One of the ways of assessing the quality of a university is in terms of the amount of research funds the faculty bring into the university. An administrator at Princess University knows that, historically, Princess professors have brought in an average of $15870 per person in research funds per year, with a standard deviation of 550.She also knows that professors at arch?rival Duchess University have attracted an average of $13925 per person per year, with a variance of 245025.The administrator takes a sample of 44 Princess professors and 39 Duchess professors and records their grant amounts for the most recent year.

9) One of the ways of assessing the quality of a university is in terms of the amount of research funds the faculty bring into the university. An administrator at Princess University knows that, historically, Princess professors have brought in an average of $15870 per person in research funds per year, with a standard deviation of 550. She also knows that professors at arch-rival Duchess University have attracted an average of $13925 per person per year, with a variance of 245025. The administrator takes a sample of 44 Princess professors and 39 Duchess professors and records their grant amounts for the most recent year. a) What is the probability that in these samples the Princess mean exceeds the Duchess mean by at least $2000? (5 marks) b) The administrator suspects that the difference between Princess and Duchess professors has narrowed because of recent hiring at Duchess. If, in fact, the true difference is now $1550, what is the probability that the administrator detects that the difference is narrowing if she samples 50 faculty from each university? (a = .05) (6 marks)