Get documents about "Basic Operations Correlation Coefficient
Shared by: derong123
-
Stats
- views:
- 9
- posted:
- 2/1/2010
- language:
- English
- pages:
- 14
Document Sample
Appendix GC GC-1
Graphing Calculator Appendix
This appendix contains some keystroke suggestions for many graphing calcu-
lator operations that are featured in this text. The keystrokes are for the TI-83/
TI-83 Plus calculators. The descriptions in the margin are the same as those
used in the text and are arranged alphabetically. Please see your manual for
additional information about your calculator.
Basic Operations Numerical calculations are performed on the home screen. You can always re-
turn to the home screen by pressing 2nd QUIT. Pressing CLEAR erases
the home screen.
To evaluate the expression –2(3 + 5) – 8 ÷ 4, use the
following keystrokes.
(–) 2 ( 3 + 5 ) – 8
÷ 4 ENTER
Note: There is a difference between the key to en-
ter a negative number, (–) , and the key for
subtraction, – . You cannot use these keys
interchangeably.
The 2nd key is used to access the commands
written in gold above a key. For instance, to evaluate
49 , press 2nd 49 ) ENTER .
The ALPHA key is used to place a letter on the
screen. One reason to do this is to store a value of a
variable. The following keystrokes give A the value
of 5.
5 STO ALPHA A ENTER
This value is now available in calculations. For in-
stance, we can find the value of 3a2 by using the fol-
lowing keystrokes: 3 ALPHA A x2 . To
display the value of the variable on the screen, press
2nd RCL ALPHA A.
Note: When using the ALPHA key, only capital letters are available on TI-
83 calculator.
Correlation The value of the correlation coefficient for a regression equation calculation is
Coefficient not shown unless the DiagnosticOn feature is enabled. To enable this fea-
ture, press 2nd [catalog] D (scroll to DiagnosticOn) ENTER
ENTER
The correlation coefficient will appear on the screen along with a regression
equation.
GC-2 Graphing Calculator Appendix
Evaluating Functions There are various methods of evaluating a function
but all methods require that the expression be en-
tered as one of the functions Y1 to Y7. To evaluate
2
x
f ( x ) = ----------- when x = –3, enter the expression into,
-
x–1
for instance, Y1 and then press VARS 11
( (–) 3 ) ENTER .
Note: If you try to evaluate a function at a number
that is not in the domain of the function, you will
get an error message. For instance, 1 is not in the
2
x
domain of f ( x ) = ----------- . If we try to evaluate the
-
x–1
function at 1, the error screen at the right appears.
Evaluating logarithms Logarithms base 10 can be found using LOG .
For instance, to find the value of 19 log ( 2 ) + 1 , press
19 LOG 2 ) + 1 ENTER .
Natural logarithms can be found using LN . For
instance, to find 3 ln ( 4 ) – 2 , press 3 LN 4
) – 2 ENTER .
Evaluating Variable To evaluate a variable expression, first store the val-
Expressions ues of each variable. Then enter the variable expres-
2
sion on the home screen. To evaluate s + 2sl when
s = 4 and l = 5, use the following keystrokes.
4 STO ALPHA S ENTER 5 STO ALPHA L
ENTER ALPHA S x2 + 2 ALPHA S
ALPHA L ENTER
Financial Financial calculations on a TI-83 Plus are accessed by pressing APPS and se-
Calculations lecting Finance. On a TI-83, press 2nd FINANCE. The process for all
financial calculations are the same: Select the finance functions, enter the val-
ues of the known variables, enter 0 for all other values, use the arrow keys to
select the unknown variable, press ALPHA SOLVE. Here are some examples.
Compound Interest Future Value Suppose an investment of $10,000 is made
at an interest rate of 6.25% compounded daily. To find the value of the invest-
ment in 5 years, access the finance functions (see above). Select TMV Solver
(TMV is an abbreviation for time value of money), press ENTER . Use the up
and down arrow keys to select the variables. For this problem, N = 5, I% = 6.25,
PV = -10000, PMT = 0, FV = 0, P/Y = 1, and C/Y = 365. Now move the cursor
to FV and press ALPHA SOLVE. The little square next to FV indicates that
value was calculated.
When using the finance
functions of a TI-83 calculator,
payments are entered as
negative numbers, as we did for
PV in the calculator screen at
the left.
The value of the investment in 5 years will be $13,668.01.
Appendix GC GC-3
Compound Interest Present Value An investor wants to have $10,000 in 4
years. To find how much the investor must place in an account today that
earns 8% interest compounded quarterly, select TMV Solver and press
ENTER . Use the up and down arrow keys to enter values for the variables.
For this problem, N = 4, I% = 8, PV = 0, PMT = 0, FV = 10000, P/Y = 1, and
C/Y = 4 . Now move the cursor to PV and press ALPHA SOLVE. The little
square next to PV indicates that value was calculated
• Note that the result is
a negative number. This
Take Note is the amount that must
The meaning of the variables in be deposited (paid) into
a financial calculation are: the account.
N: For an annuity, the
number of payments; for The investor must place $7261.74 in the account.
compound interest, the
number of years Monthly Car or Mortgage Payment A cabinet maker finances $12,500 for a car
I%: Annual interest rate
at an annual interest rate of 8.3% compounded monthly for 5 years. To find the
PV: Present value
FV: Future value monthly payment, select TMV Solver and press ENTER . Use the up and
P/Y: Number of payments down arrow keys to enter values for the variables. For this problem, N = 60
per year. For compound ( 5 ⋅ 12 ), I% = 8.3, PV = -12500, PMT = 0, FV = 0, P/Y = 12, and C/Y = 12. Now
interest, P/Y = 1
C/Y: Number of
move the cursor to PMT and press ALPHA SOLVE. The little square next to
compounding periods PMT indicates that value was calculated.
per year
PMT: END BEGIN - Select
END if payments are made at
the end of a period; select
BEGIN if payments are made at
the beginning of a period
The monthly payment is $255.25.
To calculate a monthly mortgage payment, follow the same steps as above.
The present value is the amount of the mortgage.
Calculate APR Suppose a management intern purchases an MP3 player for
$250 and finances the purchase at an 8% simple interest rate for 12 months. The
monthly payment is $22.50. To find the APR, select TMV Solver and press
ENTER . Use the up and down arrow keys to enter values for the variables.
For this problem, N = 12, I% = 0 (this is the APR, not the simple interest rate),
PV = 250, PMT = – 22.5 , FV = 0, P/Y = 12, and C/Y = 12. Now move the cur-
sor to I% and press ALPHA SOLVE. The little square next to I% indicates that
value was calculated.
The APR is 14.45%.
GC-4 Graphing Calculator Appendix
Calculating Mortgage or Loan Payoff An electrician has a 30-year mortgage
at an annual interest rate of 6.5% and makes monthly payments of $1580.17. To
find the mortgage payoff after making payments for 5 years (60 months), select
TMV Solver and press ENTER . Use the up and down arrow keys to enter
values for the variables. For this problem, N = 300 (360 – 60), I% = 6.5%,
PV = 0 , PMT = – 1580.17 , FV = 0, P/Y = 12, and C/Y = 12. Now move the
cursor to PV and press ALPHA SOLVE. The little square next to PV indicates
that value was calculated.
The mortgage payoff is $234,027.43.
Graph To graph a function, use the Y= key to enter the expression for the func-
tion, select a suitable viewing window, and then press GRAPH . For instance,
3
to graph f ( x ) = 0.1x – 2x – 1 in the standard viewing window, use the fol-
lowing keystrokes.
Y= . Θ
1 X,T,Θ,n ^ 3 – 2 Θ
X,T,Θ,n – 1
ZOOM (scroll to 6) ENTER
10
–10 10
–10
Note: For the keystrokes above, you do not have to scroll to 6. Alternatively,
use ZOOM 6. This will select the standard viewing window and automat-
ically start the graph. Use the WINDOW key to create a custom window for a
graph.
x
Some special functions such as e , ln ( x ) , and log ( x ) can be graphed by using
– 2x
the keys for these functions. For instance, to graph f ( x ) = 0.25e – 4 , press
Y= . 25 2nd ex (–) 2 Θ
X,T,Θ,n ) ZOOM (scroll
to 6) ENTER
10
–10 10
–10
Appendix GC GC-5
Intersect The INTERSECT feature is used to solve a system of equations. To illustrate
2x – 3y = 13
this feature, we will use the system of equations .
3x + 4y = – 6
Note: Some equations can be solved by this method. See Solve an Equation
below. Also, this method is used to find a number in the domain of a func-
tion for a given number in the range. See Find a domain element below.
Solve each of the equations in the system of equations for y. In this case, we
2 13 3 3
have y = -- x – ----- and y = – -- x – -- .
- - - -
3 3 4 2
2 13
Use the Y-editor to enter -- x – ----- into Y1
- -
3 3
3 3
and – -- x – -- into Y2. Graph the two func-
- -
4 2
tions in the standard viewing window. (If
the window does not show the point of
intersection of the two graphs, adjust the
window until you can see the point of in-
tersection.)
Press 2nd CALC (scroll to 5,
intersect) ENTER .
Alternatively, you can just press 2nd
CALC 5.
First curve? is shown at the bottom
of the screen and identifies one of the two
graphs on the screen. Press ENTER .
Second curve? is shown at the bot-
tom of the screen and identifies the sec-
ond of the two graphs on the screen.
Press ENTER .
Guess?, shown at the bottom of the
screen, asks you to use the left or right ar-
row key to move the cursor to the approx-
imate location of the point of intersection.
(If there are two or more points of inter-
section, it does not matter which one you
choose first.) Press ENTER .
The solution of the system of equations
is ( 2, – 3 ) .
GC-6 Graphing Calculator Appendix
Solve an Equation To illustrate the steps, we will solve the equation
2x + 4 = – 3x – 1 . The idea is to write the equation as the system of equations
y = 2x + 4
and then use the steps for solving a system of equations.
y = – 3x – 1
Use the Y-editor to enter the left and right
sides of the equation into Y1 and Y2.
Graph the two functions and then follow
the steps for Intersect.
The solution is –1, the x-coordinate of the
point of intersection.
Find a domain element For this example, we will find a number in the domain
2
of f ( x ) = – -- x + 2 that corresponds to 4 in the range of the function. This is
-
3
2
like solving the system of equations y = – -- x + 2 and y = 4 .
-
3
Use the Y-editor to enter the expression
for the function in Y1 and the desired
output, 4, in Y2. Graph the two functions
and then follow the steps for Inter-
sect.
The point of intersection is (–3, 4). The
number –3 in the domain of f produces an
output of 4 in the range of f.
Math Pressing MATH gives you access to many built-in functions.
Frac function The Frac function converts a decimal to a fraction. The fol-
1
lowing keystrokes will convert 0.125 to the fraction -- .
-
8
.125 MATH 1 ENTER .
Factorial To find 8!, press 8 MATH . Use the right arrow key to highlight
PRB and then use the down arrow key to select 4:. Press ENTER ENTER .
Appendix GC GC-7
Permutations To find the permutations of 8 objects chosen 3 at a time, P ( 8, 3 ) ,
press 8 MATH . Use the right arrow key to highlight PRB (this is the menu
that contains counting and probability functions) and then use the down ar-
row key to select 2:. Press ENTER 3 ENTER
Combinations To find the combinations of 10 objects chosen 4 at a time,
C ( 10, 4 ) , press 10 MATH . Use the right arrow key to highlight PRB and then
use the down arrow key to select 3:. Press ENTER 4 ENTER
Additional built-in functions under MATH can be found by pressing
MATH . For instance, to evaluate – – 25 , press (–) MATH
1 (–) 25 ) .
See your owner’s manual for assistance with other functions under the
MATH key.
Radical expressions To evaluate a square root expression, press 2nd
2
. For instance, to evaluate .15 p + 4p + 10 when
p = 100,000, first store 100,000 in P. Then press 0.15
2nd ALPHA P x2 + 4
ALPHA P + 10 ) ENTER .
To evaluate a radical expression other than
a square root, access x by pressing
MATH . For instance, to evaluate 4 67 ,
press 4 (the index of the radical) MATH
(scroll to 5) ENTER 67 ENTER .
Statistics Mean, Median, Quartiles and Standard Deviation The values of the mean,
median, quartiles, and standard deviation are calculated for a data set by se-
lecting the 1-Var Stats function, which is one of the options that can be
accessed by pressing the STAT key.
GC-8 Graphing Calculator Appendix
For instance, the results of an exam given to 20 students are given below.
96, 72, 76, 47, 92, 58, 66, 79, 44, 62, 76, 78, 50, 52, 90, 84, 63, 65, 75, 86
To calculate the mean, median, quartiles, or standard deviation for this data,
press STAT to access the statistics menu. Press 1 to Edit or enter data. To de-
lete data already in a list, press the up arrow to highlight the list name. For in-
stance, to delete data in L1, highlight L1. Then press CLEAR and ENTER .
Now enter each test score under L1, pressing ENTER after each entry. Use the
up and down arrow keys to change a value.
To calculate the mean, median, quartiles, or standard deviation, press STAT
to access the CALC menu. The press 1 ENTER . The results are
shown below.
Notice the down arrow by n = 20. This indicates that more values follow. Use
the down arrow key to scroll through all the values. For the calculation, we
have the mean is 70.55, the sample standard deviation is approximately 15.35,
the population standard deviation is approximately 14.96, the first quartile
(Q1) is 60, the median is 73.5, and the third quartile (Q3) is 81.5. The lowest test
score is 44 and the greatest test score is 96. Using these numbers, the range is
96 – 44 = 52 .
Linear Regression A study is done to determine the number of grams of sugar
that will dissolve in a liquid at various temperatures. The data is shown below.
Temperature, x (in oC) 20 35 50 60 75 90 100
Grams of sugar, y 50 80 120 145 175 205 230
All calculations and graphs involving statistical data begin by entering the
data using the Edit option which is accessed by pressing STAT .
For the data above, press STAT to access the statistics menu. Press 1 to Edit
or enter data. To delete data already in a list, press the up arrow to highlight
the list name. For instance, to delete data in L1, highlight L1. Then press
CLEAR and ENTER . Now enter each value of the independent variable in
L1, pressing ENTER after each entry. Use the up and down arrow keys to
change a value. When all values of the independent variable are entered, press
. This will put you in the next column to enter the values of the depen-
dent variable in L2.
Appendix GC GC-9
Create a scatter diagram Press 2nd STATPLOT (use the down arrow key
to select Plot1, Plot2, or Plot3) ENTER . Use arrow keys to move the
cursor to On and then press ENTER . The first graph type is for a scatter dia-
gram. Move the cursor over that symbol and press ENTER . Be sure that
Xlist and Ylist are the names of the lists into which you stored data. You
can change these by pressing 2nd and then selecting the appropriate list,
L1 through L6. Prepare to graph the data by adjusting the viewing window by
pressing WINDOW and entering appropriate values. Now press GRAPH .
250
0 110
0
Note: You can tell that STAT PLOTS is active by
pressing Y= . For one screen at the right, ob-
serve that PLOT1 is highlighted indicating it is
active. To turn STAT PLOTSoff, use the up arrow
key to highlight it and then press ENTER . Now
use the arrow key to move the cursor to the right
of the equal sign for Y1.
Find a linear regression equation Press STAT (scroll to 4) ENTER
2nd L1 , 2nd L2 , VARS 1 1 ENTER .The
values of the slope and y-intercept of the linear regression equation will be dis-
played on the screen. If DiagnosticOn is enabled (See Correlation coeffi-
2
cient.), then the coefficient of determination r and the correlation coefficient
r are also shown.
Note: If data is stored in L1 and L2, the keystrokes 2nd L1 ,
2nd L2 are not necessary. The keystrokes VARS 1 1 ENTER
place the regression equation in Y1. These keystrokes are not necessary but
are helpful if you need to graph the regression equation or evaluate the
equation at a given value of the independent variable. See below for more
details.
Other regression equations can be calculated. For instance, to find a regression
x
equation of the form y = ab , called an exponential regression equation, enter
the data, select ExpReg from the CALC menu under the STAT menu and
then press ENTER ENTER .
GC-10 Graphing Calculator Appendix
Note: Because the data was entered into L1 and L2, it was not necessary to
include them in ExpReg. We did include the optional Y1. This is good prac-
tice because it makes evaluating and graphing a regression equation much
easier.
Graph a regression equation Press STAT 250
(scroll to 4) ENTER 2nd L1
, 2nd L2 , VARS 1
1 ENTER . This will store the regression equation
in Y1. Now press GRAPH . It may be necessary to 0 110
0
adjust the viewing window.
Evaluate a regression equation Complete the
steps to graph a regression equation but do not
graph the equation. To evaluate the equation
when x = 50, press VARS 11 (
50 ) ENTER .
Table There are three steps in creating an input/output table for a function. First use
the Y= editor to input the function. The second step is setting up the ta-
ble, and the third step is displaying the table.
To set up the table, press 2nd TBLSET.
TblStart is the first value of the independent
variable in the input/output table. ∆Tbl is the
difference between successive values. Setting this to
1 means that, for this table, the input values are –2,
– 1 , 0, 1, 2 ... . If ∆Tbl= 0.5, then the input values of
would be –2, –1.5, –1, –0.5, 0, 0.5, ... .
Indpnt is the independent variable. When this is set to Auto, values of the
independent variable are automatically entered into the table. Depend is the
dependent variable. When this is set to Auto, values of the dependent vari-
able are automatically entered into the table.
To display the table, press 2nd TABLE. An in-
2
put/output table for f ( x ) = x – 1 is shown at the
right.
Once the table is on the screen, the up and down ar-
row keys can be used to display more values in the
table. For the table at the right, we used the up arrow
key to move to x = –7.
Appendix GC GC-11
An input/output table for any given input can be
created by selecting Ask for the independent vari-
able. The cursor will be at X=. Enter values for x and
press ENTER . The values will be displayed in the
table. The table at the right shows an input/output
4x
table for f ( x ) = ----------- for selected values of x. Note
-
x–2
the word ERROR when 2 was entered. This oc-
curred because f is not defined when x = 2.
Note: Using the table feature in Ask mode is the
same as evaluating a function for given values of
the independent variable. For instance, from the
table at the right, we have f ( 4 ) = 8 .
Trace Once a graph is drawn, pressing TRACE will place
a cursor on the screen and the coordinates of the
point below the cursor are shown at the bottom of
the screen. Use the left and right arrow keys to move
the cursor along the graph. For the graph of
3
f ( x ) = 0.1x – 2x + 2 , shown at the right, we have
f ( 4.8 ) = 3.4592 .
In TRACE mode, you can evaluate a function at any
value of the independent variable that is within Xmin and Xmax. To do this,
first graph the function. Now press TRACE (the value of x) ENTER . For the
graph at the left below, we used x = –3.5. If a value of x is chosen outside the
window, an error message is displayed.
In the example above where we entered –3.5 for x,
the value of the function was calculated as 4.7125.
This means that f ( – 3.5 ) = 4.7125 . The keystrokes
2nd QUIT VARS 1 1 MATH 1
ENTER will convert the decimal value to a fraction.
When the TRACE feature is used with two or more graphs, the up and down
arrow keys are used to move between the graphs. The graphs below are for the
3
functions f ( x ) = 0.1x – 2x + 2 and g ( x ) = 2x – 3 . By using the up and down
arrows, we can place the cursor on either graph. The right and left arrows are
used to move along the graph.
GC-12 Graphing Calculator Appendix
Window The viewing window for a graph is controlled Ymax
by pressing WINDOW . Xmin and Xmax are the Xscl Yscl
minimum value and maximum value, respec- Xmin Xmax
tively, of the independent variable shown on
the graph. Xscl is the distance between tic
marks on the x-axis. Ymin and Ymax are the Ymin
minimum value and maximum value, respec-
tively, of the dependent variable shown on the graph. Yscl is the distance be-
tween tic marks on the y-axis. Leave Xres as 1.
Note: In the standard viewing window, the distance between tic marks on
the x-axis is different from the distance between tic marks on the y-axis. This
will distort a graph. A more accurate picture of a graph can be created by us-
ing a SQUARE viewing window. See ZOOM.
Y= The Y= editor is used to enter the expression
for a function. There are ten possible functions, la-
beled Y1, Y2, Y3, . . ., Y0, that can be active at any one
Take Note 2
Although there are ten possible time. For instance, to enter f ( x ) = x + 3x – 2 as Y1,
functions, only seven can be use the following keystrokes.
displayed at one time, as shown
on the calculator screen at the Y= Θ
X,T,Θ,n x2 + Θ
3 X,T,Θ,n – 2
right. Note: If an expression is already entered for Y1, place the cursor anywhere
on that expression and press CLEAR .
2v – 1
To enter s = -------------- into Y2, place the cursor to the
3
-
v –3
right of the equal sign for Y2. Then press ( 2
Θ
X,T,Θ,n – 1 ) ÷ (
Θ
X,T,Θ,n ^ 3 – 3 )
Note: When entering an equation, the independent variable, v in the expres-
Θ
sion above, is entered using X,T,Θ,n . The dependent variable, s in the ex-
pression above, is one of Y1 to Y0. Also note the use of parentheses to ensure
the correct order of operations.
Observe the black rectangle that covers the equal
sign for the two examples we have shown. This rect-
angle means that the function is ‘active.’ If we were
to press GRAPH , then the graphs of both functions
would appear. You can make a function inactive by
using the arrow keys to move the cursor over the
equal sign of that function and then pressing ENTER . This will remove the
black rectangle. We have done that for Y2, as shown above. Now if GRAPH is
pressed, only Y1 will be graphed.
It is also possible to control the appearance of the
Default
graph by moving the cursor on the Y= screen Bold graph
to the left of any Y. With the cursor in this position, Shade above graph
Shade below graph
pressing ENTER will change the appearance of the Animate graph
Animate graph
graph. The options are shown at the right. Dashed graph
Appendix GC GC-13
Zero The ZERO feature of a graphing calculator is used for various calculations: to
find the x-intercepts of a function, to solve some equations, and to find the zero
of a function.
x-intercepts To illustrate the procedure for finding x-intercepts, we will use
f(x) = x2 + x – 2.
First, use the Y-editor to enter the expression for the function and then graph
the function in the standard viewing window. (For some functions, it may be
necessary to adjust this window so that the intercepts are visible). Once the
graph is displayed, use the keystrokes below to find the x-intercepts of the
graph of the function.
Press 2nd CALC (scroll to 2 for zero
of the function) ENTER .
Alternatively, you can just press 2nd
CALC 2.
Left Bound?, shown at the bottom of the
screen, asks you to use the left or right ar-
row keys to move the cursor to the left of
the desired x-intercept. Press ENTER .
Right Bound?, shown at the bottom of
the screen, asks you to use the left or right
arrow keys to move the cursor to the right
of the desired x-intercept. Press ENTER .
Guess?, shown at the bottom of the
screen, asks you to use the left or right ar-
row keys to move the cursor to the ap-
proximate location of the desired x-
intercept. Press ENTER .
The x-coordinate of an x-intercept is –2.
Therefore, an x-intercept is (–2, 0).
To find the other x-intercept, follow the same steps as above. The screens for
this calculation are shown below.
A second x-intercept is (1, 0).
Solve an equation To use the ZERO feature to solve an equation, first rewrite
the equation with all terms on one side. For instance, one way to solve
3 3
x – x + 1 = – 2 x + 3 is to first rewrite the equation as x + x – 2 = 0 . Enter
3
x + x – 2 into Y1 and then follow the steps for finding x-intercepts.
GC-14 Graphing Calculator Appendix
Find the real zeros of a function To find the real zeros of a function, follow
the steps for finding x-intercepts.
Zoom Pressing ZOOM allows you to create some preset viewing windows. This
key also gives you access to ZBox, Zoom In, and Zoom Out. These functions
allow you to redraw a selected portion of a graph in a new window. Some win-
dows used frequently in this text are shown below.
Related docs
