### Dr. Sarah's Circle Sampling Problem - Work together in a group of 2. Adapted from How Do You Know?

• Turn to page 149 in How Do You Know? There are 60 circles, with diameters that are shown in the table. Compute the actual average diameter of the circles and SHOW WORK.

• Choose one person to close their eyes and drop a marker onto p. 149 while the other person reads the directions. Mark down the diameter of each circle that is hit, until you have 10 numbers (you may hit the same circle more than once). Compute the average diameter of your sample and SHOW WORK. Why is this not a random sample of 10 circles (ie which circles are more likely to be chosen)? What are the biases?

• The other person should label the circles 01, 02,...,60 on their circle page. Then, by starting anywhere you like on the table on page 139 of How Do You Know? and underlining 2 digit numbers across each row, use the Table of Random Digits to choose a sample of 10 circles (the same circle may be chosen more than once). Mark down the diameter of each circle that is chosen on this page here. Compute the average diameter of your sample and SHOW WORK. Note that this is a random sample of 10 circles because every circle has the same chance of being chosen by the table of random digits.

When I performed one trail run of the two methods for 10 circles, the average diameter from the pencil drop method was closer to the actual average than was the average for the random sample method. Assume that I did everything correctly. The pencil drop method could obtain an average diameter closer to the true average even though the pencil drop method is a biased method because, for such a small sample size of 10, as in our discussion of coin tossing, chance and probability say that anything can happen.
• If I performed a trial run of the two methods to choose 1 billion circles from the 60 total circles, which resulting average diameter should be closer to the true average? Hint: Think about our discussion of the long-term behavior of coin tossing.