Manipulations with Sums. Prove the following algebraic identities involving sums, useful in demonstrating properties of some sample summaries.   Emergency Calculation. Graduate student Rosa...

Manipulations with Sums. Prove the following algebraic identities involving sums, useful in demonstrating properties of some sample summaries.

Emergency Calculation. Graduate student Rosa Juliusdottir reported the results of an experiment to her advisor who wanted to include them in his grant proposal. Before leaving to Reykjavik for a short vacation, she left the following data in her advisor’s mailbox: sample size n = 12, sample mean

and sample variance s²
= 34. The advisor noted with horror that the last measurement X12 was wrongly recorded. It should have been 16 instead of 4. It would be easy to fix X and s², but the advisor did not have the previous 11 measurements nor the statistics training necessary to make the correction. Rosa was in Iceland, and the grant proposal was due the next day. The advisor was desperate, but luckily you came along.