mathbf { A }\)
In test B, Niluki uses \(n = 80\) and
if \(D _ { 80 } \geqslant 5\) she rejects \(\mathrm { H } _ { 0 }\)
if \(D _ { 80 } \leqslant 3\) she does not reject \(\mathrm { H } _ { 0 }\)
if \(D _ { 80 } = 4\) she takes a second random sample of size 80 and if \(D _ { 80 } \geqslant 1\) in this second sample then she rejects \(\mathrm { H } _ { 0 }\) otherwise she does not reject \(\mathrm { H } _ { 0 }\)
Find the size of test \(\mathbf { B }\)
Given that the actual proportion of defective components is 0.06
find the power of test \(\mathbf { A }\)
find the expected number of components sampled using test \(\mathbf { B }\)
Given also that, when the actual proportion of defective components is 0.06 , the power of test \(\mathbf { B }\) is 0.713
suggest, giving your reasons, which test Niluki should use.