TI-82/83 Users Guide

Sampling Without replacement

A common area of probability experiments is related to "shuffling" the order of a set, or sampling from a set without replacement. In the Ti-82/83 series calculator, both can be done with a similar approach. I begin with the assumption that you have read the earlier guides on generating random digits and putting them into a list. There are minor differences in how the two calculators will perform this task, and we will use the format of the 83, but make notes about any significant differences required for the -82.

__ The problem of the Ten Hats__ We will illustrate the method around a well known statistical problem. Assume ten gentlemen enter an establishment and check their hats. The hat check clerk looses the check tickets, and when the men depart (in a group) she randomly hands each man one of the ten hats. We will explore the number of men who get back their own hat.

We begin by assigning a number to each of the men, and what would be easier than 1 through 10. Place each of these numbers in List 1, and then enter them again in List 2. Think of List L1 as the men, and L2 as their hats in proper order. Now we will randomly shuffle the order of the hats in L2 and then see who gets back their own hat.

__ Shuffling the List__ To shuffle the order of the elements in a list, we will create a list of random numbers in List L3 using the methods of previous lists. The screen copies below show the entry to produce the random digits and the appearance of the lists BEFORE we shuffle.

The next step is to use the SORTA command under the STAT/Edit menu. This command will sort the random numbers into order. The magic happens when we associate each of the numbers (hats) in List 2 with one of the random numbers in List 3. As the sorting process carries the random digits into numerical order, it will randomize the ten "hats" in List 2. The commands to do this are shown with the results of the shuffle in the next two screen displays.

Observe in the home screen display that the L3 comes first since this is the set we are ordering, and L2 will be carried along by the method. In MY example we see that none of the first seven people got their correct hats returned.

__ Counting the results:__ For a small sample such as this one, we could look at the lists and manually count how many of the people got their correct hats. For future reference on more complicated problems, however, we will illustrate a method by which the calculator will count the number of correct matches. On the Ti-83 the method is slightly easier. Using the "sum" command under the STATS/Math menu, and the equal sign under the [2nd][MATH/TEST] menu enter the command as shown in the next screen. You see that my data had two matches (numbers 8 and 10 matched). An alternative that works on BOTH calculators is shown in the screen below middle. By entering the command L1=L2 after the L4= prompt, when enter is pressed, the results, as shown on the screen at right will be zeros for the non-matches, and ones for the matches.

For long lists, the method on the right can be counted by using the "sum(L4)" command.

** Samples without Replacement. ** A similar method is used if you wished to pick four or five objects at random from a set. We could repeat the exact sequence above to select, for example, the first five numbers drawn at random from a set of ten by simply ignoring those after the first five.