1. A test for BCRA ½ test is 99% successful, i.e., if a patient carries the BCRA ½ mutation, it will detect it in 99% of all tests, and if you are the patient doesn't carry it, it will be right 90% of the times, that is negative. A patient takes the test and the result is positive.

a. What probability of carrying the mutation do you report to the patient?

b. If 100 patients are tested, how many are expected to carry BCRA ½ mutation and the test will come out negative?

2. A courier service advertises that its average delivery time is less than 48 hours for intercity deliveries. A random sample of times for 16 deliveries to an address across town was recorded. Previous analyses had shown that the distribution of the delivery time is normally distributed with a standard deviation of 4. The average delivery time of the sample is 49.5 hours, the standard deviation of the sample is 5.8 hours.

a. Is this sufficient evidence to support the courier's advertisement, at 95% level of confidence?

b. The law gives the right to a customer to get a full refund for the service if the delivery time is longer than 51.3 hours. What is the risk that the courier has delivery time of 51.3 hours or bigger and does not detect it and it is not detected using the current sampling setting? Hint: this is a type II error.

c. Suppose the courier needs to estimate the delivery time within 4 hours. Provide a quantitative proposal on how this could be achieved. What is the sample size that the courier would use in the estimation?