Canada's profile in the international physics community got a huge boost as renowned "superstar" cosmologist Stephen Hawking (shown in the photo below floating on a zero-gravity jet) accepted a research post at the country's "crown jewel" of theoretical physics study, The Perimeter Institute in Waterloo, Ontario. The ratio of mental age (MA) divided by chronological age (CA) and multiplied 100 is called "Intelligence Quotient (IQ). It is suggested that the average IQ of top civil servants, professors, and research scientists is 140. Suppose the standard deviation is 5.

a. Suppose a full professor from a Canadian university is selected at random. What is the probability that the IQ of the selected Canadian professor is below 130? State any necessary assumptions you have made to compute this probability.

b. Suppose the assumption(s) made in part a was not justifiable. A researcher decided to take a random sample of 81 full professors from the Canadian universities system.

(i) What is the sampling distribution of the sample mean x? Explain. (ii) Find the mean and standard deviation of the sampling distribution of x. (iii) What is the probability that the sample average IQ of this sample is less than 130? Compare your answer with answer given in

a. Summarize your findings. (iv) Suppose that Stephen Hawking's score was 187. If you include this score in the sample of 81 Canadian professors, do you think Canadian professors' average IQ will improve significantly? (v) Will the probability in (iii) change if we change the wording from "less than" to "less than or equal to"? Why or why not? (vi) Can you calculate the probability that the sample average IQ is exactly 135? Justify your answer. (vii) If the sample mean x is actually 130, what can be said about the claim that 140? What conclusion might you draw? (viii) What is the probability that the sample mean differs from the population mean by more than 2? (ix) Within what limits would you expect the sample average to be, with prob- ability 0.95?

c. The total IQ score x, is the sum of the individual 81 selected professors. (i) What kind of sampling probability distribution do you expect the total scores to have? Explain. (ii) Provide the mean and standard deviation of the probability distribution of the total score x,. (iii) Find the probability that the total score will be between 11,000 and 11,400, inclusively.