3. Consider a test that measures the level X of a chemical in the blood of a patient to decide whether a disease is pres

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3. Consider a test that measures the level X of a chemical in
the blood of a patient to decide whether a disease is present.
Suppose that X = W if the patient does not have the disease and X =
1 + W if they do, where W ∼ N(8, 1 9 ) i.e. the usual level of the
chemical in the blood is Normal distributed around 8 and with a
variance of 1 9 and the disease increases the level by 1.
(a) If the null hypothesis is that a patient is healthy and the
alternative is that they have the disease, write down H0 and HA in
terms of possible values of θ where X = θ + W .
(b) Design a level α = 0.05 test to decide between H0 and
HA, using the pivot approach. i.e. find κ such that you reject H0
when X > κ. Hint: Start by calculating α = P(X > κ|H0).
(c) Find the the probability β of concluding the patient
does not have the disease when it is present.
(d) If you observe x = 8.6, is there enough evidence to reject
H0 at significance level α = 0.01?
(e) If you would like the probability of missing a present
disease to be less than 5%, what is the smallest significance level
that you can achieve? Comment on your result. Hint: evaluate the
critical value κ corresponding to this value of β.