LISTS AND DATA ENTRY
SECTION 1
Lists and Data Entry
Data is stored in the TI-83 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 L1 to L6 (Fig 1). A better method however is
to use the STAT 1:Edit to go to the List Editor 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 unit we will describe how to use the List Editor to
define and manipulate lists.
A list can be created from an existing list by placing the cursor on the list
name at the top of a column and entering a formula, e.g. L1 + 10. (See
Fig 3.) If this formula is entered in quotes then the new list will be
automatically recalculated when data items in the original list are
altered.
An auto-calculating list is marked by a dot alongside the list name. In
this example L2 is not auto-calculating but L3 is. Notice the change
when the first data item is altered (Fig 4).
Fig 3
Fig 4
MATHEMATICS
1
LISTS AND DATA ENTRY
Example 1
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 STAT 1:Edit and enter the data as L1 (Fig 5).
Now with the name of L2 highlighted with the cursor enter the formula,
using L1 as the temperature in º Fahrenheit (Fig 6). You can choose to
make L2 auto-calculating by putting the formula into quotes.
Fig 5
Fig 6
This calculation could have been done on the home screen as shown
(Fig 7). Once again, by putting the formulae into quotes the list would
become auto-calculating. The result of this is shown in Fig 8.
Notice in this example that L2 is auto-calculating but L3 is not.
Fig 7
2
Fig 8
MATHEMATICS
LISTS AND DATA ENTRY
Exercises
1.
Create a list L1 using {4,8,11,14,15,17,20}.
Create new lists L1 – 7, 3 × L1, L12.
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)
36.00
67.24
92.90
141.73
483.86
887.15
1783.97
2796.46
3666.05
(8km = 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 order.
On the TI-83, the median of any list can be found using LIST MATH (Figs
1 and 2).
Fig 1
Fig 2
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.
4
MATHEMATICS
MEDIAN, QUARTILES AND BOXPLOTS
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 TI-83 to view the statistical results for a list of data, use STAT
CALC 1: 1-Var Stats (Fig 3) followed by the list name (Fig 4).
Fig 3
Fig 4
Several statistics appear (Fig 5). Use the down arrow several times to
see the quartiles and median (Fig 6).
Fig 5
Fig 6
Also use the VARS 5: Statistics PTS menu (Fig 7) to bring quartiles to the
home screen (Fig 8). (These variables are only available after the 1-Var
Stats command has been used.)
Fig 7
Fig 8
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
Ensure that the Y= screen is cleared (Fig 9) or that all functions are
deselected and the Format screen is as shown (Fig 10). To create this
plot use STAT PLOT, and select a plot 1, 2 or 3. Select On. For Type
select the fifth icon and choose the Xlist to be the List where the data is
stored, in this case LI (Fig 11).
Set an appropriate window by using ZOOM 9: ZoomStat (Fig 12).
Fig 11
Fig 12
The Boxplot should be displayed (Fig 13).
Press TRACE and use the cursor keys to read the extremes, quartiles and
median (Fig 14).
Fig 13
Fig 14
Boxplots are particularly effective for comparison of two or more sets of
data.
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.
6
MATHEMATICS
MEDIAN, QUARTILES AND BOXPLOTS
Solution
Enter the new scores into another list, such as L2.
Each class is set up in a STAT PLOT (Fig 15).
Fig 15
Fig 16
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).
Any data item value less than Q1 – 1.5(Q3 – Q1) or greater than Q3 +
1.5(Q3 – Q1) is called an outlier. A modified boxplot will not put
outliers on the whisker, but plot them as single points. The whisker will
go to the smallest and largest data items that are not outliers.
Example 5
Suppose one student from the original class takes the test late and
scores 22. Create a modified box plot for the original class.
Solution
Add a score of 22 to L1 and choose the modified boxplot icon, icon 4 in
the STAT PLOT menu; then choose how the outliers will be denoted
(Fig 17). We see a different boxplot with the outlier plotted separately
and the next value,50, at the end of the whisker (Fig 18).
Fig 17
Fig 18
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.
Test for outliers.
Create a boxplot or modified 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 TI-83 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 T1-83 (Fig 1).
Fig 1
2.
Enter the STAT PLOT Menu and choose PLOT 1.
On this screen switch On plot 1.
Choose the TYPE of graph to be a scatter graph, the 1st of the 6
icons.
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 (L1 and L2).
Finally choose how the data points will be shown on the graph (Fig
2).
Fig 2
3.
In order to draw the graph to an appropriate range on the axes,
choose ZOOM 9:ZoomStat (Fig 3).
Your scattergraph is drawn (Fig 4).
Fig 3
Fig 4
The range on the axes has been set automatically by the calculator.
You can see what this range is by pressing the WINDOW button
(Fig 5).
You may also want to adjust the values selected to those of your
choice (Fig 6).
Fig 5
4.
To see the graph again press GRAPH.
Individual points on the graph can be interrogated using the
TRACE button and the cursor arrows (Fig 7).
Fig 7
10
Fig 6
MATHEMATICS
TWO VARIABLE STATISTICS
5.
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.
Solution (ii)
Pearson’s product–moment correlation coefficient simplifies
algebraically to a more useful form given by:
r=
1.
sxy
sxxsyy
=
∑ xy − ∑
x
∑y
n

(∑ x )  
(∑ y ) 
2
2
∑x − n  ∑y − n 



2
2
The various statistics used in this formula can be obtained on the
TI-83.
Choose STAT, CALC 2: 2-Var Stats (Fig 8).
Enter the Names of the list holding the Data, i.e. L1 and L2. Notice
that the names are separated by a comma (Fig 9).
Fig 8
Fig 9
These various statistics are displayed. Notice that to see them all
you must scroll down the screen (Fig 10).
Fig 10
MATHEMATICS
11
TWO VARIABLE STATISTICS
2.
The product–moment correlation coefficient, r, can now be
calculated, either manually using the appropriate values from the
above screen or using the calculator.
In the CATALOG screen scroll down until you reach
DiagnosticOn, press ENTER twice (Fig 11).
Fig 11
Now choose STAT, CALC 4:LinReg (ax+b) (Fig 12).
Next enter the List names for the data sets using a comma to
separate the names (Fig 13).
Fig 12
Fig 13
On pressing ENTER the following display (Fig 14) is given:
Fig 14
3.
12
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.
MATHEMATICS
TWO VARIABLE STATISTICS
The general equation of the least squares regression line of y on x
is given by
∑x∑y
sxy ∑ xy − n
where
and a = y − bx
=
=
b
y = ax + 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 enter these values on the Y= screen
manually, or call them up as follows.
• Choose Y= (Fig 15)
Fig 15
• Now choose VARS 5:Statistics (Fig 16),
followed by EQ 1:RegEQ (Fig 17).
Fig 16
Fig17
• This has automatically ‘called up’ the regression equation into
Y1=.
• This can be seen by pressing Y= (Fig 18).
Fig 18
MATHEMATICS
13
TWO VARIABLE STATISTICS
• To see the regression line on the graph press GRAPH (Fig 19).
Fig 19
14
MATHEMATICS
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 TI-83
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 L1 and L2 (Fig 1):
Fig 1
Calculate the mean and standard deviation.
Use STAT CALC 1: 1-Var Stats (Fig 2) followed by the list name (Fig 3 ).
MATHEMATICS
15
MEAN AND VARIANCE OF DISCRETE RANDOM VARIABLES
Fig 2
Fig 3
This gives
Fig 4
The mean is 2.5 and the standard deviation is 1.204.
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.
16
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 STATCALC operations.
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 TI-83 has six registers (called lists) for storing data. A list can hold
up to 999 data values. Option 1 of the STAT EDIT 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. The standard way to clear lists is using option 4
in the STAT EDIT menu. You must specify which lists you wish to clear,
otherwise you will get an error message.
Press
[STAT] 4 [2nd] [L1] [ENTER]
[STAT] 4 [2nd] [L2] [ , ] [2nd]
[ L3 ] [ENTER]
See
Explanation
To clear more than
one list at a time,
put a comma between
the named lists.
MATHEMATICS
17
SUMMARY STATISTICS FOR A SINGLE VARIABLE
There are other ways of clearing lists which are sometimes faster. One
method uses option 4, ClrAllLists, in the MEM menu. As the name
suggests ClrAllLists clears all of the data stored in all of the lists.
Press [2nd] [MEM] 4 [ENTER]
If you examine the list screen
you will see that all data has
been cleared from each list.
A third method of clearing lists is as follows.
Press [STAT] 1.
The display in the screenshot shows some
typical data values.
Press [ ∆ ] to move the cursor over L1 at the
top of the screen. Press [CLEAR] [ENTER]
and all the values in L 1 disappear. To clear
L2, position the cursor over L2 at the top of
the screen and press [CLEAR] [ENTER].
Repeat for the other lists.
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
Press [STAT ] and you will see the screen shown.
Now press 1 or [ENTER] to see the lists.
The data are entered one at a time, pressing
[ENTER] after each item is entered.
Press 2 [ENTER]
10 [ENTER]
11 [ENTER]
etc until all the data are entered.
Once all the data have been entered the calculator is ready to provide
you with the various summary values.
18
MATHEMATICS
SUMMARY STATISTICS FOR A SINGLE VARIABLE
You can obtain these values by:
• selecting 1 - Var Stats from the STAT CALC menu
• entering the list to which it is to be applied (in this case L1 )
• pressing [ENTER] to confirm your selection.
You should obtain the results in the diagrams which follow.
Press
See
Explanation
[STAT]
[ ]
This is the STATCALC menu. You
need to select option 1:
1-Var Stats.
1
[2nd] [L1]
1 - Var Stats is pasted to the Home
screen. The list to be summarised,
L 1, is entered.
[ENTER]
The ‘one variable’ summary
statistics are displayed.
Use the [ ∇ ] key to scroll down
to see more.
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. Sx and σx are measures of
how widely spread the data are. (Sx is the value obtained when n–1 is
used to calculate the standard deviation and σx is the value obtained
when n 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.
MATHEMATICS
19
SUMMARY STATISTICS FOR A SINGLE VARIABLE
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.
20
MATHEMATICS
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 L1 and the corresponding
frequencies in list L2.
You now need to instruct the calculator to perform a 1-Var Stats
summary of the values in lists L1 and L2.
Press
See
Explanation
[STAT] [ ] 1
[2nd] [L1] [ , ] [2nd]
[L2 ]
Select 1 - Var Stats and paste
it to the Home screen.
Notice the order:
L1 values first, L2
(frequencies) second.
[ENTER]
Confirm the selection.
The summary statistics
are displayed.
[∇]
Scroll downwards to display
all the information.
MATHEMATICS
21
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, STATPLOTS.
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 TI-83.
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 L1 and enter the
earnings for men in L 2.
Press [2nd] [STAT PLOT].
Each time you wish to set up one or more plots,
it is a good idea to begin by switching off all the
plots; then switch on the plot which you wish
to be displayed.
22
MATHEMATICS
BOXPLOTS
For the example above two plots must be set up: Plot 1 for the women’s
earnings and Plot 2 for the men’s earnings.
Select option 1 by pressing either 1 or [ENTER].
Set up Plot 1 as shown ensuring that all the other
plots are off.
Now move the cursor to the top of the screen and press [ ] to select
Plot 2 and press [ENTER].
Now set up Plot 2 as shown.
The boxplots are now set up and are ready to be displayed.
From the [ZOOM] menu select 9 : ZoomStat.
Two boxplots are displayed on the graphing screen.
Plot 1 is at the top of the screen and Plot 2 is beneath it.
Using TRACE with boxplots
The five values which should be marked on a boxplot are:
min, Q1, Median, Q3, max
Press the [TRACE] key which is located in the top row of the keyboard.
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.
Modified boxplots
The TI-83 allows you to draw what are called modified boxplots which
are selected using the fourth icon when you are setting up the statistical
plots.
MATHEMATICS
23
BOXPLOTS
The boxplots obtained are similar to regular boxplots except that points
more than 1.5 times the interquartile range are plotted individually
beyond the end marks of the boxplot. You can trace these points, which
are known as outliers.
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 Sx , the population standard deviation is
denoted by σx). 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 TI-83 can be used to investigate data and summary statistics as follows.
Enter the following data into L 1:
4
5
7
6
7
8
8
32
5
14
9
5
14
20
21
6
Set up the [STATPLOT] as shown.
Draw a Boxplot showing this data
by pressing Zoom 9.
From your Boxplot how can you describe the data set? Symmetrical,
skewed or what?
Calculate the statistics on the data.
From [STAT] and CALC choose 1- Var
Stats and enter L 1.
24
MATHEMATICS
BOXPLOTS
A lot of information will appear!
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.
MATHEMATICS
25
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, STATCALC 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 L1, L2 and L 3.
Consider first the relationship between the Mathematics mark and the
Physics mark.
Set up Plot 1 as shown, ensuring that all other
plots are off.
From the [ZOOM] menu choose 9 : ZoomStat.
The graph shown will appear.
26
MATHEMATICS
MARKS IN EXAMS
As the data looks linear we could perform linear
regression on it. Press [STAT], choose CALC and
then 4: LinReg(ax+b).
We want to perform linear regression on L1 and L2 and put the resulting
equation into Y1 so that we can plot it on the graph.
Enter L1 , L2 and Y 1 as the argument and press
[ENTER]. Y1 can be found under [VARS],
Y-VARS, 1: Function. The display gives the
values of a and b.
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 is stored in Y1.
The fitted line can be seen by pressing [GRAPH].
Making predictions with the Line of Best Fit
Pressing the [TRACE] function allows you to
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.
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
27
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 STATCALC functions.
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 d.
2.
See if you can find a quadratic equation connecting the speed S
and the Braking Distance d.
3.
Find the equation connecting the Speed S and the Total Distance d.
4.
Use the equation in (3) to predict the overall stopping distance for
cars travelling at speeds of
(i) 55 mph
(ii) 73 mph
28
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 L1, L2 , L3
and L4 respectively.
1.
The following screenshots illustrate the steps involved in
entering the data in lists L 1 – L4.
setting up the plot for L1 against L2.
obtaining the plot which is a straight line.
finding the line of best fit.
obtaining the equation of the line.
2.
The following screenshots again illustrate the steps involved in
setting up the plot for L1 against L3.
calculating the quadratic fit to the
data.
The quadratic is
shown.
MATHEMATICS
29
HIGHWAY CODE
3.
Simply adding the results in the previous two parts will give you
the result for the total distance. (Y3 = Y1 + Y2)
Obtain S = 0.016d2 + 0.263d + 0.6
30
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 STATCALC operations.
Level:
S3 or S4
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 L1 and L2 as shown.
Set up Plot 1 as shown ensuring that all other
Plots are off.
From the [ZOOM] menu choose
9 : ZoomStat
MATHEMATICS
31
FIRE DAMAGE
The graph shown will appear.
As the data looks linear we could perform linear
regression on it. Press [STAT], and choose CALC
and then select 4: Linreg(ax+b)
We want to perform linear regression on L1 and L2 and put the resulting
equation into Y1 so that we can plot it on the graph.
Enter L1 , L2 and Y1 as the arguments and press [ENTER].
(Y1 can be found under [VARS], Y-VARS, 1 : Function.)
The display gives the values of a and b.
What has the calculator actually worked out?
The calculator has worked out the line of best fit using a process called
Linear Regression. The equation connecting distance to fire damage is
stored in Y1 .
Press [GRAPH] to show the fitted line.
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
[TRACE] [ ∆ ] and enter a 5 to obtain the result.
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.
32
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 STATCALC and PwrReg.
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
33
BREAKING STRENGTH OF CABLES
Enter the data in lists L 1 and L2 as shown.
Press [STAT] CALC and select A: PwrReg
and paste it to the home screen.
Enter the arguments as shown and press
[ENTER] to obtain the results.
The equation representing the breaking
strength of cables is given by y = 2.95x 3.40
This equation is stored in Y1.
Press [2 nd] [STATPLOT] and set up Plot 1
as shown.
Press [ZOOM] and select 9: ZoomStat
The graph shows the curve and the data points.
The equation is a good fit to the data.
We can now use the equation to estimate the breaking
strengths for cables with diameters not in the table.
We can estimate the breaking strength
of cables with diameters 2.25 and 3.65
by calculating Y 1(2.25) and Y 1(3.65) as shown.
34
MATHEMATICS
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
35
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) = abx.
Calculator work: STATCALC; Use of ExpReg 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) = ab n where n is the number of years after records are
started.
Enter the data in lists L 1 and L2 as shown.
Press [STAT] CALC and select 0: ExpReg
and paste it to the home screen by
pressing [ENTER].
36
MATHEMATICS
BREEDING GULLS
Enter the arguments as shown and press
[ENTER] to obtain the results.
The equation representing the number
of breeding pairs is given by
P(n) = 385.259 × 1.095n
This equation is stored in Y1.
Press [2nd] [STATPLOT] and set up
plot 1 as shown.
Press [ZOOM] and select 9: ZoomStat
The graph shows the curve and the
data points.
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.
We can estimate the number of breeding
gulls in the next two years by calculating
Y1(11) and Y 1(12) as shown.
Task
The data shows the number of pairs of breeding herons. Fit an equation
of the form P = abn 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
Pairs
290
343
404
477
563
664
MATHEMATICS
37
PENDULUM LENGTHS AND PERIODS
SECTION 12
Pendulum lengths and periods
(This section is adapted from a topic in the handbook for the TI-83)
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 L 1.
Enter the corresponding times in L2 .
38
MATHEMATICS
PENDULUM LENGTHS AND PERIODS
Press [2nd] [STAT PLOT] and set up
Plot 1 as shown ensuring that all other
plots are off.
From the [ZOOM] menu choose 9: ZoomStat
The graph shown will appear. This is a scatter
plot of the length against time data.
Since the scatter plot of length against time appears to be approximately
linear, fit a line to the data.
We want to perform linear regression on L1 and L 2 and store the
resulting equation in Y1 so that we can plot it on the graph.
Press [STAT] CALC and select 4: LinReg.
Enter L1 , L2 and Y1 as the arguments and
press [ENTER].
Y1 can be found under [VARS], Y-VARS,
1: Function.
The display gives the values of a and b.
What has the calculator actually worked out? The calculator has worked
out the theoretical line of best fit using a process called Linear
Regression.
Residuals are calculated and are stored
automatically in the list name RESID,which
becomes an item on the LIST NAMES menu.
(The residual is the difference between the y
value of a plotted point and the y value on the
regression line, for the same x value.)
Press [GRAPH]. The regression line and the
scatterplot are both displayed.
MATHEMATICS
39
PENDULUM LENGTHS AND PERIODS
The regression line appears to fit the central portion of the scatter plot
well. However a residual plot may provide more information about this
fit.
Press [STAT] 1 to select 1:Edit. The stat list editor is displayed.
Press [ ] and [ ∆ ] to move the cursor onto L3.
Press [2nd] [INS]. The unnamed column is displayed in column 3 L 3, L 4,
L5 and L6 shift right one column. The Name = prompt is displayed in the
entry line, and the alpha-lock is on.
Press [2nd] [LIST] to display the LIST NAMES menu.
If necessary press [ ∇ ] to move the cursor onto the list name RESID.
Press [ENTER] to select RESID and paste
it to the stat list editor’s Name= prompt.
Press [ENTER] and RESID is stored in
column 3 of the stat list editor.
Press [ ∇ ] repeatedly if you wish to examine the residuals.
Notice that the first three residuals are negative. They correspond to the
shortest pendulum string lengths in L 1. The next five residuals are all
positive and three of the last four are negative. The latter correspond to
the longer string lengths in L1. Plotting the residuals will show this
pattern more clearly.
Press [2nd] [STAT PLOT] 2 to select Plot 2
and select the arrangement shown in the diagram.
RESID is simply typed into the Ylist.
Press [Y=] to display the Y= editor. Deselect Y1 and switch off Plot 1.
Press [ZOOM] 9 to select 9: ZoomStat from
the ZOOM menu and Plot 2 is displayed as
shown. This is a scatter plot of the residuals.
40
MATHEMATICS
PENDULUM LENGTHS AND PERIODS
Notice that the pattern of the residuals is a group of negative residuals,
then a group of positive residuals, then a group of negative residuals.
The residual pattern indicates a curvature associated with the data set
for which the linear model did not account. The residual plot indicates a
downward curvature, so a model that curves down with the data would
be more accurate. Perhaps a function such as a square root would fit.
Try a power regression to fit a function of the form y = axb.
Press
Press
Press
2.
Press
[Y=] to display the Y= editor.
[CLEAR] to clear the linear regression equation from Y1.
[ ∆ ] [ENTER] to turn on Plot 1. Press [ ] [ENTER] to turn off Plot
[ZOOM] 9 and the original scatterplot appears again.
Select PwrReg from the STAT CALC menu.
The arguments are L1, L2 and Y1 as before.
Press [ENTER] to calculate the power regression
which is entered in Y1. Again the residuals are
stored automatically in the list name RESID.
Press [GRAPH] and the regression line and the
scatter plot are displayed.
The new function y = 0.192x0.523 appears to fit the data well. To get more
information we could again examine a residual plot.
Press [Y=] to display the Y= editor.
Press [ ] [ENTER] to deselect Y1.
Press [ ∆ ] [ENTER] to turn off Plot 1. Press [ ] [ENTER] to turn on Plot
2.
Press [ZOOM] 9 to select ZoomStat from the
Zoom menu. Plot 2, a scatter plot of the
residuals, is displayed.
MATHEMATICS
41
PENDULUM LENGTHS AND PERIODS
The new residual plot shows that the residuals are random in sign, with
the residuals increasing in magnitude as the string length increases.
With this model, the largest residual is about 0.041 and the smallest
negative residual is about –0.027. All other residuals are less than 0.02
in magnitude.
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.
To predict the periods for a pendulum with
string lengths of 20cm and 50cm continue
with the steps (explained earlier) to paste
Y1 onto the home screen. Obtain Y1(20)
and Y 1(50) as shown.
Note: Since a string length of 50cm exceeds the lengths in the data set,
and since residuals appear to be increasing as string length increases, we
would expect more error with this estimate. This latter estimate is called
extrapolation.
42
MATHEMATICS