LISTS AND DATA ENTRY
SECTION 1
Lists and Data Entry
Data is stored in the fx-9750G in LISTS. There are several ways to create
a list. From the home screen curly brackets can be used to store a data
set in a list, with a name from List 1 to List 6 (Fig 1). A better method
however is to select the LIST (4) icon from MAIN MENU to go to the List
Mode and enter the data directly into a column; this method is rather
like using a spreadsheet on the computer (Fig 2).
Fig 1
Fig 2
In the rest of this section we will describe how to define and manipulate
lists.
V
V
You can move between lists using the
keys and between cells
within a list with the V
keys. Input a value and press EXE to
store it in the list. The cursor automatically moves down to the next cell.
V
You can use the result of an expression as list input. To put the result
of 2 + 3 into the next cell you press 2 + 3 EXE and 5 appears in
the cell.
V
V
You can batch input a series of values. Highlight the list header you want
to use (using the
{ , input the
V keys) then press SHIFT
values you want, pressing
between each value. Press SHIFT } after
the final value.
,
Press EXE to store all the values in your list. You can also use list names
inside a mathematical expression to input values into another list.
MATHEMATICS
1
LISTS AND DATA ENTRY
V
highlight LIST 3 using the
V
Example 1
To store in LIST 3 the result of adding the values in LIST 1 to the values
in LIST 2 you:
keys
V
Fig 3
press OPTN F1
(LIST) 2 EXE
(LIST)
(List)
F1
1
+
F1
Fig 4
V
press
1
x
OPTN
5
÷
F1
8
(LIST)
F1
V
Example 2
If LIST 1 contains distances in kilometres and you want to convert these
to miles and store them in LIST 2 you:
V
highlight LIST 2 using the
keys
Fig 5
(List)
EXE
Fig 6
Example 3
Create a list for the following data set of average temperatures in New
Zealand, given in º Fahrenheit.
Jan
63
May 53
Sept 52
Feb 62
June 50
Oct 54
Mar
July
Nov
61
48
59
Apr
Aug
Dec
57
49
61
Convert these temperatures to º Centigrade, using the formula
(F – 32) × (5/9).
Solution
To create a list use the LIST icon from MAIN MENU and enter the data as
List 1 (Fig 7).
2
MATHEMATICS
LISTS AND DATA ENTRY
V
Fig 7
V
V
Now with List 2 highlighted with the
keys enter the
formula, using List 1 as the temperature in º Fahrenheit (Fig 8).
Fig 8
This calculation could have been done on the RUN (1) from MAIN
MENU screen as shown (Fig 9). (As before, List is obtained using
OPTN
F1
F1 .)
Fig 9
Exercises
1.
Create a list List 1 using {4,8,11,14,15,17,20}.
Create new lists List 1 – 7, 3 × List 1, List 12.
2.
Create a list showing the mean distance from the sun in millions of
miles to each planet. Then create a new list showing the mean
distance in millions of kilometres.
Planet
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
Pluto
Mean distance from sun
(Millions of miles)
36.0
67.2
92.9
141.7
483.9
887.2
1784.0
2796.5
3666.1
(8 km = 5 miles)
MATHEMATICS
3
MEDIAN, QUARTILES AND BOXPLOTS
SECTION 2
Median, Quartiles and Boxplots
The median of a set of data is one measure of the average or centre of
the data. When the data are arranged in order, there should be an equal
number of data items above and below the median. If the set has an odd
number of items, then the median is one of the items. If the set has an
even number of items, then the median is the mean of two items.
Example 1
Find the median for this set of 24 test marks.
100
83
66
100
81
65
97
80
63
95
77
60
92
75
58
88
71
54
85
70
51
83
69
50
Solution
Since there is an even number of test marks the median will be the mean
of the middle pair of numbers in an ordered list. In other words, the
median will be the mean of the 12th and 13th data items if this list is
placed in ascending order.
On the fx-9750G, the median of any list can be found
Either from RUN(1) in the MAIN MENU.
∇
∇
∇
Assuming the list you want to work on is in List 1 by
pressing OPTN F1 (LIST) F6 ( ) F4 (Med) F6
( ) F6 ( ) F1 (List) 1 ) EXE
Fig 1
Or from LIST(4) in the MAIN MENU.
V
V
V
Assuming the list you want to work on is in List 1
and List 2 is empty use
to
V
highlight cell 1 in List 2 then press as above from
and including OPTN . This puts the median of List
1 in cell 1 of List 2. (Don’t put it at the bottom of
List 1 as that will change the contents of List 1 and
will affect any more work you do on it.)
Or see after the next section on Quartiles.
4
MATHEMATICS
Fig 2
MEDIAN, QUARTILES AND BOXPLOTS
The lower quartile, Q1, of a data set can be described as the median of
the lower half of the items and the upper quartile, Q3, as the median of
the upper half. If the median is one of the elements, it is not included
in either half.
Example 2
Find the quartiles for the same set of test scores.
Solution
The lower half of this data set has 12 items, so Q1 is the mean of the 6th
and 7th items, in this case 64. Similarly, the upper quartile Q3 is the
mean of the 18th and 19th items, or 86.5.
On the fx-9750G to view the statistical results for a list of data, select the
STAT(2) option from the MAIN MENU (Fig 3), press F2 (CALC) F6
F1
(SET) and make sure the top line has 1Var as List 1 then EXIT
(1Var) (Fig 4).
Fig 4
Several statistics appear (Fig 5). Use
quartiles and median (Fig 6).
Fig 5
Fig 6
V
Fig 3
several times to see the
Fig 7
The Max, Min and Quartiles of a data set are often displayed in a box and
whisker plot (boxplot).
MATHEMATICS
5
MEDIAN, QUARTILES AND BOXPLOTS
Example 3
Create a boxplot of these test marks.
∇
Solution
From the MAIN MENU select STAT(2) or EXIT
if you are
EXIT
continuing from the last section.
Press F1 (GRPH) F6 (SET) and make sure the screen looks like Fig 8.
To set the whole screen, press F1 (GPH1)
F6 ( ) F2 (BOX)
F1
(List 1) ( ) F1 (1).
Then press EXIT F1 (GPH1); see Fig 9.
V
V
V
Fig 8
Fig 9
The Boxplot should be displayed (Fig 10).
Press SHIFT F1 (TRACE) and use the cursor keys to read the extremes,
quartiles and median (Fig 11).
Fig 10
Fig 11
If you do not get the screen shown in Fig 11, then make adjustments
using SHIFT MENU (SET UP) to have Graph Func On, Coord On, Grid
Off, Axes On, etc.
Boxplots are particularly effective for comparison of two or more sets of
data.
6
MATHEMATICS
MEDIAN, QUARTILES AND BOXPLOTS
Example 4
Suppose another class sitting the same test score these marks
94
70
93
64
90
61
84
54
81
53
81
48
78
40
75
32
74
Create two boxplots to compare their performance with the original
class.
Solution
Enter the new scores into another list, such as List 2.
You now need to SET GPH2 to List 2 as follows:
Go into STAT(2) from the MAIN MENU.
Press F1 (GRPH) F6 (SET) F2 (GPH2)
to Graph Type
F6
F1 (1) EXIT
( ) F2 (BOX)
(List 2)
F2
You can now press F1 (GPH1) to see the original boxplot and then
EXIT and F2 (GPH2) to see the second boxplot or F4 (SEL) and
make sure StatGraph1 and StatGraph2 are both set on DrawOn (see Fig
12), then press F6 (Draw) and both boxplots will be displayed at the
same time (GPH1 at the top of the screen); see Fig 13. You can
V
V
∇
V
TRACE
SHIFT
F1
to look at the features of each boxplot; see Figs 14 and 15.
Fig 12
Fig 13
Fig 14
Fig 15
The difference between the third quartile Q3 and the first quartile Q1, is
called the interquartile range. It measures the spread of the middle
50% of the data. In these examples, the interquartile ranges of the
marks are 22.5 for the original class (example 1) and 29 for the second
class (example 4).
MATHEMATICS
7
MEDIAN, QUARTILES AND BOXPLOTS
Exercises
1.
Find the minimum, maximum, median and quartiles for the
running speeds of the following creatures, given in mph.
Create a boxplot for the data.
Cheetah
Coyote
Rabbit
Snail
2.
70
40
35
0.03
Cat
Hyena
Pig
Man
29
40
11
28
Lion
52
Greyhound 40
Tortoise
0.18
Find the minimum, maximum, median and quartiles for heights of
the following pupils. (Measurements in cm.)
159 161 163 164 165 168 168 169 171 172 173 174
175 175
3.
A company has two machines that fill bottles of soft drinks.
Samples from each machine show the following number of
millilitres per can.
Machine 1: 320, 319, 319, 321, 318, 317, 319, 316, 315, 320
Machine 2: 318, 321, 315, 315, 314, 315, 318, 317, 320, 313
Create a boxplot for each machine.
Sketch one above the other.
Describe the performance of the two machines.
8
MATHEMATICS
TWO VARIABLE STATISTICS
SECTION 3
Two Variable Statistics
Objectives
After completing this unit you should be able to use the fx-9750G to:
• draw scattergraphs
• calculate and assist you in your interpretation of Pearson’s
product–moment correlation coefficient.
• determine the least squares regression line of y on x given by
y = ax + b.
• predict values using this regression line and comment on their
reliability.
Example
The table below shows the test results for 10 students in both Maths and
Physics.
Maths
65
45
40
55
60
50
80
30
70
65
Physics
60
60
55
70
80
40
85
50
70
80
(i)
Draw a scattergraph for this data and comment on the relationship
observed.
(ii) Calculate the Pearson’s product–moment correlation coefficient.
(iii) Find the least squares regression line for this data.
Solution to (i)
1.
Enter the data into the Lists on your fx-9750G (Fig 1).
Fig 1
2.
Go into STAT(2) from the MAIN MENU and press F1 (GRPH) F6
(SET) and make sure you set the Graph Type to Scatter. Do this by
highlighting Graph Type and press F1 (Scat).
MATHEMATICS
9
TWO VARIABLE STATISTICS
Choose which data set is to be on the x-axis and which on the
y-axis by entering the appropriate list name (List 1 and List 2).
Finally choose how the data points will be shown on the graph (Fig
2) by highlighting Mark Type and choosing F1, F2 or F3. When you
have completed this stage press EXIT .
Then press F1 (GPH1) to see the graph (Fig 3). If you want to
change the settings for showing the axes for grid lines or if you
wish to switch Label to Off, press SHIFT MENU (SETUP) and make
your choices then press EXIT .
Fig 2
3.
Fig 3
V
You can look at the coordinates of individual points by pressing
SHIFT
F1
(Trace) and
; see Fig 4.
Fig 4
4.
10
Interpretation. Generally, the higher the Maths mark the higher
the Physics mark, and vice versa. Marks scored for Maths and
Physics appear to be correlated.
MATHEMATICS
TWO VARIABLE STATISTICS
Solution to (ii)
Pearson’s product–moment correlation coefficient simplifies
algebraically to a more useful form given by:
x y
xy − ∑ n∑
sxy
∑
=
r=
2
2
sxxsyy

(∑ x )  
(∑ y ) 
2
2
∑x − n  ∑y − n 

 

The various statistics used in this formula can be obtained on the
fx-9750G. If, when you have drawn the graph, your screen looks
like Fig 5, press F6 ( ) giving Fig 6 then F4 (2VAR).
∇
1.
2.
Fig 5
Fig 6
Fig 7
Fig 8
V
You can then scroll down
Fig 9).
and pick out the terms (Fig 7, Fig 8,
Fig 9
The product–moment correlation coefficient, r, can now be
calculated, either manually using the appropriate values from the
above screen or using the calculator.
Press
EXIT
to return to the List screen then
F1
(GPH1) (Fig 10).
Fig 10
MATHEMATICS
11
TWO VARIABLE STATISTICS
Press F1 (x) to give Fig 11 which gives the computed coefficients
for the correlation of y on x and the product–moment correlation
coefficient r.
Fig 11
3.
The value of Pearson’s product–moment correlation coefficient, r,
is now seen. In this example, r = 0.7365.
This would indicate that although there is a positive correlation it
is not very strong.
The general equation of the least squares regression line of y on x
is given by
∑x∑y
sxy ∑ xy − n
=
and a = y − bx
y = ax + b where b =
2
(∑ x )
sxx
2
x
−
∑
n
The calculator has already evaluated a and b.
They are a = 0.7108 and b = 25.196.
So the regression line has equation y = 0.7108x + 25.196.
To draw this line on the graph press
and the graph is drawn; see Fig 12.
Fig 12
12
MATHEMATICS
F6
(DRAW) when at Fig 11
MEAN AND VARIANCE OF DISCRETE RANDOM VARIABLES
SECTION 4
Mean and Variance of Discrete Random Variables
Let X be a discrete random variable taking values
x1, x2,....., xn with probabilities p 1, p2,..... pn. The
variance of X, denoted by σ2, is the number
σ2 = (x 1 – µ )2 p1 + (x 2 – µ )2 p2 + .......... + (x n – µ )2 pn
The standard deviation σ of X is the square root of
the variance.
When given the probability distribution of a random variable X, the
fx-9750G list facility can be used to find the mean and standard
deviation.
Example 1
Let the output of the random variable X denote the number of defective
computer parts in a shipment of 400.
The following table gives the probability density function (pdf) of X:
X
0
1
2
3
4
5
pdf(X)
0.02
0.2
0.3
0.3
0.1
0.08
Compute the mean and standard deviation.
Solution
Input the values shown in List 1 and List 2 by going into STAT(2) from
the MAIN MENU(Fig 1):
Fig 1
Fig 2
Fig 3
Calculate the mean and standard deviation.
Press F2 (CALC) F6 (SET) so that you can set the calculator correctly.
You need to have 1Var XList as List 1 and 1Var Freq as List 2 (Fig 2).
The press EXIT and F1 (1Var) to give Fig 3 where you can see the
mean is 2.5 and the standard deviation is 1.204.
MATHEMATICS
13
MEAN AND VARIANCE OF DISCRETE RANDOM VARIABLES
Exercises
1.
A random variable X has a probability density function given by:
X
–2
–1
0
1
2
pdf(x)
0.3
0.2
0.1
0.3
0.1
Compute the mean and standard deviation.
2.
Bluetits always lay three eggs.
The number of eggs which hatch, X, has the following probability
distribution:
X
0
1
2
3
pdf(x)
0.1
0.3
0.4
0.2
Compute the mean and standard deviation.
14
MATHEMATICS
SUMMARY STATISTICS FOR A SINGLE VARIABLE
SECTION 5
Summary statistics for single or variable data
Introduction:
An introduction to Statistics on the graphic
calculator. This topic looks at a single variable.
Maths content:
Basic simple Statistics.
Calculator work: Use of LISTS and extracting statistics.
Level:
S3 or S4
In this section you will enter some sets of data into the statistical
registers (or lists) of the calculator and calculate a number of different
statistics for the data.
The fx-9750G has six registers (called lists) for storing data. A list can
hold up to 255 data values. Option 4 of the MAIN MENU takes you to
the list screen where lists can be entered and edited.
There are 3 main stages involved in using the statistical keys to find the
values required. These are:
• checking and clearing data lists
• entering the data into one or more of the lists and editing where
necessary
• doing the calculations
Before entering new data into your calculator it is good practice to clear
away any existing data.
An easy way is to choose LIST(4) from the MAIN
MENU and then highlight the list to be deleted
and press F4 (DEL.A) F1 (YES); see Fig 1.
Fig 1
∇
Alternatively, go into STAT(2) from the MAIN MENU, highlight the list to
be deleted and press F4 (DEL.A) F1 (YES) or if DEL.A is not on
screen press F6 ( ) so that it is.
MATHEMATICS
15
SUMMARY STATISTICS FOR A SINGLE VARIABLE
The following simple example illustrates how to find a number of
statistics for a set of data.
Example 1
The marks obtained by pupils in a Geography class test (out of 12 ) were
2, 10, 11, 3, 5, 8, 12, 7, 8, 8
Entering the data
Choose STAT(2) from the MAIN MENU and
you will see the screen shown (Fig 2).
The data are entered one at a time, pressing
[EXE] after each item is entered.
Press
2 [EXE]
10 [EXE]
11 [EXE]
etc until all the data are entered (Fig 3).
Fig 2
Fig 3
Once all the data have been entered the calculator is ready to provide
you with the various summary values.
∇
You can obtain these values by pressing F2
(CALC) from the list screen (if it is not showing
press F6 ( ) as needed then F2 (CALC)).
Now press F6 (SET) and make sure the 1Var
XList is List 1 and the 1Var Freq is 1 (Fig 4). Press
(Fig 5) F1 (1VAR) and you will see the
EXIT
results – scroll down with
to see the full list;
see Figs 6–8.
Fig 4
V
Fig 5
Fig 6
Fig 7
Fig 8
These results are explained as follows:
x is the mean of the values in the list. Σx is the sum of the values. Σx2 is
the sum of the squares of the values. xσn–1 and xσn are measures of how
widely spread the data are. (xσn–1 is the value obtained when n–1 is used
to calculate the standard deviation and xσn is the value obtained when n
16
MATHEMATICS
SUMMARY STATISTICS FOR A SINGLE VARIABLE
is used ). n is the number of values in the list. minX and maxX are the
lowest and highest values in the list.
Med is the median – the middle value when the data are sorted into
ascending order.
Q1 and Q 3 are known as the quartiles.
Example 2
A popular brand of battery is sold in packs of four. A price check was
made in eleven different large stores and produced the following
results:
Store
Price
Store
Price
W H Smith
Woolworth
Currys
Boots
Dixons
Rackhams
£
£
£
£
£
£
Superdrug
Tesco
Sainsbury
Great Mills
Quick Buy
£
£
£
£
£
3.49
3.09
3.49
3.29
3.39
3.79
3.29
2.99
3.29
3.49
3.76
Enter these prices into one of the data lists of the calculator and
produce summary statistics for the data.
Example 3
Two groups of rats were provided with different diets, one group having
a restricted diet and the other permitted free eating. A note was made
of the number of days that the rats in each group lived and this data is
shown in the lists.
Length of lives of rats on a restricted diet and free eating:
Restricted 1136 901 1327 1220 789 1181 604 1085 1045 211 974
Free eating 675 791 630 731 547 768 387 702 736 836
Notice that these are not paired data – there is no link between, for
example, the first values in each list.
Produce summary statistics for these sets of data.
MATHEMATICS
17
SUMMARY STATISTICS FOR A SINGLE VARIABLE
Calculating the mean and other statistics from frequency data
Example 4
The example shows the daily temperatures at midday during the month
of June one year.
Temp. (°C)
Frequency
12
13
14
15
16
17
18
Temp. (°C)
1
2
2
3
3
3
1
19
20
21
22
23
24
Frequency
4
2
3
3
2
1
Enter the temperature values in list List 1 and the corresponding
frequencies in list List 2.
You now need to instruct the calculator to perform a 1-Var Stats
summary of the values in lists List 1 and List 2.
Explanation
To choose the calculation
route.
Press
F1
See
(CALC)
Fig 9
To set what you want each
list to represent.
F6
(SET)
Fig 10
Make sure the 1Var XList is
List 1 and the 1Var Freq is
List 2.
Fig 11
18
MATHEMATICS
SUMMARY STATISTICS FOR A SINGLE VARIABLE
Explanation
To display the summary
statistics
Press
See
EXIT
F1
(1VAR)
Fig 12
V
and to see the complete set
of statistics scroll down
several times.
Fig 13
Fig 14
MATHEMATICS
19
BOXPLOTS
SECTION 6
Boxplots (1)
Introduction:
This unit explains how to enter data into lists,
set up a plot and then to display the plot.
Maths content:
Statistics – boxplots.
Calculator work: LISTS, StatGraph.
S3 or S4
Level:
(This section is adapted from a feature in Tapping into Mathematics with
the TI-83 Graphic Calculator, (eds) Barrie Galpin and Alan Graham,
Addison Wesley, 1997.)
There are three main stages involved in obtaining boxplots on the
fx-9750G. These are:
• Entering the data
• Setting up the plots
• Displaying the plots.
Example
The gross weekly earnings including overtime for 17 chefs and cooks in
£s are shown in the table.
Women
Men
165 210 110 235 152 128 172 136
147 275 233 188 165 330 130 200 249
Construct a boxplot for each set of data.
Start by entering the data into the calculator.
Enter women’s earnings in List 1 and enter the
earnings for men in List 2.
Press
20
F1
(GRPH)
MATHEMATICS
F6
(SET).
Fig 1
BOXPLOTS
Now adjust/set StatGraph 1 screen to Fig 2.
[Use
to highlight Graph Type and then
choose from menu at the bottom of the screen
F2
(Box) to get Med Box, then
to XList
and F1 (List 1) then
to Frequency and
(1)].
F1
V
V
Fig 2
V
Press EXIT and repeat the above process to set
the StatGraph 2 screen to Fig 3.
Fig 3
Then press
(Fig 4).
EXIT
to return to this screen
Fig 4
You now need to select which graphs to display
so press F4 (SEL) and using
and F1
(On) twice and F2 (Off) once, set the screen
like Fig 5.
V
Now press F6 (DRAW) to draw the boxplots,
as in Fig 6. Two boxplots are displayed on the
graphing screen. StatGraph 1 is at the top of the
screen and StatGraph 2 is beneath it.
Fig 5
Fig 6
Using TRACE with boxplots
The five values which should be marked on a boxplot are:
min, Q1, Median, Q3, max
TRACE
Press SHIFT F1 .
Using the right and left cursor keys you can display the above five values
on the screen one at a time.
Using the up and down cursor keys moves the
cross from one boxplot to the corresponding
point of the other boxplot. The display at the
top of the screen shows which graph the trace
refers to and a flashing cross shows the position
of the value displayed at the bottom of the
screen.
Fig 7
MATHEMATICS
21
BOXPLOTS
Boxplots (2)
(This section is adapted from an item in graphiTI 6 and 7. graphiTI is
the newsletter of the TI user group at The Centre for Teaching
Mathematics, University of Plymouth.)
Not all the measures (mean, median and mode) are suitable for all types
of data. For symmetrical data the best measure of average is the mean and
the best measure of spread is the standard deviation. (The sample
standard deviation is denoted by xσn–1, the population standard deviation
is denoted by xσn). For skewed data the best measure of average is the
median and the best measure of spread is the interquartile range (Q3 –
Q1). Encouraging students to see the shape of the data before calculating
the statistics will ensure that they pick out the appropriate measures.
The fx-9750G can be used to investigate data and summary statistics as
follows.
Enter the following data into List 1 (it can be good practice to delete all
the lists first – use F4 (DEL.A) F1 (YES)):
4
5
7
6
7
8
8
32
5
14
9
5
14
20
21
6
Set up the StatGraph 1 as shown in Fig 8
Fig 8
and the selection as shown in Fig 9.
Fig 9
Draw the Boxplot by pressing
F6
(DRAW).
Fig 10
From your Boxplot how can you describe the data
set? Symmetrical, skewed or what?
Calculate the statistics on the data by
pressing F1 (1VAR).
Fig 11
22
MATHEMATICS
BOXPLOTS
A lot of information will appear!
V
Use
to scroll down all the information.
Fig 12
Fig 13
What is the appropriate measure of average, the mean(x) or the median
(Med)?
In the following exercise, find a suitable measure for the average of the
data.
1.
The following are the sizes of 28 families with children:
3
4
2.
3
5
4
7
4
3
5
6
4
5
5
7
4
5
6
6
4
4
5
3
5
6
7
6
The following are the number of nights stayed in Britain by a
sample of 22 overseas visitors in 1996:
1
6
3.
6
5
3
7
2
9
3
11
3
12
1
14
4
15
6
17
7
20
4
22
5
25
The data is the temperature in degrees Centigrade at a weather
centre for two weeks in May:
10
11
13
13
9
16
12
14
15
12
10
15
8
10
Either a boxplot or a histogram of the data is the best way to see if
the data is skewed. [To see a histogram change the Graph Type in
StatGraph1 to (Hist), then EXIT , F1 (GPH1) which gives the Set
Interval screen – you may need to adjust the values then F6
(DRAW)].
Fig 14
Fig 15
Fig 16
MATHEMATICS
23
MARKS IN EXAMS
SECTION 7
Marks in exams: Linear regression
Introduction:
A look at scattergraphs and the line of best fit
for sets of data.
Maths content:
Straight line fit to data.
Calculator work: Using LISTS, StatGraph operations, drawing
scattergraphs.
S3 or S4
Level:
The graphic calculator can be used to plot scatter graphs and to
determine the equation of the line of best fit for linear data.
Example
13 pupils sat tests in Mathematics, Physics and English. The results are
shown in the table below:
Maths
Physics
English
74 61 40 38 62 58 31 48 50 35 20 80 24
69 63 37 27 53 60 27 30 62 43 20 72 14
38 50 72 82 28 57 68 51 21 70 92 16 96
Enter the data into lists List 1, List 2 and List 3.
Fig 1
Consider first the relationship between the
Mathematics mark and the Physics mark.
Set up StatGraph 1 as shown in Fig 2.
F1 (GPH1).
Press EXIT
Fig 2
The graph shown will appear; see Fig 3.
Fig 3
24
MATHEMATICS
MARKS IN EXAMS
As the data looks linear we could perform linear
regression on it. Press F1 (x).
The display gives the values of a and b.
Fig 4
What has the calculator worked out?
The graphic calculator has worked out the
theoretical line of best fit using a process
called Linear Regression.
The equation relating the Mathematics mark
to the Physics mark can be stored in Graph
Func by pressing F5 (COPY) EXE .
The fitted line can be seen by pressing
F6
(DRAW).
Return to MAIN MENU by pressing MENU then
select GRAPH(5) and F6 (DRAW).
Fig 5
Making predictions with the Line of Best Fit
TRACE
Pressing F1 gives the trace function, letting
you work out a good estimate for any student
who may have missed any exams. By using the
cursor keys you can display the x and y
coordinates at any point along the line.
Fig 6
One student was absent for her Physics test. If she scored 54 in her
Mathematics test, what mark would you give her for Physics?
Repeat the previous steps to find equations to represent the
relationships between:
(i)
(ii)
the Mathematics and English marks
the Physics and English marks.
What mark should the absent student be awarded if she missed her exam
in English?
Remember: always plot a scatter diagram of your data first.
MATHEMATICS
25
HIGHWAY CODE
SECTION 8
Highway Code
Introduction:
This topic investigates the stopping distance of a
car which is made up of the thinking distance
and the braking distance.
Maths content:
Fitting lines and curves to data.
Calculator work: LISTS and StatGraph operations.
Level:
S3
The Highway Code gives the following data for the shortest braking
distances of a car (with good brakes on a dry road) travelling at different
speeds. The total distance is made up from the distance travelled before
the driver realises what’s happening (thinking distance) and the
distance travelled in bringing the car to a stop (braking distance).
Stopping Distances
Speed
(mph)
20
30
40
50
60
70
Thinking
distance(m)
6
9
12
15
18
21
Braking
distance(m)
6
14
24
38
55
75
Total
(m)
12
23
36
53
73
96
(Average length of car = 4m; Source – Highway Code)
1.
Display Speed against Thinking Distance and find the equation
connecting S and dthi.
2.
See if you can find a quadratic equation connecting the speed S
and the Braking Distance db.
3.
Find the equation connecting the Speed S and the Total Distance
d tot .
4.
Use the equation in (3) to predict the overall stopping distance for
cars travelling at speeds of
(i) 55 mph
(ii) 73 mph
26
MATHEMATICS
HIGHWAY CODE
The formula obtained in the above questions contribute towards road
safety in a number of ways. They are used to provide advice to drivers
on the distance to leave between vehicles; they can also be used by road
drivers in considering safe visibility distance and in devising safe speed
limits for different types of roads.
Solutions
Enter the speeds, thinking, braking and total distances in lists List 1, List
2 , List 3 and List 4 respectively.
1.
Fig 1
Fig 2
The following screenshots illustrate the steps involved in
entering the data in the lists.
setting up the plot for List 1 against List 2.
Fig 3
obtaining the plot which is a straight line.
Fig 4
finding the line of best fit.
Fig 5
obtaining the equation of the line.
2.
The following screenshots again illustrate the steps involved in
Fig 6
setting up the plot for List 3 against List 1.
Fig 7
calculating the quadratic fit to the
data (choosing F3 (x^2)).
Fig 8
Fig 9
The quadratic is
shown.
MATHEMATICS
27
HIGHWAY CODE
3.
The following screenshots illustrate the steps involved in setting
up the plot for List 4 against List 1
Fig 10
Fig 11
calculating the quadratic fit to the data
Fig 12
and displaying the coefficients.
Write the equation down using the calculated coefficients, and by
comparing this with the results from 1 and 2 you will see that this
third equation is the sum of the two from 1 and 2.
4.
Copy the equation
S = 0.016dtot2 + 0.263dtot + 0.6
as Y1 = 0.016X2 + 0.263X + 0.6
in the TABLE(7) from MAIN MENU.
Fig 13
Press F5 (RANG)
and put 55 for the start value
and 73 for the end value
and 18 for the pitch.
Fig 14
then press EXE and
the results.
F6
(TABL) to see
Fig 15
(You may have to alter SETUP to get the same as Fig 15.)
28
MATHEMATICS
FIRE DAMAGE
SECTION 9
Fire Damage
Introduction:
The relationship between the distance from the
site of a fire to the nearest fire station and the
amount of damage caused by the fire is
investigated.
Maths content:
Linear regression.
Calculator work: Use of LISTS and StatGraph operations.
S3 or S4
Level:
An insurance company decided to investigate the relationship between
the distance from the site of a fire to the nearest fire station (miles) and
the amount of damage caused by the fire in thousands of dollars. It
investigated a sample of 15 major residential fires in a particular
suburban area, obtaining the data shown in the table below.
Distance x
3.4
Fire Damage y 26.2
1.8
17.8
4.6
31.3
2.3
23.1
3.1
27.5
5.5
36.0
0.7
14.1
Distance x
2.6
Fire Damage y 19.6
4.3
31.1
2.1
24.0
1.1
17.3
6.1
43.2
4.8
36.4
3.8
26.1
3.0
22.3
Consider the relationship between distance and fire damage.
Enter the data into lists List 1 and List 2 as shown.
Fig 1
Set up the graph by pressing F1 (GRPH)
(SET) and make the settings like this.
F6
Press
EXIT
,
F1
(GPH1) to see the graph.
Fig 2
MATHEMATICS
29
FIRE DAMAGE
As the data looks linear we could perform linear
regression on it. Press F1 (x).
Fig 3
The display gives the values of a and b.
Fig 4
What has the calculator actually worked out?
The calculator has worked out the line of best fit using a process called
Linear Regression.
Press
F6
(DRAW) to show the fitted line.
Fig 5
Use the line to estimate the amount of damage that
would be caused by a future residential fire at a site
5 miles from the nearest fire station. To do this press
(x) F5 (COPY) EXE and the equation
F1
connecting distance to fire damage is stored in Y1.
Fig 6
Then press
and select icon GRAPH(5) from the MAIN MENU then
MENU
TRACE
(DRAW) and you can use TRACE by pressing SHIFT F1 and then
the cursor keys to set the x-value. You may need to alter the View
F6
Window settings
value.
V-Window
SHIFT
F3
to give you access to the appropriate x
How sensible do you think it would be to estimate from the graph, the
amount of damage for a fire at a distance 10 miles from the nearest fire
station? If you think it is not sensible explain why.
30
MATHEMATICS
BREAKING STRENGTH OF CABLES
SECTION 10
Breaking strength of cables
Introduction:
The breaking strength of cables depends on the
diameter of the cable. It is required to find an
equation connecting the diameter and the
breaking strength.
Maths content:
Fitting equations to data.
Calculator work: Use of STAT and Pwr.
S5 or S6
Level:
Cables are tested under laboratory conditions to determine their breaking strength. Weights are attached to the cable and this weight is steadily
increased until the cable breaks. The breaking strength of the cable is a
function of the diameter of the cable.
The results obtained in a number of experimental trials are shown in the
table below.
Diameter
of the cable
x(mm)
Maximum weight held by
the cable before breaking
y(kg)
1
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
2.85
11.41
35.51
65.43
122.07
202.81
326.51
498.46
700.36
Engineers believe that an equation of the form y = kxn could be used to
represent the data for the cables. This equation once determined could
be used to estimate the breaking strengths of cables whose diameters
are not listed in the table above.
MATHEMATICS
31
BREAKING STRENGTH OF CABLES
Enter the data in lists List 1 and
List 2 as shown.
Fig 1
Press F1 (GRAPH) F6 (SET) and ensure
F4
the settings are as shown then EXIT
(SEL)
and make sure the graph selection is as shown
and press F6 (DRAW) to see the scatter
diagram. Then F6
(Pwr) will give the
F3
equation representing the breaking strength
of cables as y = 2.95x3.40 (Fig 6).
Fig 2
Fig 3
Fig 4
This equation is stored in Y1 by pressing
(COPY) EXE .
F5
Press F6 and you can see the equation gives a
good fit to the data (Fig 7).
We can now use the equation to estimate the
breaking strengths for cables with diameters not
in the table.
Fig 5
Fig 6
Fig 7
Return to MAIN MENU MENU and select the
TABLE(7) icon. You will see the equation
(Fig 8) and to evaluate particular points press
(TABL) and enter the two x values 2.25
F6
and 3.65 as shown in Fig 9. If you prefer to use
the rounded formula values you can change Y1
when you go into TABLE to 2.95 X,θ,T ^ 3.4 EXE
and then F6 to input 2.25 and 3.65 to evaluate
the functions. (You may have to alter SET UP to
get the same as Fig 9.)
32
MATHEMATICS
Fig 8
Fig 9
BREAKING STRENGTH OF CABLES
Task
The data below shows the breaking strength for a number of cables.
Fit an equation of the form y = kxn to the data and use it to
estimate the breaking strengths for d = 1.65 and d =4.75
Diameter
1
2
3
4
5
6
Breaking strength
3.26
15.17
30.49
63.28
145.51
202.02
MATHEMATICS
33
BREEDING GULLS
SECTION 11
Breeding Gulls
Introduction:
A study is made over a number of years of the
number of pairs of gulls breeding on a nature
reserve. Students are required to fit an
equation to the data so that future breeding
numbers can be estimated.
Maths content:
Functions of the form f(x) = aebx.
Calculator work: STAT; Use of Exp function.
S5 or S6
Level:
The number of pairs of breeding gulls estimated each year in a nature
reserve is recorded over a 10-year period. The figures are given in the
table below:
Year
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
Number
422
462
505
554
606
664
727
796
871
954
To model the population scientists believe that pairs breed according to
the formula P(n) = aebn where n is the number of years after records are
started.
Enter the data in lists List 1
and List 2 as shown.
Fig 1
Fig 2
Press F1 (GRPH) F1 (GPH1)
to see the scatter diagram.
Fig 3
34
MATHEMATICS
BREEDING GULLS
∇
Press F6 ( ) F2 (Exp) and the equation
representing the number of breeding pairs
is given by P(n) = 385.259 × e0.0907n
Press F6 (DRAW) to see the curve plotted
on top of the scatter diagram.
Fig 4
Fig 5
Using the equation for values outside the range of the table is called
extrapolation and generally this procedure is not recommended except
for estimation purposes.
To calculate estimates of other year values,
e.g. 11 for 2001 and 12 for 2002, you should
return to MAIN MENU (press MENU ) and
choose the GRAPH(5) icon.
Enter the equation of the curve at Y1 [if the
Graph Func at the top of the screen does not
show Y= then press F3 (TYPE) F3 (Y=).]
Fig 6
ex
385.259 SHIFT In 0.0907 X,θ,T EXE
You can now do either or both of the following:
a) Press
(DRAW) to display the graph and
F6
TRACE
V
V
then press SHIFT F1 followed by the cursor
keys
to change the x values and note
the corresponding y values. [If the screen does
not show the range of values you need, you should
change them in V-Window – in this example
settings of Xmin:10, max:13, scale:1, Ymin:800
max:1300, scale:50 show the appropriate section
of the curve.]
b) Return to MAIN MENU and choose the TABLE(7)
icon. You will see the Y1= 385.259e0.0907X and
by pressing F5 (RANG) and setting Start to 11 and
End to 12 pitch to 1 EXE you set up the two
x values you are interested in. Then pressing
(TABL) shows the required results.
F6
EXE
[Note: if you don’t set a suitable range and go
directly to F6 (TABL) you can alter the x values
on screen to the ones you want.]
Fig 7
Fig 8
MATHEMATICS
35
BREEDING GULLS
Task
The data shows the number of pairs of breeding herons. Fit an equation
of the form P = ae bn to the data and use it to estimate the figures for
1997 to 2000.
Year
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
36
Pairs
290
343
404
477
563
664
MATHEMATICS
PENDULUM LENGTHS AND PERIODS
SECTION 12
Pendulum lengths and periods
Introduction:
The mathematical relationship between the time
of swing and the length of the string for a
simple pendulum is to be established.
Maths content:
Fitting curves to data.
Calculator work: LISTS, STATPLOTS, STATCALC operation.
Level:
S4 or S5
A group of students are attempting to determine the mathematical
relationship between the length of a pendulum and its period (the time
taken for one complete swing of the pendulum). The students make a
simple pendulum from string and washers and then suspend it from the
ceiling. They record the pendulum’s period for each of 12 string
lengths. The results are shown in the table.
Length (cm)
6.5
11.0
13.2
15.0
18.0
23.1
24.4
26.6
30.5
34.3
37.6
41.5
Time(sec)
0.51
0.68
0.73
0.79
0.88
0.99
1.01
1.08
1.13
1.26
1.28
1.32
Enter the 12 string lengths in list List 1.
Enter the corresponding times in List 2.
Fig 1
MATHEMATICS
37
PENDULUM LENGTHS AND PERIODS
The calculator can fit various models to your experimental data. Since it
will also draw the model graph on your data points you might be
tempted to judge ‘by eye’ which is the best model. If one model is very
poor compared to another this may well be good enough. But it would
be useful if we could make the judgement based on a numerical
calculation, particularly if the models are both quite good. One way to
make this judgement is to compare the correlation coefficients for each
model. The nearer the correlation coefficient is numerically to 1 the
better the fit.
∇ ∇ ∇
Press F1 (GRPH) F1 (GPH1) to see the
scatter diagram. Select the model you
want the calculator to fit
F1 (x) for the linear model y = ax + b
or F6 ( ) F1 (Log) for the model y = a + b lnx
or F6 ( ) F2 (Exp) for the model y = aebx
or F6 ( ) F3 (Pwr) for the model y = axb.
Fig 2
For each one you should note on paper the constants a, b for the rule
and the correlation coefficient r for comparison purposes. After each
one you can press F6 (DRAW) to see the model fitted to the scatter
diagram.
If you want to see each model in turn displayed on the scatter diagram
F1
you can press EXIT
(GPH1) before you select the next model.
The models should given these results:
Fig 3
Fig 4
Fig 5
Fig 6
Fig 7
Fig 8
Fig 9
Fig 10
from which you can see that the model y = 0.192x 0.522 is the best one.
38
MATHEMATICS
PENDULUM LENGTHS AND PERIODS
To predict other periods for different string lengths you should return
to MAIN MENU and in GRAPH(5) enter the chosen model and either
DRAW and TRACE to investigate other points or use TABLE(7) from
MAIN MENU to calculate the particular values you want.
Now that you have a good model for the
relationship between length and period you can
use the model to predict the period for a given
string length.
Fig 11
Fig 12
Note: Since a string length of 50cm exceeds the lengths in the data set,
we would expect more error with this estimate. This latter estimate is
called extrapolation.
MATHEMATICS
39