Source: 40 Puzzles and Problems in Probability and Mathematical Statistics (Interesting book by Wolfgang Schwarz)
Inspiration: This problem demonstrates clearly the shortcomings of out grading system through exams
a) What is your prediction for the outcome of this second measurement if standard deviation = 3?
b) Answer the same question if standard deviation = 20?
Update (Sep 15, 2012):
Minor change in the question as suggested by Akshay Soni
Update (5th Feb 2013):
Solution posted by Akshay Soni (IITB Mech Senior Undergraduate) , AB and Nikhil Simha R (Amazon India SDE, CSE IITB 2012 Alumnus) in comments! Rephrased and improved formatting of the solution and posted by me in comments!