Example 1: In a certain shire in England, a poll is taken to find out the percentages of Whigs and Tories. Of a simple random sample of 400 residents, 120 say they are Whigs; the rest are Tories. What should we guess is the percentage of population of that shire that is Whigs? And how do we use the standard error?

Example 2: Of 2500 Purdue faculty, a simple random sample of 100 is polled. Forty percent say they are voting for Abhyankar for faculty president. What is a 95% confidence interval for the percentage of the whole Purdue faculty voting for Abhyankar?

Example 3: Of the 10 million people in a state, we sample 1600 at random and ask the number of shirts (or blouses, or tops, or whatever) they own. Suppose the 10 million have an average of 10 shirts, with a standard deviation of 4 shirts; but that 30% have at least 12 shirts. What is the:

1. probability that the average number of shirts owned by people in the sample is at least 12?
2. probability that at least 500 people in the sample own at least 12 shirts?

Beginnings of solutions for Example 3:

1. In this case the box model is a box of 10 million tickets, each having the number of shirts of a different person on it. The problem gives us information that we usually don't have: The average of this box is 10, with an SD of 4.
2. In the case we are counting the number of people who have at least 12 shirts; someone having 12 shirts is now the same as someone who has 35. So the box model is now a 0-1 box, with 10 million tickets. And again the problem gives us information we don't usually have: the part of the box that is 1's is 30%.