The Texas Instruments
TI–73 Graphing Calculator
on/off To turn on, press ON
To turn off, press 2nd ON
Note: Large screen will hold 7 lines of text.
There are 4 zones of keys:
graphing keys
menu keys
advanced functions
scientific calculator
Most keys perform more than one function:
secondary function Æ
x-1
b ¨ primary function
c
† function (yellow on TI-73), press 2nd key and
secondary function To choose the secondary
notice the arrow in the cursor.
†
alpha letter To enter an alphabetical letter, go to the TEXT menu. Use
the arrows to select the text to insert, then press ENTER. When
done, arrow down to Done.
memory To check the memory or reset the calculator, press 2nd 0 .
If you are proctoring an exam and you have to make sure that the
students have not programmed formulas into the calculator, then
you have to clear the memory. Usually, this would be the only
reason you would ever do anything this drastic.
†
You would press 7 to select Reset and clear the memory.
contrast
Mode
To adjust the contrast:
To make lighter, press 2nd —
(repeat as needed)
To make darker, press 2nd ∆
(repeat as needed)
The MODE key will show the settings (current ones are highlighted):
Normal Sci
The way numbers will be written.
Float 0123456789 Float is floating decimal up to 10 digits, numbers
represent the fixed number of decimal places.
Degree Radian
How angles are measured.
A»b/c b/c
How fractions > 1 will be written.
Autosimp Mansimp Determines if fractions will automatically be reduced
to lowest terms, or if this has to be done manually.
Start out with Normal, Float, Degree, either A»b/c or b/c, and Mansimp.
To exit MODE menu, press CLEAR or 2nd QUIT to return to home screen.
Minus vs Negative There are two very similar keys that must be kept separate; the negative key
(–) and the minus key — . Try 3 + 4 – -2 ENTER . You should see the correct answer
9. Now try 9 * – 7 ENTER with the incorrect minus sign. Note the error message. If you
Schoaff, 2004
TI-73 Graphing Calculators, p. 1
press 2 the cursor will be placed at the error. Change the minus to a negative for the
correct answer.
Multiplication The multiplication symbol is not needed between coefficients and variables, such
as 5X, or between a factor and parentheses.
Try 5(73–29) ENTER .
Functions
Try 20
parentheses are used to enclose the argument of the function √(20)
Try 5-1
(Use the x-1 key.)
Exponents
†
Square roots and other functions are entered the way we write them.
Power functions look a little different. You can square numbers with the x2 key.
Press 1 7 x†2 and you’ll see 172.
To cube 17, you either enter 1 7 ^ 3 , and you’ll see 17^3, or enter 1 7 MATH 3
,
and you’ll see 173. Press ENTER for the answer.
For exponents > 3, such as 26, you use the ^ before the exponent. 26 is keyed in as 2 ^ 6
.
INT ÷
This operation does integer division. For example, 17 int÷ 3 gives an answer 5 r 2. By
contrast, 17 ÷ 3 gives 5.666666667. If you press the F ´ D key, then return, you will get
17
2
or 5 .
3
3
Fractions
To enter a fraction, enter the numerator, press the b/c key, then enter the
denominator. Then either press ENTER or press the right arrow to exit the fraction. You can
†
† also press the b/c key first, enter the numerator (which allows you to enter an operation such
as 5+3), then press the down arrow to enter the denominator, then exit the fraction.
†
If you chose Mansimp under the MODE menu, then a fraction such
12
as
will have an arrow next to it. To reduce the fraction, either
56
press the SIMP key, then enter a factor, then press enter. You can
also press the SIMP key without a factor, then press enter. The
fraction will be reduced by the smallest common factor. You will see Fac=2 and the resulting
6
fraction
with an arrow indicating that it can be further simplified.
28
The graphing calculators will also return answers in either decimal or
fraction form. For example, if you enter 5÷6 * 3÷4, then the answer
†will be in decimal form. Choose F´D and Enter to convert the
625
5 3
result to the fraction
. If you entered * , the result will be
1000
6 4
Schoaff, 2004
†
†
TI-73 Graphing Calculators, p. 2
the fraction
15
, which can be converted to a decimal by using the F´D key.
24
Using the calculator to find the Greatest Common Divisor (GCD) of 2 numbers:
18
example:†The GCD is used to reduce fractions, such as
:
24
1. Change the MODE to Mansimp.
18
2. Enter 18 b c 24 to display the fraction. Notice that when you press Enter, you get Ø
.
24
This means that the fraction can be reduced.
3. Press SIMP then Enter. This simplifies the fraction by dividing by the first integer that goes
into both numbers. You can see that the first common factor was 2, but that the fraction can
still be simplified.
4. Repeat this procedure, until the fraction is completely simplified.
18
3
= with common factors: 2·3
24
4
The product of the prime factors is the greatest common divisor. The GCD(18, 24) = 2·3 = 6.
You can also use the gcd function from the MATH menu. Choose gcd( then enter 18,24). The
result is 6.
Another way to illustrate this is using Venn diagrams. The prime factors of 18 are 2·3·3 and the
prime factors of 24 are 2·2·2·3. Use a Venn diagram where one set contains the factors of
18, one set contains the factors of 24, and the intersection contains the common factors.
18
24
3
2
3
2
3
Using the GCD to find the Least Common Multiple (LCM) of 2 numbers
example: The LCM is used most often when you want to find the least common denominator for
two fractions. To find the LCM of 70 and 112
1. Determine the GCD of both numbers.
Enter 70 b c 112 and determine common factors: 2 · 7 = 14. GCD(70, 112) = 14.
2. Calculate the product of both numbers.
70 · 112 = 7840
Schoaff, 2004
TI-73 Graphing Calculators, p. 3
3. Divide the product by the GCD. This is the LCM. LCM(m, n) =
m⋅n
gcd(m, n)
7840 divided by 14 = 560. LCM(70, 112) = 560.
You can also use the lcm function from the MATH menu. Choose lcm( then enter 70,112). The
result is 560.
Using the Venn diagram method:
70
112
5
2
7
2
2
2
The GCD is the product of the factors in the intersection. The LCM is the product of all the
numbers in the Venn diagram: 2·2·2·2·5·7 = 560.
Convert menu
Using the TI-73 calculator to convert measures.
example:
To convert 5 inches to centimeters, just follow these steps:
1.
Enter 5.
2.
3.
Press 2nd CONVERT then choose Length.
Select inch, then select cm, and now press Enter.
5 inches = 12.7 cm
To convert 70 inches to yard, feet, inches:
1.
Enter 70.
2.
3.
4.
Press 2nd CONVERT then choose Length.
Select inch, then select yard. Press Enter.
You get 70 inches = 1.944444444 yards.
70 inches – 1 yard (36 inches) = 34 inches.
5.
34, Press 2nd CONVERT then choose Length. Select inch, then select ft. Press
Enter.
You get 34 inches = 2.833333333 ft.
34 inches – 2 ft (24 inches) = 10 inches.
70 inches = 1 yard 2 ft 10 inches.
6.
7.
Schoaff, 2004
TI-73 Graphing Calculators, p. 4
Previous answer The last answer calculated is kept in a register ready to use. Try ÷ 2
ENTER. The previous answer will be divided by 2.
When you want to take a functional value of the previous answer, use the ANS key (press
2nd
(–)
). To find ANS , try 2nd x2 2nd (–) ) ENTER .
Math menu Press MATH to see the math menu. To choose
something from a menu, either enter the desired number or use
the down arrow to select and press enter. Leave the menu by
3
pressing CLEAR or 2nd QUIT . To compute 6 27 , enter 6,
then go to MATH , 4 is the cube root function, then enter 27.
Compute 5! (5 factorial) Factorial is in the probability submenu
(PRB). Press 5 MATH then use the right arrow to get to PRB.
Choose 5 for !.
Ê 5ˆ
, the combination of 5 items taken 2 at a time, enter
Ë 2¯
5, then go to the PRB submenu and use choice 4, nCr (combination of n items taken r at a
time). After nCr, enter 2. You should get an answer of 10.
To compute
Ê 7ˆ
What is Á ? You can use 2nd ENTRY (above the ENTER key) to get back your previous
Ë 3¯
entry. Now just edit to read 7 nCr 3.
Recursive calculations:
a. What is the value of $1000 invested at 6% interest compounded annually for 10 years?
To figure this out for 1 year, you would have to calculate the interest on $1000 and add it to the
$1000, or 1000 + .06*1000. This is the same as:
1000 * 1.06 ENTER
Balance after 1 year
Now you can repeat this calculation:
ANS * 1.06 ENTER
Balance after 2 years
now just press ENTER
Balance after 3 years
keep pressing enter until you get to 10 years.
b. Suppose that $1000 was invested for you when you were born at 6% interest compounded
monthly. How much would you have when you are 18 years old (college age)?
For this solution, realize that 6% is the annual interest, not the monthly interest. The amount of
.06
interest you would get monthly is
.
12
Ê .06 ˆ
1000Á1+
Amount after 1 month
˜
Ë 12 ¯
†
TI-73 Graphing Calculators, p. 5
Schoaff, 2004
†
Ê .06 ˆ
ANSÁ1+
Amount after 2 months
˜
Ë 12 ¯
ENTER
Amount after 3 months
and so on. You would have to repeat this calculation 12*18 times to get the final answer.
You might have noticed that you are just multiplying the previous answer by the same factor.
Since you†need to multiply by this factor 12*18 times, you can use the formula
Ê .06 ˆ(12*18)
1000Á1+
, which is keyed in as 1000(1+.06/12)^(12*18)
˜
Ë 12 ¯
c. How many years would it take to have $2000 if you invested $1000 at 6% interest compounded
quarterly? Recall and edit the last entry (press 2nd ENTER ); use trial and error to close in on
†
the answer.
Graphic solutions: How many years will it take for our $1000 to double if invested at 6%
compounded annually? This can be solved by graphing the function y = 1000*1.06x and
the line y = 2000. Now determine where these two graphs intersect.
Format
Go to 2n d ZOOM, set CoordOn, GridOff, AxesOn, LabelOff,
ExprOn.
Entering functions
First press 2nd Y#= to go to STAT PLOT. Press
4, ENTER to clear any plots. Press Y#= on the top row. If there are any functions
already there, go to each line and press Clear. To enter X in a function, use the key
With the cursor on the Y1 row, enter 1000 * 1.06^x ENTER .
With the cursor on Y2, enter 2000.
Range
x.
†
Press WINDOW and set the viewing window. Press ENTER after each entry.
Xmin = 0
Xmax = 20
Xscl = 2
Ymin = 1800
Ymax = 2200
Yscl = 100
(Note, DX is set automatically.)
Press GRAPH . First Y1 is graphed, then Y2.
Tracing Press TRACE . The values of x and y are displayed at the
bottom of the screen. Use the four arrow keys to move the
blinking cursor to the point of intersection. The left and right
arrows move along one graph, the up and down arrows switch
between graphs. The equation for the function you are on is
shown at the upper left.
Schoaff, 2004
TI-73 Graphing Calculators, p. 6
Zooming
Press ZOOM , then option 2. Move the cursor to the desired point for the center of
the zoomed-in window (in this case, the point of intersection), then press ENTER . You
now have a close-up view.
Another way to zoom in on the desired point is by pressing ZOOM
, then option 1. This draws a box around the section you want
expanded. Move the cursor to the left and above the desired area,
then press ENTER . Use the arrows to move the cursor below and
right of the area, then press ENTER . The box you drew is now
expanded to fill the screen.
Trace to find the point of intersection. How long will it take for your money to double?
Graph Style: You can change the style of the graph on the Y= menu by using the left arrow to
move to the left of Y1. The default is a solid line. Now press the ENTER key. You can
now see the menu choices: bold line, shade above, shade below, a bubble with a trail, a
bubble without a trail, and dotted line. Experiment.
Shading graphs To shade the region above one graph and below another, we must first define
and activate two functions. Press Y#= to set Y1 = X and Y2 = –3X + 1. Clear any other
functions if they were defined. Press ZOOM and select Standard. You should see the two
graphs drawn.
To shade the region above Y1 and below Y2,
on the Y= menu, use the left arrow to go to
the left of Y1 =. Press enter twice to shade
above the graph. Arrow down to the Y2 =
icon and change to shade below. Press
GRAPH . The crosshatched area is the desired region.
To get only the desired region, turn off the shading in Y=, then enter the following sequence
of keys:
DRAW
Select 5: Shade(
it needs two variables
Y-VARS
(2n d APPS, then 2: Y-Vars)
Select 1 for Y1
Enter a comma
(to the left of 4)
Y-VARS
Select 2: Y2
Press )
then ENTER
Schoaff, 2004
TI-73 Graphing Calculators, p. 7
Show graphically the region y ≤ x3 – 8x and y ≥ x +
1.
Enter the two functions as Y1 and Y2.
Graph them in the standard window [–10, 10] by [–10,
10]. You can set this window by pressing ZOOM 6.
Shade appropriately.
Sketch the graph or print out using the Graph Link.
Statistics To get to the Statistics Menu, press STAT . Notice the
submenus: Ls, OPS, MATH, and CALC. To edit the lists, press
LIST.
Clearing data Begin by clearing the data already in memory.
To clear a single list, press STAT, then OPS, then Select 3: ClrList.
Enter the lists to be cleared from the STAT menu, Ls.
To clear all lists, go to MEM (2n d 0) and select 6: ClrAllLists.
To clear a particular list, go to LIST, then go to the label at the top of the list. Press Enter to
select, then press Clear.
DO NOT GO TO THE MEM MENU AND DELETE LIST L1, L2, ETC. Use this
method only to delete lists you name yourself and do not want to keep.
Also clear any functions in the Y ! = menu for graphing.
To erase the functions, place the cursor to the right of each = sign, then press CLEAR .
To turn off a function, place the cursor on the equal sign and press enter. (If the = is not
black, the function will not graph.)
Press 2nd QUIT to return to the main menu.
You can also go to MEM and delete any lists, functions, programs, etc. there. WARNING:
If you delete the lists L1 , L2 , etc., then you won’t have an L1 , etc.
Entering data Pull up the LIST menu.
You will see six columns labeled L1, L2, etc.
Type in some paired data point(s) followed by ENTER . For
example, in L1 enter 2, 3, 4, 5 and in L2 enter 5, 4, 3, 2.
Graphing data Next go to the PLOT menu by pressing 2nd Y = .
Choose 1: Plot1. In Plot1, choose On, Type: scatterplot (1st
choice), Xlist: L1 , Ylist: L2 , Mark: +.
To graph the data, first set the range by choosing ZOOM and then
choose 7: ZoomStat.
Schoaff, 2004
TI-73 Graphing Calculators, p. 8
Curve fitting We have five options for fitting a curve to the data.
line: Select STAT, CALC
Choose LinReg This fits a linear function: y = ax + b
Press ENTER
Note the values of a, and b.
The default lists are L1 , L2 . If you are using any other lists, you need to specify them.
Press enter.
On the TI-73, from the catalog choose DiagnosticOn to get the r value. The
closer r is to 1 or –1, the better the fit. Press enter twice until you see Done
on the home screen. Now press 2n d ENTRY twice to get back to
LinReg(ax+b), press enter. You should see r=–1, which means the values
are inversely proportional and you have a perfect fit.
To graph the regression line to compare it to your data, press Y = and position yourself on
an empty line. Now choose V A R S , Statistics. Use the right arrow to go to EQ, then
choose RegEQ. This pastes the regression equation into Y1. Press GRAPH to see both the
points and the line. (The + symbol for the data allows you to distinguish the points.)
You can do it all in one step from the STAT, CALC menu.
Choose LinReg again, and then before pressing ENTER , enter L1 ,
L2 ,Y1 . This will calculate the values for a and b, and also enter the
resulting equation in Y1 . Actually, since L1 and L2 are the default
lists, you could merely enter LinReg Y1 . (To get L1 and L2 , go to
STAT Ls. To get Y1 , go to VARS, 2:Y-Vars, then 1:Y1 .) Now if
you go to Y=, you will see the regression equation was entered automatically.
Other statistical regression choices are Med-Med (ax+b), quadratic (ax2 + bx + c) and
exponential (a * bx). You can also choose Manual-Fit which allows you to select 2 points
on the screen using the arrow keys, and then joins them with a line.
Go back to STAT PLOT and turn off Plot1. Or, on the TI-73, this can be done from the Y= menu
by using the up arrow to any highlighted PLOT and pressing enter.
Function values This method can also be used to get Y-values for entered X-values.
Press Y ! = and enter a function you want to evaluate in Y2, for example.
Press 2nd QUIT to return to the main menu.
Now enter a value and store it in X.
To get the corresponding Y-value, pull up Y2 from the Y-Vars menu.
Table
The TI-73 will give a table of values for functions entered in the Y=
list. Go to TBLSET , which is 2n d WINDOW . Here you can set the
initial value to be displayed in the table and the increment. If you set
the Independent variable (x) to Ask, then when you go to TABLE
nothing will show. Enter an x value and the corresponding y value
will appear.
Schoaff, 2004
TI-73 Graphing Calculators, p. 9
Euclid’s method for finding the GCD
To find the GCD of 105 and 975 as follows:
1.
Use integer division to divide the larger number by the smaller number.
105) 975
2.
Now integer divide the previous divisor by the remainder.
30 )105
3.
= 9 r 30
= 3 r 15
Repeat until you have a remainder of zero.
15) 30
=2r0
4.
The last divisor will be the GCD.
GCD(105, 975) = 15.
The reasoning behind Euclid’s method is based on a sequence of equalities using the division
algorithm:
975 = 9 · 105 + 30
so any divisor of 975 and 105 must also divide 30
105 = 3 · 30 + 15
so any divisor of 105 and 30 must also divide 15
30 = 2 · 15
so 15 is the GCD.
Problem 1
Find the GCD of the following numbers using Euclid’s algorithm. Show all intermediary steps.
a.
GCD(220, 2924)
b.
GCD(14595, 10856)
Problem 2
You may have heard of this problem before. A very wealthy family offered their son the following
choices if he agreed to do his daily chores for an entire month. These were his choices:
Plan A: They would give him 1 penny the first day, 2 pennies the second day, 4 pennies the third
day, 8 pennies the fourth day, and continue doubling the previous day's allowance until the
end of the 31-day month.
Plan B: Pay him a fixed amount of $1,000,000.
Which offer should he accept: Plan A or B?
1.
To constantly double the previous amount, you can use the constant function of the calculator.
Press 2n d SET , choose Single and enter C1 = *2.
2.
3.
Press 2n d QUIT to return to the home screen. Enter 1 CONST and continue pressing CONST
as the counter counts the number of times you multiply by 2.
How much will be earned in cents on the seventh day?
How does the counter value n compare to the day number?
Schoaff, 2004
TI-73 Graphing Calculators, p. 10
4.
$1,000,000 = ___________________ pennies. Continue using the constant key to determine
the single day where Plan A will pay more than $1,000,000 in pennies. On what day will that
occur? Record the amount in pennies and in dollars.
Day
Pennies Paid
Cumulative Total
1
1
1
2
2
1+2=3
3
4
1+2+4=7
4
8
7+8=15
5
16
31
6
X
5.
6.
7.
Enter the above chart into lists as follows: Name a list DAY to represent the day number of
the 31-day month. To save time in entering the numbers 1-31, you can use the sequence
command. On the home screen, press 2n d STAT ’ OPS’ 7:seq( type in X+1,X,0,30,1)
STO˛L DAY . You should now have the numbers 1-31 in the list DAY.
To the right of DAY, name two new lists PPAID (pennies paid) and CUMTP (cumulative
total in pennies).
Using the table above, find a formula that would generate the pennies paid on day X. Enter it
into list PPAID.
Write a formula to express the cumulative total paid in pennies from day 1 to day X. Enter it
into list CUMTP.
To get a better idea of the shape of the graph, define Plot 1 as a scatterplot, Xlist: DAY, Ylist:
CUMTP. Enter y = 100000000 for Plan B. Determine where the two plans are equivalent.
Problem #3
The total girth of a package cannot exceed 108 inches. Girth is measured by taking the total
distance around a package and adding the height (2L + 2W + H). For a square-based package
with a girth of 108 and L = W = x, then h = 108 – 4x. The volume is H*L*W = (108 – 4x)x2.
For a cylindrical package, you measure the circumference and height. If the radius = x, the
circumference is 2πx, and the height = 108 – 2πx. The volume is H*Area of base = (108 –
2πx)πx2.
By graphing these functions on the same axes, we can see which container will have the greater
volume. Set Y1 = (108 – 4X)X2 and Y2 = (108 – 2πX)πX2. To enter X2, use the X x2 keys. To
enter π, use the 2nd F´D keys. Set your window for X to go from 0 to 28, Y from 0 to 16000.
Use the TRACE key to determine the container with the greatest volume.
†
Schoaff, 2004
TI-73 Graphing Calculators, p. 11
a. For what interval of X does the square-based package have a greater volume? (Graph of Y1
above Y2 ?)
b. For what interval of X does the cylindrical package have a greater volume? (Graph of Y2 above
graph of Y1 ?)
c. Which shape has the greatest volume and for what value of X does this occur? (What is the
greatest y value shown?)
Problem #4
A hazardous waste protection agency tested a square tract
of land, as depicted by the grid to the right. They
detected dangerous substances in the region below a line
that crossed the vertical axis at 1 and had a slope of 2, and
below a line that crossed the vertical axis at –2 and had a
slope of –3. Show this region on the adjacent grid.
That is, show graphically the region defined by y < 2x
+ 1 and y < –3x – 2.
Enter the two functions as Y1 and Y2, shading
accordingly.
Graph them in the standard window.
Give the coordinates of a point inside the contaminated
region.
Give the coordinates of a point on the boundary of the contaminated region.
Give the coordinates of a point in the uncontaminated region.
Simulations with Graphing Calculators
These exercises can be done using software or graphing calculators. Warning, both calculators and
software have a default value of the seed for the random command. This means they will all
generate the same values. To avoid this on calculators, first store an individualized integer into rand.
For example, you can store your social security number into rand and then each time rand is
accessed a unique sequence is generated. This only needs to be done when the calculator is new or
when it has been reset. (i.e., 123456789Ærand) For software, generate some number of random
numbers before you began the simulation. This has to be done each time the computer is started.
1.
To simulate tossing a coin 6 times: First decide how to represent
heads and tails; for example, let 1 = heads and 0 = tails. In this case
we want 6 random integers, 0 or 1. The format is: randInt(0,1,6)
sample result Æ {0 1 1 0 0 0} or 2 heads and 4 tails.
randInt( is in the MATH menu, Æ to PRB. The comma is left of 4.
The TI-73 also has coin( in the MATH menu, Æ to PRB.
coin(6) will give the same result.
2.
To simulate throwing a die 6 times, you can use randInt(1,6,6)
Schoaff, 2004
TI-73 Graphing Calculators, p. 12
sample result Æ {4 4 3 6 2 6}.
The TI-73 also has dice(in the MATH menu, Æ to PRB. dice(6) will give the same result.
3.
To simulate throwing two dice 6 times, first decide how to do this. If
you use randInt(2,12,6) then each of the sums is equally probable.
But this is not the case when you throw two dice because certain
numbers are more likely to come up. For example, 1+11 = 11+1 =
12, so Pr(12) = 2/36; but 1+6 = 2+5 = 3+4 = 4+3 = 5+2 = 6+1 = 7,
so Pr(7) = 6/36. The correct function would be randInt(1,6,6) +
randInt(1,6,6)
sample result Æ {9 6 6 7 7 5}
On the TI-73 you can also use dice(6) + dice(6) or dice(6, 2) will give the same result.
Problem 5
a. How would you simulate picking a marble from a jar containing two red marbles, one blue
marble, and one green marble? How many random integers would you use? How do you
make the probability of picking a red marble more likely than picking a blue or green
marble?
b. What changes would you make if you were picking two marbles out of the jar
simultaneously?
Problem 6
Suppose the school cafeteria gives a coupon with each soft drink purchase. The coupon has one of
the letters W, I, N. When you buy a soft drink you are equally likely to get each letter. When you
have all three letters you win a free drink. Model this situation and study the variation in the
number of drinks that will be needed. In particular, what is the average number of drinks you must
buy to win a free drink? How much variation is there from the average?
a. Describe how you can model the buying of one drink.
b. Describe how you can model one trial of the experiment. That is, model buying drinks
until you have won a free one.
c. Run fifty trials of the experiment and record the 50 values for the variable.
d. Make a histogram for the fifty values generated above. What are the shape, center, and
variation?
e. Complete the following statement based on your data: To get a free drink, the average
number of drinks needed is _________, and 90% of the time it will take between ______
and ________ drinks.
f. Instead of the letters being equally likely, make one of them less likely. For example,
make the N occur with the probability 0.2 or 0.1. Explain how you would do this.
Problem 7
Suppose that in a large city there are 20% who are African-American, 10% Hispanic, and 70%
Others. Five people are randomly selected from the city to be on a planning committee. What is
the probability that all three groups are represented in the sample? Estimate the probability by
simulation. The following steps will guide you.
a. Explain how you can model the selection of one person in the city.
b. Explain how you can model one trial of the experiment.
c. What event is of interest in the problem?
Schoaff, 2004
TI-73 Graphing Calculators, p. 13
d. Do 50 trials of the experiment. Show the random numbers, how you interpret them, and
the conclusion about the event of interest in each trial.
e. Based on your trials, what is the probability estimate?
Problem 8
Random Numbers & the Birthday Problem: In a group of people, how likely is it that two people
have the same birthday? How does the size of the group of people affect the answer? If you ignore
leap years, you can represent each possible birthday by a random number
from 1 to 365.
a. First, simulate the birthdays of 50 people. Now this is very
difficult to analyze because the list runs off the page. You can
make your task easier by storing the list in L1. Now you can sort
your list in L1, then just scroll down the list to determine how
many duplications occur.
randInt(1,365,50) STO 2nd STAT L1, Enter. Then 2nd STAT
ÆOPS, SortA(2nd STAT L1), Enter. Then to see the list, LIST .
Did any two people have the same birthday? (I had a pair of 86's.)
Describe your results.
b. Another method for generating birthdays on the graphing calculator was published in the
Mathematics Teacher. This author has students generate random numbers from 1 to 12
and store them in L1. Then students generate random numbers from 1 to 31 and store
them in L2. Then SortA(L1, L2) will sort both lists together according to L1 (L2 will not
be sorted). Students then go through the lists to find pairs while
disregarding all impossible dates, such as
2/30, etc.
Use this procedure to create and sort a second
set of 50 "birthdays." (I had three February
3's.) Compare these results to your results in
part (a) above.
Create and sort a third set of 50 "birthdays" using either method.
Of the 3 sets, how many contain at least one pair with the same birthday?
c. Do two or three simulations each for birthdays of the following numbers of people.
Describe the results.
40 people
30 people
20 people
10 people
d. Based on your simulations above, what is the smallest size group for which you feel the
probability is 50% or greater that at least two people share the same birthday? Justify
your answer.
Problem 9
Simulating an epidemic: To simulate this in the classroom, assign a number to each person and
have everyone stand up. For demonstration purposes, we shall use n = 10 people. The larger the
number, the better the statistics generated. We want to generate random integers from 1 to n, count
how many people are infected after each time period (depending on the disease, this could be 1 day,
Schoaff, 2004
TI-73 Graphing Calculators, p. 14
1 week, 1 month, etc.), and record this number in a table (see below). For our demonstration, we
will use randInt(1,10). To start press Enter once to get the first random integer. result Æ 9
Person #9 is now infected and should sit down. In row #1 of the table, enter 1.
Press Enter once. result Æ 5 Person #5 is now infected and should sit down. In row #2 of
the table, enter 2 for the number of people infected.
Press Enter twice because each of these infected people could infect two additional people.
result Æ 2 5 Person #2 is now infected and should sit down. Person #5 was already
infected. In row #3 of the table, enter 3 for number infected.
Press Enter three times because we have 3 infected people. result Æ 10 5 9 Person #10 is
now infected and should sit down. . Persons #9 and #5 were already infected. In row #4 of
the table, enter 4 for number infected.
Press Enter four times because we have 4 infected people. result Æ 2 4 5 8 Persons #4
and #8 are now infected and should sit down. . Persons #2 and #5 were already infected. In
row #5 of the table, enter 6 for number infected.
Press Enter six times because we have 6 infected people. result Æ 6 7 2 4 5 3 Persons
#6, #7 and #3 are now infected and should sit down. . Persons #2, #4 and #5 were already
infected. In row #6 of the table, enter 9 for number infected.
Press Enter nine times because we have 9 infected people. result Æ 4 9 10 4 6 2 10 4
5 All of these people were already infected. In row #7 of the table, enter 9 for number
infected.
Repeat for row 8. result Æ 9 9 1 We can stop there because #1 was the last person left
standing and is now infected. In row #8, enter 10 for the number infected.
time
1
2
3
4
5
6
7
8
random #
9
5
2, 5
10, 5, 9
2, 4, 5, 8
6,7,2,4,5,3
4,9,10,4,6,2,10,4,5
9, 9, 1....
number infected
1
2
3
4
6
9
9
10
With a different size group, you would need more or fewer time periods. Having students sit
down makes it easier to count how many are infected. You could also list all the number and
cross them off as people are infected. Typically the number newly infected starts slowly, then
increases rapidly, and then slows down again as most of the population is infected.
To analyze this data, enter the time in L1 in the calculator. Enter the number infected in L2.
Go to 2nd STAT PLOT , turn off all plots except for Plot 1. Turn Plot 1 On. Arrow down to
Type and choose the first plot, a scatter plot, press Enter. For Xlist, on the TI-73 press 2nd
STAT and choose L1 , Enter. For Ylist, follow the above directions to get L2 . Choose the +
mark.
Now press Y= to clear out any equations there. Press ZOOM , then ZoomStat to set the
window. For our small sample, the data looks sort of linear. (You can experiment with other
regression equations to see what fits best.) For a linear regression, go to STAT , CALC,
LinReg(ax+b). L1 and L2 are the default, so we don't have to enter any lists. We do want the
Schoaff, 2004
TI-73 Graphing Calculators, p. 15
equation pasted into Y1 , so on the TI-73 go to 2nd VARS , Y-Vars, Y1 . Press Enter to choose
Y1 , then Enter to calculate the linear regression equation. The r-value is .98 for our limited
data, which indicates a good fit. Press GRAPH to see both the data and the equation.
Statistics with Graphing Calculators
Exploring the different graphs possible:
1.
Enter data:
Go to LIST . Arrow to the blank list, press
ENTER . Go to 2nd TEXT , then use the
arrows to enter ST for Student, then
Done, then Enter twice.
Student
Angella
Bob
Carol
David
Jean
Lisa
Peter
Matthew
Verbal Score
502
590
505
560
487
585
590
480
Math Score
444
550
542
590
493
620
600
553
You can enter text into the ST list, or enter numbers. To enter the first
name as text, press 2nd TEXT , first use the arrow keys to enter a " mark,
then select the letters, then Ø to Done, then Enter. Now the calculator
identifies the rest of that list as text, so you do not need to start with a "
mark. I would suggest using only the first letter of each name. Otherwise,
just enter the numbers 1, 2, 3, 4, 5, 6, 7, 8 for the students.
2.
Similarly insert and label the column for Verbal Scores VS and the
column for Math Scores MS. Enter the SAT scores from the chart above.
Press 2nd QUIT to return to the home screen.
3.
Statistical computations:
Go to 2nd STAT CALC, 1-Var Stats, then 2nd STAT Ø to VS, Enter,
Enter.
These same computations are available individually under 2nd STAT MATH. then Ø to
mean( , stdDev( , median( , min( , max( . The TI-73 also finds the mode if there is one.
Record the statistics in the table below. Repeat for the Math Score.
Verbal Score
Math Score
Record the Mean:
Record the Standard Deviation:
Record the Median:
Record the Minimum:
Record the Maximum:
Calculate the Range:
Schoaff, 2004
TI-73 Graphing Calculators, p. 16
4.
Box Plots: First press Y= and clear out any functions there. Then go to 2nd PLOT select Plot
1, Enter. Press Enter to turn Plot 1 On, Ø to Type, then Æ to the box plot icon (also called
box and whiskers), Enter. Then Ø to Xlist.
Go to 2nd STAT Ø to VS, Enter.
Press ZOOM, then Ø to ZoomStat. You now see the box plot for
the verbal scores.
Repeat the above steps, using Plot 2, for the Math Scores. Go to
ZoomStat again to graph the two box plots together. (It has to
resize the window for the second plot.)
What statistics are used to create the box plots? Use TRACE to help determine this. Use the
↑Ø arrows to go between plots.
The "whiskers" are the long lines extending from each box. A point that is at an extreme,
where the distance between that point and the median is greater than 1.5 times the distance
between Q1 and Q3, is called an "outlier." Looking at the data, do you think that one of the
whiskers is exceptionally long due to an outlier? You might have noticed in the PLOT menu
there were two box plots. The one with a dot at the end of one whisker does not connect
outliers, but shows them as separate points.
5.
Calculations with the Data: Return to the statistics editor. (LIST). Make a new list called CS
for combined scores. Position the cursor on the new label and enter the following:
Go to 2nd STAT Ø to MS, Enter. Then type +, then go to 2nd STAT Ø to VS, Enter, Enter.
The CS column now has a list of the sums of each score. If you arrow up to the name of the
list, you will see {946, 1140, 1047, …}. If you change anyone's score in MS or VS, CS will
have to be recalculated because it contains a list, not the formula. To attach a formula to CS,
after CS= type a " quotation mark, then follow the above instructions to enter the formula.
When you press Enter, the same values appear in the column, but when you arrow up to the
name, you will see the formula instead of the list. Now if you change anyone's score in MS
or VS, the value in CS changes automatically.
What is Angella's (1) combined score? Who has the highest combined score?
6.
Histograms: Turn all the plots off by going to the PLOT menu and
choosing PlotsOff. Now go back to the PLOT menu and choose
Plot 1 again. Turn Plot 1 on, choose the histogram. First graph the
Verbal Scores, then the Math Scores, then the Combined Scores.
Remember to use ZoomStat to get a good window. If you change
the Xscl in Window, you will change the width of the columns.
(Note that you cannot graph more than one histogram at a time with the graphing calculator.)
How might a school guidance counselor use this graph?
One of the choices in the PLOT menu is a bar graph, which is different from a histogram
because it uses categories on the x-axis. The bar graph can be used with up to 7 categories
and up to 3 data lists. It can't be used for this particular set of data because we have 8
categories (names).
Schoaff, 2004
TI-73 Graphing Calculators, p. 17
7.
Sorting: To display the combined scores in ascending order, remember that we have to
include the other three lists too to keep the correct scores with the correct person.
Go to 2nd STAT, OPS. Choose SortA(, then go to 2nd STAT Ø to CS, Enter, comma, then
go to 2nd STAT Ø to ST, Enter, comma, then go to 2nd STAT Ø to VS, Enter, comma, then
go to 2nd STAT Ø to MS, Enter, ) Enter.
What are the benefits of sorting? How is this useful?
8.
Scatter Plot: A scatter plot can give information on whether the two scores are related. For
example, if the points seem to go from lower left to upper right, we consider the scores to have
a positive correlation. In other words, there is some relationship between getting high math
scores and high verbal scores. On the other hand, if the general trend is from upper left to
lower right, there is a negative correlation. This would suggest that students, who score high
in math, score low in verbal, and vice versa.
Do you believe there should be a positive, negative, or no correlation
between math and verbal SAT? We can test this conjecture with our
small sample.
Go to PLOT, turn off all other plots except for Plot 1. Type should
be scatter plot (dots), Xlist: VS, Ylist: MS, any Mark you prefer.
Press ZOOM, ZoomStat and look at the graph. Is there a correlation?
Does it make a difference if Xlist is MS and Ylist is VS? Switch them and regraph.
9.
Determine the regression line:
Go to 2nd STAT CALC, choose LinReg(ax+b), Enter. Now whatever is currently your
Xlist (2nd STAT Ø to MS), Enter, comma, whatever is currently your Ylist (2nd STAT Ø to
VS), Enter, comma, 2nd VARS, Y-Vars, Y1 , Enter, Enter.
On the home screen, you should see the calculated values for a and b, and the correlation
coefficient r. If you do not see a value for r and r2 , then do the following: Press 2nd
CATALOG (above PRGM on TI-73). Arrow down to DiagnosticOn, press Enter. Then press
2nd ENTRY twice to get back to the LinReg command, and press Enter.
The closer r is to +1 or –1, the better the fit between the function and the data. The r-value for
this function tells you that a line is not a particularly good match for this data. Now press
GRAPH to see the regression line graphed with your data. Because the regression line has a
positive slope, you can see that your data has a positive correlation.
Problem #10
Pie Charts
A pie chart, or circle graph, represents parts of one whole. Before making a pie chart, make sure
that you know what the whole circle represents. You should be able to state that the entire circle is
100% of ____________. This is the biggest mistake that students make in using a pie chart — the
sum of the segments is not one whole thing. For example, it would be wrong to use a pie chart to
represent the population of the US over several 10-year intervals from 1900 to 1990, since one
segment represents 100% of the population in that year. One person may be represented in several
segments if they were alive in 1950, 1960, 1970, 1980, and 1990.
(NOTE: Most people use pie charts incorrectly. If in doubt, don't!!)
Use the following data on the types of periodicals published in the U.S. in 1980 and 1990.
(Source: Statistical Abstract of the United States, 1991)
Schoaff, 2004
TI-73 Graphing Calculators, p. 18
Periodicals
Weekly
Semi-monthly
Monthly
Bimonthly
Quarterly
Other
1980
10236
1716
645
3985
1114
1444
1332
1990
11092
553
435
4239
2087
2758
1020
1.
Clear L1 and L2 . Do this by going to LIST, then ↑
to the label, press CLEAR, then Enter.
In L1 enter the labels. Position your cursor in the
first data position, then 2nd TEXT. First enter a "
quotation mark, then W for Weekly, then Done,
Enter. Similarly enter "S for Semi-monthly, "M
for Monthly, "B for Bimonthly, "Q for Quarterly,
"O for Other. (The quotation mark will not show
up in the list.)
Why should the category Periodicals not be entered as data? Should the “Other” category
be included? Why or why not?
Now in L2 enter the numbers for 1980.
2.
To make a pie chart for the periodicals in 1980, go to 2nd PLOT. Turn off all plots except for
Plot 1. For Plot 1, Type is circle graph (2nd row, 1st column), CategList: L1 , Data List: L2 .
Use either Number or Percent.
Press Y= and clear all functions. Press Press ZOOM, ZoomStat. to set the window.
Also make a pie chart for the periodicals in 1990 using L3.
3.
You can now ask questions like the following:
a. Which type of periodical is represented by the largest portion of each pie chart?
b. Which type of periodical showed the largest decrease from 1980 to 1990?
c. Which two types of periodicals showed large increases from 1980 to 1990?
4.
You can also display this same data as a pictograph. From the PLOT menu, Type would be the
icon in row 1, column 3. CategList: L1 , Data List: L2 , Scale about 500, Choose vertical or
horizontal, then choose which icon you want. Press ZOOM, ZoomStat.
5.
Another way to display the data from both 1980 and 1990 simultaneously is with a bar graph.
From the PLOT menu, Type would be the icon in row 1, column 4. CategList: L1 , DataList1:
L2 , DataList2: L3 . Choose vertical or horizontal, then choose 2 because there are only 2 data
lists. (A default will appear in DataList3, ignore it.) Press ZOOM, ZoomStat. The bar graph
makes it easier to compare two values.
Schoaff, 2004
TI-73 Graphing Calculators, p. 19
Problem #11
The World's Fastest Men & Women
Is there a pattern to the times recorded for the
gold medallists in the men's 100-meter run
from past Olympics? If there is a pattern, can
you determine what the winning time in
Barcelona, 1992 might have been? Don't look
up the winning time yet! What about Atlanta
in 1996?
A scatterplot is a good way to see if there is a
pattern in the relationship between the year of
the Olympics and the gold medal winning
time. This scatterplot is easily created by hand,
with computer software, or with graphing
calculators. Before getting started with the
graphing calculator, check the MODE settings
and put them all on the default settings (the
ones on the left).
Entering data: First clear all lists by pressing
2nd MEM ClrAllLists. Now enter the data
( LIST on TI-73). Under L1, enter the year of
each entry and press ENTER after each one.
Then go to the top of the List L2 and enter
each time for each year in the same order.
When finished press 2nd QUIT to return to the
home screen.
Setting the window: Before drawing the
scatterplot, set the WINDOW because the
scatterplot is drawn on the same graphics
screen you have used to graph functions. Our
X data range from 1896 to 1988, so set Xmin
to 1890 and Xmax to 2000 (or similar figures)
and Xscl to 20. Because the winning times
range from 12 down to 9.99, set Ymin to 9 and
Ymax to 13 and Yscl to 0.5.
Men’s 100 Meter Run Gold Medallists
for the Modern Olympic Games
Year Name Time
(Sec.)
1896 Thomas Burke, United States
12.
1900 Francis W. Jarvis, United States 10.8
1904 Archie Hahn, United States
11.
1908 Reginald Walker, South Africa 10.8
1912 Ralph Craig, United States
10.8
1920 Charles Paddock, United States 10.8
1924 Harold Abrahams, Great Britain 10.6
1928 Percy Williams, Canada
10.8
1932 Eddie Tolan, United States
10.3
1936 Jessie Owens, United States
10.3
1948 Harrison Dillard, United States 10.3
1952 Lindy Remigino, United States 10.4
1956 Bobby Morrow, United States
10.5
1960 Armin Hary, Germany
10.2
1964 Bob Hayes, United States
10.0
1968 Jim Hines, United States
9.9
1972 Valeri Borzov, USSR
10.14
1976 Hasely Crawford, Trinidad
10.06
1980 Allan Wells, Great Britain
10.25
1984 Carl Lewis, United States
9.99
1988 Carl Lewis, United States
9 99
Drawing the scatterplot: First clear all functions out of the Y= screen. Next go to PLOT by
pressing 2nd , then Y= . At Plot 1, press ENTER . Under Plot 1, choose ON, then for TYPE choose
the first icon for scatterplot. Next choose L1 for Xlist and L2 for Ylist (unless you named them
something else).
then L1 , 2nd STAT then L2 .
In order to see your points better, choose the first or second icon under Mark. Then press the
GRAPH key.
2nd STAT
Finding the line of best fit: The TI-73 can determine very quickly the regression line for these data.
From the Home screen ( 2nd QUIT ), press STAT . Then under CALC, choose LinReg(ax +b). L1
and L2 are the default lists, so you do not have to enter anything unless you named the lists
Schoaff, 2004
TI-73 Graphing Calculators, p. 20
something else. To graph the line of best fit, we need to enter it into Y1 . To do this, while still on
the home screen, before pressing Enter after LinReg(ax+b), do the following:
Press 2nd VARS , choose Y-Vars, then choose Y1 .
Press ENTER and you will see values for a and b. Record these values below.
a=
b=
The value of b represents the y-intercept, a represents the slope of the regression line. Now press
GRAPH and you will have the scatterplot and the line of best fit.
With this mathematical model you can now make a number of predictions about Olympic times.
1.
What might have been a good winning time in Barcelona?
There are several ways to answer that question. One is to trace the function, get X as close to 1992
as you can, and read Y. Another is to store 1992 into X, go to VARS, Y-VARS, choose Y1 , and
press Enter. A third method is to use the TABLE menu.
2.
Now go to a reference book or the internet. Who won in Barcelona? What was the actual
winning time?
3.
What time might have been expected for the winner at the Atlanta Games in 1996?
4.
What was the actual winning time?
5.
Notice from the table on the previous page that there is no time listed for some years. Why
not?
Use your mathematical model to predict what the winning times might have been, had there
been Olympics in those years.
6.
Using the model, what should we have expected for Australia in 2000? What was the actual
winning time?
7.
What should we expect for the winning time in Greece in 2004?
Schoaff, 2004
TI-73 Graphing Calculators, p. 21
Problem #12 Fat Versus Calories
The following table contains the number of calories and grams of fat for selected fast foods. Does
the scatterplot of these variables for each item look linear? Is there a positive correlation between fat
and calories — the more fat, the higher the number of calories?
Fast food item
Grams of fat
Burger King Whopper ......................33
McDonald's Big Mac.........................34
Wendy's Big Classic…........................28
Arby's Roast Beef ............................19
Hardee's Roast Beef .........................17
Roy Roger's Roast Beef.....................11
Burger King Whaler .........................26
McDonald's Filet-O-Fish....................23
Arby's Chicken Breast Sandwich........32
Burger King Chicken Tenders............12
Church's Fried Chicken (2 pc.)...........35
Hardee's Chicken Filet Sandwich.......20
Kentuckv Fried Chicken (2 Pc.)..........31
Kentucky Fried Chicken Nuggets......17
McDonald's Chicken Nuggets...........18
Roy Roger's Chicken (2 pc.)...............35
Wendy's Chicken Fillet Sandwich .....24
Calories
.............584
.............572
.............500
.............365
.............338
.............335
.............478
.............415
.............567
.............223
.............487
.............431
.............460
.............281
.............286
.............519
.............479
To Graph, remember to do the following:
•
Clear all graphs in the Y= screen and turn off all other PLOTS.
•
Enter the data as sets of ordered pairs with x as the number of grams of fat and y as the
number of calories.
•
Select an appropriate viewing window, such as [10,40] by [200,600]. Choose Xscl and Yscl
so you don't have more than 20 hash marks on each.
•
Choose LinReg(ax +b) and put your equation in Y1 or Y2.
1.
The standard form of the regression line uses a as slope and b as the y-intercept, so an
approximate equation of a line through the data is:
y=
2.
What kind of correlation is there between fat and calories — positive, negative, or neither?
3.
If you found out that a particular meal at Denny's had 575 calories, what would you expect the
grams of fat to be?
4.
If you found out that a White Castle hamburger contained 15 grams of fat, what would you
expect the calories to be?
Schoaff, 2004
TI-73 Graphing Calculators, p. 22
Would You Eat The Potato Salad?
A single bacterium can find its way to a scoop potato salad on a family picnic. This bacterium
doubles every minute. One million bacteria are enough to make you sick. Would you eat the
potato salad 30 minutes later?
1. Use M&Ms or beads or beans to represent the bacteria and complete the following table. See
how many bacteria there are each minute up to 7 minutes.
TIME
0 minutes
1
M&MS
EXPONENTIAL
FORM
PATTERN
1
-Do you see a pattern to describe how the number of bacteria grows? Use the third column to
show the pattern.
-Describe this pattern using exponents. (Use fourth column)
2. What is the function that describes this pattern?
3.
Put this function in the graphing calculator in y=. Look at the graph.
-
Describe the appearance of the graph.
-
Describe how it changes over time.
4. Use the table created by the graph to find out how many bacteria there will be in 30 minutes.
5. How would the function change if the bacteria tripled every minute?
6. How would the function change if we started with 4 bacteria?
Schoaff, 2004
TI-73 Graphing Calculators, p. 23