The primary objective of the second section is to introduce the idea of randomness and to solve random sample problems. To get a real feeling for what randomness is, have the students generate their own random number tables. One way to do this is to put ten wooden squares, each of which has one of the digits 0 9 inclusive on it, in a container and shake. Have a student draw a digit and write it on the board. Making it obvious that the sample has been returned, shake and repeat for maybe twenty numbers. Since each time each digit has an equal chance at being picked, this is called a random selection of digits, and what you are beginning to do is to generate a random digit table on the board.
After doing this much together on the board, have students generate their own. Use Form E, #1. They can work in pairs and save time, increase interest and use less sets of squares for generating their numbers.
When the tables are done, use one of them to randomly pick a committee from the class. One way to do this is as follows.
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1. List all students’ names on the board. Have students write the list on back of their worksheets.
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2. Number each name. Say it goes from 01 to 20. Since we have 20 students, we need to read in groups of two digits and so write our numbers 01,02, 03, . . . 20. They will see the reason for this as soon as they start reading the table to select a committee.
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3. Have a student point arbitrarily to a spot on the table you are using. Explain that we will start there and read the numbers from the table. However, before we start to read the numbers, we have to decide if we want to read horizontally, vertically or diagonally. It doesn’t matter, but once we pick a way we should stay with it until the task is done. An overhead projector might be helpful here.
Consider the table below. If you were starting at the third row second digit, we could read 78, 85, 53, 32, 12, etc. In doing so we would be reading across and in groups of two digits but moving one digit at a time. If we had a larger table we might prefer to move two digits at a time and so starting from the same place we could read 78, 53, 21, 22, 21, 17, 73, 70, 23, 25, 33 etc.
row:
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1
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14073
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|
2
|
43318
|
|
3
|
77853
|
|
4
|
21222
|
|
5
|
11773
|
|
6
|
70232
|
|
7
|
52333
|
|
8
|
90012
|
|
9
|
86746
|
|
10
|
64337
|
However we read it, the first number that is from 01 to 20 inclusive gives us the first member of the committee, and we keep going discarding numbers that do not have meaning for our task. If we are selecting a committee of three, we keep going until we get three numbers from 01 to 20 inclusive, and the names that correspond to these numbers are the ones on our committee. If we use the table above and keep counting the way we originally began we would read the numbers 78,85,53,32,21,12,22, 22,21,11,17. That would give us 12, 11 and 17. The names corresponding to 12, 11 and 17 would make up our class’s randomly selected committee. The task is now completed.
By the time the class has selected a committee of three, then each student selects his or her own committee, #2 on Form E, most students probably will be able to do the random sample problem #3 on Form E.
#3 Form E. A batch of 200 new cars has just been completed. Your job is to randomly select 15 of the cars for a special safety check.
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a. Describe how to do this.
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b. Select the 15 cars. Use random number table on handout.
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c. List the 15 numbers selected.
For this problem you will want a larger random digit table than the ones generated. Form F. A classroom set of copies of a random table is needed. Solution:
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a. Number all the new cars 000 to 200 inclusive. Arbitrarily select a place to start on the larger table, decide if to read across, down or diagonally and begin reading in groups of three digits. Any three digit number 000 to 200 inclusive we keep, and any others we discard. Continue until we have 15 useful numbers. The cars with these numbers will be used for the special check.
Two possible extensions might be to use this method to take a survey or to do a simulation problem. To take a survey of the student body requires that several decisions be made. One decision is what question or questions do you want to ask? Since this unit deals with numerical values, you’ll want numerical data back so you can evaluate it using the techniques from Section I. Possible questions could be “How much soda do you drink in a week?” or “What do you expect your annual income to be ten years from now?”
Another decision is how large a sample do you want? What is an adequate size? Too small and it may not be valid. Too large a sample may be too much work to do. Thirty seems to be a good size with which to work. Once you have the size of your sample, how will you go about getting a random sample, gathering the data, analyzing the results? Can you publish the results in the school newspaper?
The second extension could be this simulation problem from
Understandable
Statistics
, Brase/Brase, p13.
A single pollen grain floating on the surface of water will move randomly from the impact of the water molecules. The task is to chart the course of a pollen grain as it moves on a drop of water for seven position changes. A problem, however, is that the pollen grain is so small and its movements are so fast that you would need to use a microscope and slow motion camera to see the changes. Since you do not have this equipment, you will have to use a random number table to simulate the observed direction of the pollen grain for seven position changes.
Instructions. Allow that for each position change, the pollen grain is in the center of a circle marked in degrees as shown below. 0 degrees indicates east, 90 degrees indicates north, 180 degrees indicates west, and 270 degrees indicates south. The arrow points to the direction of change.
(figure available in print form)
Solution. Using a random number table, arbitrarily decide where to begin and in which direction you will read. Then, since there are 359 possible positions, begin reading in groups of three digits. Keep the numbers that are between 000 and 359 inclusive and discard those that are not. When you have seven such numbers, chart the position changes according to the instructions above. A possible looking solution might be as follows.
(figure available in print form)
