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Random Number Tables
A table of random digits is a long string of the digits 0,1,2,3,4,5,6,7,8,9 with these two
properties:
1. Each entry in the table is equally likely to be any of the above 10 digits.
2. The entries are independent of each other. That is, knowledge of one part
of the table gives no information about any other part.
You can think of the table as being the result of the digits 0 to 9 being placed in a hat,
then drawn out one at a time, replaced and then a second draw taken etc.
In a random number table each entry is equally likely to be any of the 10 possibilities.
Each pair of entries is equally likely to be any of the 100 possible pairs: 00, 01.....99.
Each triple of entries is equally likely to be any of 1000 possible triples: 000, 001..999.
Each quartet of entries is equally likely to be any of 10000 possible quartets:
0000...9999 etc.
Most random number tables group the digits in groups of 5 just to make the table easier
to read. Which row you choose to take your random numbers from is completely
immaterial.
How to use a random number table for experimental randomization is illustrated in the
following example.
In a nutrition experiment 30 rats are to randomly divided into 2 groups of 15 each.
Two digits are needed to label each of 30 rats, so we use labels 01, 02 ,03 ......29, 30.
Next we choose a line in a random number table. If the line starts as:
69051 64817
87174 09517 84534 06489 87201
97245
The first 15 two-digit groups in this part of the line are
69 05 16 48
17
87
17
40 95 17 84
53 40 64 89
All numbers above 30 must be discarded and all repeats need to be discarded, so from
this first 15 we have only 05, 16, 17. If we continue on this line we get the following
pairs:
87 20 19 72 45
From these we take 20 and 19, so we must continue onto other lines until we get 10 more
numbers between 01 and 30 inclusive.
These rats will form the experimental group and the other 15 will be the control group.
In Math 11, chapter 5 we use several functions that are based on the random number
table. These functions simulate statistical experiments such as : coin tosses, game
spinners, dice rolls etc.
Part 2a STATISTICAL SIMULATIONS
The TI-83 calculator has a random number menu which identifies the different random
functions it can perform.
> > PRB
MATH
1: rand(
ENTER
This is a “random-number generator” for a number between 0
and 1 or you could have it pick a number in any range by entering
x (mult. symbol) and range number. Numbers can be decimals.
For example to pick a number between 0 and 10 you would enter
x
10
ENTER
To produce the 15 “random Integers” needed in the rat experiment you would choose
5: randInt( ENTER
and the cursor will prompt you for information about( lowest
number, highest number , how many). We would enter
ENTER
01
30
,
15
,
This will give us a list of 15 random numbers between 1 and 30.
If, for example we got 3 repeats you could try again but ask for 3 numbers instead of 15.
Anytime you wish to save a list for future use press
number; to save in L1 you would enter
STO>
You can see your saved list by pressing
STAT
STO>
2nd
d
and
and then the list
1
ENTER
Most of the work in chp. 5 deals with “simulating binomial experiments.” To do this
you would choose;
7:randBin( ENTER
and the cursor will prompt you for information about( number of
trials or # of items, probability, number of simulations or
experiments)
If we wished to toss 1 coin 12 times and see how many times a
head would appear we could consider 2 ways of doing this;
1st (1, .5, 12) as
1
.5
,
12
,
1
This will give us a string of 12 numbers, either 1 for a “head” or 0
for “no head”. We would need to count how many “1”s to find
the total # of heads. This could be a simulation for asking 1
question to 12 different people to which the answer is yes or No
2nd (12, .5, 1) as 12
,
.5
,
1
This will give us one number for an answer, which represents
the total number of heads that showed up in our one toss of 12
coins. This is the more efficient way of doing the simulation.
If we wanted to simulate repeating an experiment 50 times, then all that has to be
changed is the last number, from 1 to 50. Our entry would look like this;
ENTER
7:randBin(
12
then
,
.5
,
50
This will give you a string of 50
numbers, each number representing the number of “heads”, “yes answers” or whatever
you were simulating. This is where it is advisable to store and save the list:
STO>
2nd
1
ENTER
or instead of 1 , the # of the list that you wish to save to. If you wish to “sort” the list
press
choose 2:SortA( , press ENTER
STAT
The prompt will ask for the list, type in list #(always press
number)and then press ENTER
2nd
before the list
To prepare a “Histogram” of the results, students should review Gr. 10 technology:
2nd
ENTER
Y=
1
ENTER
This opens up the menu for statistical graphs.
This will make the first graph available
Cursor down and > > to choose the Histogram
Cursor down and if your data is not in list 1, type in the appropriate list #.
(Freq: 1) is what you want ! for last line.
Adjust the window to accommodate your range of data on the x and y-axis:
Now adjust the values : x-min is smallest data value
WINDOW
x-max is largest data value
ENTER
xscl will be 1 for our data
y-min is 0
y-max is a number that represents how many
pieces of data you expect in the tallest bar
yscl is a number that will accommodate your
y-max value. If for example, your y-max is 20 a
yscl of 1 would be fine, but if it is 200, then a
yscl of 10 or 20 will be better.
After you have adjusted the Window press
values on the Histogram.
GRAPH
and use
Trace
to read
Part 2b
BINOMIAL EXPERIMENTS
“Investigations 2 and 3” can be done using the TI-83 function (7:randBin. as in the
examples in 2a. For “Investigation 3” the probability number would be .25 instead of .5.
Before beginning “Investigation 4” have students clear any data in the lists:
STAT
Choose “4:ClrList”
highlighting the list # ,then press
ENTER
CLEAR
or clear each list one at a time by
ENTER
For “Investigation 4” have students enter in L1 the sample sizes: 12, 24, 36, 48
……120
In L2 have students enter in the number of “yes” for each sample size of L1.
Have students highlight L3 and type in
2nd
2

2nd
1
ENTER
This will enter in L3 all the “experimental probabilities” for the different sample sizes.
To create the line graph for these values; Press
2nd
Y= and choose the second
choice, the line graph.
The Xlist should be L1for the sample sizes and the Ylist should be L3 for the
probabilities. To see the graph press ZOOM
9
Students could draw on the graph the “theoretical probability” of .5 by going to the
menu and at y1= typing
and then
GRAPH
.5
Y=
For “Investigation 5” have students do a succession of 7:randBin( functions where the
first experiment would be :
10
,
.34
,
1
For successive experiments where more blocks are chosen each time the only thing to
change is the #10 to 20, 30,40 ………. 80 and record your answers. (Note it is not
necessary to enter the 1, as it is understood that if not stated ,the number of times
the experiment is being done is once) It may be a good idea for students to enter the 1,
just to keep them aware of what this number refers to.
Part 3
BINOMIAL DISTRIBUTION
In “Investigations 6, 7 and 8” students will also use a random number function that will
give the theoretical probabilities for different “Binomial distributions” It is labeled:
binompdf( This function will prompt for; #of items in the experiment, probability of
getting “yes”, and in the larger bracket the ways “yes” could be distributed.
Example: Toss 4 coins or (1 coin 4 times) and see the theoretical probabilities of how
getting a head(yes) will be distributed. If 4coins are tossed at once the possible results
could be ;
0 heads, 1 head, 2 heads, 3 heads or 4 heads therefore {0,1,2,3,4} are the ways the #
of possible heads could be distributed
To access this function;
and from this menu find position(after # 9)
2nd
VARS
0: binompdf( , now press
ENTER
For “investigation 6” respond to the prompt; with ( 4, .5,{0,1,2,3,4} The answer will be
the “theoretical probabilities” of getting the different # of heads for a 4 coin toss given
in decimal form. The answer can be stored in a list in the usual way.( these answers
should be the same as those obtained from using a tree diagram or Pascal’s Triangle)
For the “experimental” part of these investigations, in “Investigation 6 and 7” use
7:randBin(4, .5, 20 ) and store in desired list(s)
In “Investigations 6” students could store the “experimental” probabilities from a
sample space of 20 in a list and the possibilities {0,1,2,3,4} in another list . From these 2
lists they can draw the appropriate Histograms. The Xlist: will be the possibilities and
the Freq: will be the list containing the 5 “experimental” probabilities
In “Investigation 7” save the class averages for each of the “experimental”
probabilities in a list and the possibilities {0,1,2,3,4} in another list. As above from these
2 lists they can draw the appropriate Histograms. The Xlist: will be the possibilities and
the Freq: will be the list the 5 “experimental” probabilities are in.
In “Investigation 8” the procedures are the same as for 6 and 7 with the 4 being
replaced by 14 and the possibilities are {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14} in the
“theoretical probability” function ; 0: binompdf(
In the “experimental” part ( 7:randBin )of the experiment 4 is replaced by 14 and 20 by
50 and then by 500.
Part 4 90% CONFIDENCE INTERVALS
In this part students will simulate large numbers of experiments with different sample
sizes to see the range of possible “yes” answers that will belong in a 90% confidence
level.
“Investigations 9, 10, 11 and 12” all deal with collecting data from a large number of
experiments but changing the sample sizes and the probability ratio.
In each case 100 experiments are to be done but in “Investigation 9 and 10” the sample
sizes are 20, in “Investigation 11” the sample sizes are 40 and in “Investigation 12” the
sample sizes are 100.
Students can see how the “90% box plots” are made by simulating 100 experiments for a
given sample size and a specific probability ratio and then making the “90% box plot”
Example:
To create a “90% box plot” for a sample size of 20 and a probability
ratio of 70%(The probability ratio is often referred to as the “Population
Proportion”)
Simulate the 100 experiments by using 7:randBin and at the prompt enter
(20, .70, 100 ) Store the 100 numbers that will be generated in a list. Sort
the list. You want the middle 90 numbers for the “box plot” so the
beginning 5 numbers make the first whisker and the last 5 numbers make
the last whisker.
The box begins at the 6th number and ends on the 95th.
Students should display the position of the “box plot” in reference to a
number line from 0 to 20 on the bottom and the point count from 0 to1
along the top.
If each student creates a “90% box plot” for a different “Population
Proportion” and puts it on chart paper, the finished result should look like
the textbook chart on page 195.
Similarly, the charts for sample size 40 and sample size 100 can be created. Students
should be aware that the “90% box plot” is made from 90% of the data,
So if they only did 60 experiments, the “Box Plot” would be the middle 54 numbers with
3 numbers on each end for the whiskers.
It is probably more important to have students practice interpreting and using the charts,
than creating them, as the procedure is the same and complete official charts exist in the
text book.
Part 5 Statistical Formulas and Tests
The first section of this part does not have any investigations or simulations.
It deals with the formula to determine the “margin of error” when the data has
a “Normal Distribution”
The last part that deals with the “Chi-Square” statistic has several investigations.
In “Investigation 13” students can create a sampling distribution of Chi-squares. Using
7:randBin( 50,.5, 100) create a list of numbers that represent the “observed” number of
times a “yes” appears when the probability is .5. Store this list in L1. The theoretical or
“expected” outcome is 25. (50 –“observed” numbers) would represent the outcomes for
getting a “no”. We need to use the numbers in L1 to produce a list of Chi-square
numbers and save them in L2
Based on the Chi-square formula; (observed – expected)2 /expected, the chi-square
values will be found by entering the fraction for the “yes” + the fraction for the “no”
Highlight L2 and enter (L1–25)2  25 + (50–L1–25)2  25 enter
Sort L2.
Create a 90% BoxPlot with the first 90 numbers and use the last 10 for the whisker.
The first number in the Box should be 0 and the last number 2 or larger. Theoretically it
should come near 2.7.
From this Box plot we conclude that if the chi-square value is 2.7 or less, the “null
hypothesis is probably correct 90% of the time.
In “Investigation 14” students are working with the Chi-square formula
The Chi-square statistic can be found using the TI-83 if the observed data is put in one
matrix and the corresponding expected data in another matrix.
After the matrices are complete Press
STAT and cursor over to
TESTS
And cursor down to C: X2 – Test On the menu that follows if you choose Calculate the
next window will give the Chi-square value, the probability that the null hypothesis is
true and the degree of freedom. Select Draw and the chi-square graph appears with the
calculated probability area shaded in.
Other formulas that might be useful:
In “Investigation 15” and ”Investigation 16”
Degree(s) of freedom; (r – 1)(c – 1), r = # of rows and c = # of columns when the data
is organized in a table form
Calculate expected value from data in table form; For each observed data, the
expected data is found by “multiplying” the column total the data is in by the row
total the data is in and “dividing” by the total of all the data.