This exercise presents a confidence interval for the population median that requires no assumption about the population distribution other than it is essentially continuous.

(a) Explain why for a random sample of size n the sample proportion πˆ falling below themedian has expected value 0.50 and standard error σπˆ = 0.50/

√

n, and so the probability is about 0.95 that the number of observations falling below (above) the median is within n(1/

√

n) =

√

n of half the sample.

(b) For the ordered sample of size n, explain why a 95%

confidence interval for the median goes from the observation that has the index (n + 1)/2 −

√

n to the observation with the index (n + 1)/2 +

√

n. For Example 5.9 on median shelf life in a library, show that this interval is 11 to 19 years.