To get full marks for the following questions you need to convert the question from words to a mathematical expression (i.e. use mathematical notation), defining your random variables where necessary, and using correct probability statements.

A German physician, Carl Wunderlich, measured temperatures from about 25,000 people in the mid 1800s and found that the average was 37 degrees Celsius. There is some evidence that the mean body temperature has been changing over time. Assume that the body temperature of adults is approximately normally distributed with a mean of 37 degrees Celsius and standard deviation 0.40 degrees Celsius.

(a) [2 marks] What value of body temperature distinguishes the highest 10%?

(b) [2 marks] People with body temperatures above 38 degrees are said to have a significant fever. Show that approximately 0.62% of people have a significant fever.

(c) [4 marks] Suppose we take a random sample of 50 people and want to determine whether or not they have a significant fever. Let X represent the number of people in a sample of size 50 who have a significant fever. What probability distribution does X follow? Justify your answer.

(d) [2 marks] Using the probability distribution from part (c), find the probability that at least 1 person in the random sample of 50 have a significant fever.