Chapter 3, Problem 3.25P

The following method was proposed to estimate the number of people over the age of 50 who reside in a town of known population 100,000: “As you walk along the streets, keep a running count of the percentage of people you encounter who are over 50. Do this for a few days: then multiply the percentage you obtain by 100,000 to obtain the estimate.” Comment on this method. Hint: Let p denote the proportion of people in the town who are over 50. Furthermore, let ${\alpha }_1$ denote the proportion of time that a person under the age of 50 spends in the streets, and let ${\alpha }_2$ be the corresponding value for those over 50. What quantity does the method suggested estimate? When is the estimate approximately equal to p?