TI-82 Users Guide #3

Math operations in Lists

Usually we don't generate lists just for the thrill of it, but because we wanted to do something or test some idea. The Mathematical operations under the list command will allow us to extend the information we get from simulations.

Press <2^{nd}>

It is easy to save lists and recall them later, but there are some precautions. You need to make sure the list doesn't have any old data stored in it that might confuse the issue. To do this we want to clear the list we will be using. In this exploration we will be using the list L1, so lets clear it now. Press ^{nd}> <1> and the screen should add the name of the list, L1. A <2> at the end would have cleared list, L2. Now press enter and the calculator tells you that the command has been executed with the "Done" response.

Now we need to generate a list and save it to L1. We want to see how many times a heads comes up in 25 flips of a coin, so we will generate a list of 25 zeros and ones and call the ones heads. You should know from last time that the way to do this is seq(int(2*rand),x,1,25,1). To save the list as L1, all we have to do is press the STO> button and <2^{nd}>

__ Finding the SUM of terms in a list__ Now we want to know, "How many heads did we flip?". To find out, all we have to do is find the sum of all the numbers in the list, since the "tails" are zeros and will not add anything to the sum, the sum is just how many ones (heads) there were. The key strokes <2

**DESCRIPTIONS OF THELIST " MATH" OPERATIONS:**

- Min( L1) gives the smallest value in List L1
- Max(L1) gives the largest value in the list
- Mean(L1) gives the arithmetic average of a list of values.
- Median(L1) gives the number that would occur in the middle of the ordered set. If you don't know what the median is, try experimenting with several lists of numbers until you think you understand, or look in a math book for examples, or ask your teacher.
- Sum L1 adds all the values in the list
- And Prod L1 multiplies them all together. Since at least one of your values is zero in L1, Prod L1 will probably be zero, but if you wanted to see if ALL 25 flips were heads, Prod L1 would be 1 if it were, and zero otherwise.

The numbers don't have to be in a saved list. The command mean({2,4,9}) will produce 5, the average of the three numbers in the parentheses and braces, but you DO have to have the braces if the list is unnamed. The command Prod {3,4,5} will give 60.

There is one more shortcut you can use. You don't have to save a list and THEN go to the math operations. You can put the operation in front of the "seq" command and it will work without naming the list. Try entering "SUM seq(x,x,1,10,1)" The calculator should go directly to the answer 55 without showing you the sequence. This can be handy if your trying to figure out a pattern for the sums of integers starting at one:

1 =1

1+2 =3

1+2+3 =6

1+2+3+4 =10

Hmmmm, wonder if I can figure that pattern out?

Ok, that is a lot of commands for one session, so lets give you a chance to try them out and see if you understand. Remember the answers are down the page, but don't look until your really stumped. If you still have problems, see your teacher, or contact Mr Ballew by e-mail. His address is on the Mathboy's page.

- Try to use a sequence to produce the integers from 1 to 25, then find their sum.
- Find the product of the first 15 integers (big, huh?)
- Find the sum of the first 25 ODD integers (try the first 8 odds or the first five odds, what do you notice about the sums of the odd integers?)
- Simulate rolling a regular die (1-6) 30 times. What is the mean (average) of the rolls? Will it be the same every time you do it?
- Simulate picking 5 random integers between 1 and 50 from a hat. What is the biggest and smallest number you get? Do this several times and compare the different values you get for the max and min. What do you think would be the effect if you picked 10 or 20 numbers each time instead of 5?
- Repeat the simulation in problem 5 but find the mean and median of each list? Which one is bigger? Is that always true? Can you look at the list and predict which will be bigger?

All of our samples this time involved simple mathematical procedures, and drawing from a sample without replacement. What if we wanted to count how many heads we got in a row, or the number of threes we drew from the hat in problem 5, or if we wanted to simulate drawing cards from a deck and other situations where you don't return the card , so you couldn't have two ace of spades, for example? Check back soon and we'll show you how to use the List commands under the Stat menu to do some more math with Lists.

CAREFUL… WE STORED THE ANSWERS DOWN HERE… DON'T LOOK BEFORE YOU REALLLLLLY TRY TO ANSWER THE QUESTIONS

- seq(x,x,1,25,1)-> L1 should store the list… sum L1 should give 325
- This number is called 15 factorial, and can also be found by entering 15 and then pressing $and\; 4\; and\; then.\; The\; screen\; should\; show\; 15!\; And\; give\; 1.307674\dots \; E12\; ,\; which\; means\; a\; little\; more\; than\; one\; TRILLION.\; To\; see\; how\; the\; factorial\; (!)\; works,\; try\; 3!\; Or\; 4!\; And\; see\; what\; you\; get.$
- The sum of the first 25 odd integers is 25*25 = 625. This is true for any number of odd integers starting at one. 1+3+5+7+9 = 5*5=25…cute huh?
- The average SHOULD be about 3.5, but it will change from experiment to experiment, which is good, because that is what happens when you roll a real die.
- The biggest and smallest numbers will vary a lot in a sample of five, but if you take larger samples, the max should be closer to 50, and the min should be closer to 1.
- In small samples like a five draws, either value may be higher (and they may be the same, but not likely). If one number is very high compared to the rest, the average (MEAN) will probably be higher.