A method to investigate the sensitivity of the sample mean and the sample median to extreme outliers is to replace one or more elements in a given dataset by a number y and investigate the effect when...

A method to investigate the sensitivity of the sample mean and the sample median to extreme outliers is to replace one or more elements in a given dataset by a number y and investigate the effect when y goes to infinity. To illustrate this, consider the dataset from Quick Exercise 16.1:

4.6 3.0 3.2 4.2 5.0

with sample mean 4 and sample median 4.2.

a. We replace the element 3.2 by some real number y. What happens with the sample mean and the sample median of this new dataset as y → ∞?

b. We replace a number of elements by some real number y. How many elements do we need to replace so that the sample median of the new dataset goes to infinity as y → ∞?

 c. Suppose we have another dataset of size n. How many elements do we need to replace by some real number y, so that the sample mean of the new dataset goes to infinity as y → ∞? And how many elements do we need to replace, so that the sample median of the new dataset goes to infinity?