APPLIED MATHS probability2

1. In an attempt to diagnose early a certain metabolic disorder in babies a new screening test has been proposed. This test consists of determining the amount of a certain steroid in the urine. Extensive study has shown that it can be assumed that the distribution of the quantity of the steroid is N(20, 9) for unhealthy babies with metabolic disorder.
(a) The suggested screening process will classify a baby as healthy if the steroid level is below 24 units, and unhealthy otherwise.
(i) What is the probability that a baby will metabolic disorder will be misclassified as healthy?
(ii) What is the probability that a healthy baby will be misclassified as having metabolic disorder?
(b) If medical doctors want to ensure that not more than 3% of the babies with metabolic disorder will be wrongly classified as healthy, what critical steroid level should be used for classification?
(c) It is known that 10% of all new-born babies have metabolic disorder. What proportion of babies will be misclassified under each of the two criteria specified in (a) and (b) above?


2. The sizes of two large batches, A and B, of electrical items produced on two different machines are in the ratio 6 : 4. The resistances of the items in each batch are assumed to have a normal distribution, with mean 80 ohms and standard deviation 5 ohms for batch A, and mean 72.5 ohms and standard deviation 6 ohms for Batch B. The two batches are combined and thoroughly mixed.
(a) Find the mean ( m ) and standard deviation ( k) of the resistances of the items.
(b) Any item whose resistance is more than 2k ohms above or below m is regarded as defective. Find the proportion of defective items in the combined batch.
(c) A random sample of 10 items is taken. Find the probability that it contains not more than two defective items
(d) An item is randomly selected and is found to be defective. What is the probability that it originally belonged to batch B?