Let X be arandom variable that represents the level of glucose in the blood (milligrams per deciliter of blood)

Let X be arandom variable that represents the level of glucose in the blood (milligrams per deciliter of blood).

The distribution of X for a healthy person is normally distributed with men p = 85 and standard deviation o =

25. A person suffers from severe excess in insulin would have a lower level of glucose. A blood test with result of X < 40 would be used as an indicator that medication is needed.

(a) What is the probability that a healthy person will be suggested with medication after a single test?

(b) A doctor uses the average result of 2 tests for diagnosis, that 1s X. The second test will be conducted one week after the first test, so that the two test results are independent. For many. healthy persons, each has finished two tests, find the expectation and standard error of the distribution of X.

(c) The doctor suggests medication will be given only when the average level of glucoses in the 2 blood tests is less than 40, that is X¥ < 40, so to reduce the chance of unnecessary use of medication on a healthy person. Use the distribution 1n part (b)) to find the probability that a healthy person will be suggested with medication after 2 tests to verify this doctor’s theory.

Question 2 (20 marks)

Coal is carried from a mine in West Virginia to a power plant in New York in hopper cars on a long train. The automatic hopper car loader is set to put 75 tons of coal into each car. The actual weight of coal loaded into each car is normally distributed, with mean of 75 tons and standard deviation of 0.8 ton.

Coal is carried from a mine in West Virginia to a power plant in New York in hopper cars on a long train. The automatic hopper car loader is set to put 75 tons of coal into each car. The actual weight of coal loaded into ,each car is normally distributed, with mean of 75 tons and standard deviation of 0.8 ton.

(a) There are 97% of the cars will be loaded with more than K tons of coal. What is the value of K?

(b) What is the probability that one car chosen at random will have less than 74.4 tons of coal?

(c) Among 20 randomly chosen cars, what is the probability that more than 2 cars will be loaded with less than 74.4 tons of coal?

(d) Among 20 randomly chosen cars, most likely, how many cars will be loaded with less than 74.4 tons of coal? Calculate the corresponding probability.

(e) In the senior management meeting, it is discussed and agreed that a car loaded with less than 74.4 tons of coal is not cost effective. To reduce the ratio of cars to be loaded with less than 74.4 tons of coal, it is suggested changing current average loading of coal from 75 tons to a new average level, M tons.

Should the new level M be (I) higher than 75 tons or (II) lower than 75 tons? (Write down your suggestion, no explanation is needed in part (e)).