A paracetamol concentration of more than 150 mg per kilogram body weight is considered as dangerous. The measurements of paracetamol in blood tests are modelled

by a random variable with normal distribution N (μ, σ2). The standard-deviation, linked to the testing method, is supposed to be known and equal to 5 mg. For better assessment, 4 blood tests are usually made. The results are assumed to be independent realisations of the same normal distribution N (μ, σ2).

1. Give the hypotheses and the decision rule for the test deciding, at threshold 5%, whether a patient is at risk, on view of 4 blood tests. (you are a cautious doctor).

2. On a given patient, the 4 blood tests gave concentrations of 140, 133, 148, 144. Give the p-value for the test of the previous question. Is he at risk?

3. From now on, the standard-deviation is supposed unknown. Give the test statistic and the decision rule for the test deciding, at threshold 5%, whether a patient is at risk, on view of 4 blood tests.

4. For the patient of question 2, give an interval containing the p-value for the test of the previous question. What is your conclusion?