1.
11syllabussyllabusrrefefererenceence
Strand:
Statistics and probability
Core topic:
Data collection and
presentation
In thisIn this chachapterpter
11A Scatterplots
11B Regression lines
11C Time series and trend lines
Scatterplots
and time series
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452 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d
Introduction
The manager of a small ski resort has a problem. He wants to be able to predict the
number of skiers using his resort each weekend in advance, so that he can organise
additional resort stafﬁng and catering if needed. He knows that good deep snow will
attract skiers in big numbers but scant covering is unlikely to attract a crowd. To
investigate the situation further, he collects the following data over twelve consecutive
weekends at his resort.
As there are two types of data in this example, they are called bivariate data. For
each item (weekend), two variables are considered (depth of snow and number of
skiers). When analysing bivariate data, we are interested in examining the relationship
between the two variables. In the case of the ski resort data, the manager might be
interested in answering the following questions.
1. Are visitor numbers related to depth of snow?
2. If there is a relationship between visitor numbers and depth of snow, is it always true
or is it just a guide? In other words, how strong is the relationship?
3. How much conﬁdence could be placed in the prediction?
In this chapter we shall examine how questions such as these can be answered by the
appropriate presentation of data.
Depth of snow (m) Number of skiers
0.5 120
0.8 250
2.1 500
3.6 780
1.4 300
1.5 280
1.8 410
2.7 320
3.2 640
2.4 540
2.6 530
1.7 200
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C h a p t e r 1 1 S c a t t e r p l o t s a n d t i m e s e r i e s 453
1 When graphing pairs of variables, we commonly have an ‘independent variable’ and a
‘dependent variable’. Explain these terms.
2 By convention, which of the above two variable types is graphed on the horizontal
axis?
3 For the following pairs of variables, state which variable would be classiﬁed as the
dependent variable.
a age and height
b distance travelled and time taken (at a ﬁxed speed)
c temperature and elevation
d blood alcohol level and reaction time
e IQ and results on an academic test
f overtime pay and hours worked
g value of a car and its age
h time taken to travel a given distance and speed of travel
4 For each of the pairs of variables in question 3, state what happens to the value of the
second variable when there is an increase in the value of the ﬁrst variable.
5 If we collected data for each of the pairs of variables in question 3 and plotted the
points, we would obtain a set of points scattered over the Cartesian plane. Consider a
straight line which could be drawn indicating the general trend in each case.
a Draw up a set of labelled axes (no scale) for each case in question 3.
b Sketch a straight line which would show the general trend of the relationship
between the two variables.
Scatterplots
We shall now look more closely at the data collected
by the ski resort manager, and at the three questions
he posed. To help answer these questions, the data
can be arranged on a scatterplot. Each of the data
points is represented by a single visible point on the
graph.
When drawing a scatterplot, it is important to choose the correct variable to assign to
each of the axes. The convention is to place the independent variable on the x-axis
and the dependent variable on the y-axis. The independent variable in an experiment or
investigation is the variable that is deliberately controlled or adjusted by the
investigator. The dependent variable is the variable that responds to changes in the
independent variable.
Neither of the variables involved in the ski resort data was controlled directly
by the investigator but ‘Number of skiers’ would be considered the dependent
variable because it is likely to change depending on depth of snow. (The snow
0
200
400
600
800
0 2 31 4
Numberofskiers
Depth of snow (m)
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454 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d
depth does not depend on numbers of skiers.) As ‘Number of skiers’ is the
dependent variable, we graph it on the y-axis and the ‘Depth of snow’ on the
x-axis.
Notice how the scatterplot for the ski resort data shows a general upward trend.
It is not a perfectly straight line but it is still clear that a general trend or
relationship has formed: as the depth of snow increases, so too does the number
of skiers.
Once the scatterplot has been drawn, we can determine if any pattern is evident.
Worked example 1 shows how, as a general rule, as height increases so does
mass.
We can also look to see if the pattern is linear. In worked example 1, although the
points are not in a perfect straight line, they approximate a straight line.
The ﬁgures below show examples of linear and non-linear relationships.
Linear relationships
Positive relationship between variables; Negative relationship between variables;
that is, as one variable increases, the that is, as one variable increases, the
other variable also increases. other variable decreases.
The table below shows the height and mass of ten Year 11 students.
Display the data on a scatterplot.
Height (cm) 120 124 130 135 142 148 160 164 170 175
Mass (kg) 45 50 54 59 60 65 70 78 75 80
THINK WRITE
Show the height on the x-axis and the
mass on the y-axis.
Plot the point given by each pair.
1
40
50
60
70
80
100 140 160120 180
Mass(kg)
Height (cm)
2
1WORKEDExample
0
y
x 0
y
x
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C h a p t e r 1 1 S c a t t e r p l o t s a n d t i m e s e r i e s 455
Non-linear relationships
In other cases it may be that there is no relationship
at all between the two variables. Such a scatterplot
would look like the one shown on the right.
0
y
x 0
y
x
0
y
x 0
y
x
0
y
x
The table below shows the length and mass of a dozen eggs.
a Display this information in a scatterplot.
b Determine if there is any relationship between the length and mass of the eggs and state
if the relationship is linear.
Length (cm) 6.2 3.9 4.5 5.8 7.2 7.6 6.1 6.7 7.3 5.1 6.0 7.3
Mass (g) 60 15 25 50 95 110 55 75 95 35 54 96
THINK WRITE
a Display length on the x-axis and
mass on the y-axis.
a
Plot the point given by each pair.
b Study the scatterplot to see if mass
increases as length increases.
b As length increases, so does the mass of the
egg.
Study the scatterplot to see if the points
seem to approximate a straight line.
The points do not approximate a straight line
and so the relationship is not linear.
1
0
20
40
60
80
100
120
0 81 2 3 4 5 6 7
Mass(g)
Length (cm)
2
1
2
2WORKEDExample
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456 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d
An understanding of the relationship between two variables offers a tool for future
planning by businesses. Umbrellas can be stocked in preparation for the wet season,
heaters can be stocked for an on-coming winter, the stocks of ice-cream are much lower
during winter. An appreciation of these relationships provides an opportunity for a busi-
ness to gain an edge over its competitors.
The graphics calculator and
statistical data
A graphics calculator is a great labour-saving device when it comes to statistical
calculations and display of data. If you have one readily available, its use in this
chapter will greatly reduce your manual calculations.
Let us repeat worked example 2 using a graphics calculator. (Instructions here
apply to the Texas Instrument TI–83 calculator. Other brands offer similar
functions.)
1 Clear the Y= editor and turn off any existing plots by pressing [STAT PLOT]
and choosing 4: PlotsOff.
2 Press and select 1:Edit. Enter the x-data (length) in L1 and y-data (mass)
in L2.
3 Press [STAT PLOT] then press to select the options for Plot 1.
4 Turn the plot ON; select the type as a scatterplot; the Xlist as L1; and the Ylist as
L2.
5 Press and choose 9:ZoomStat.
6 The scatterplot shown in worked example 2 will be shown on the screen. Press
, then use the arrow keys to examine the plot.
7 To examine the summary statistics of the x- and y-variables, press , then
use the right arrow key to highlight CALC. Option 1 displays statistics of the
variable x (mean, standard deviation, quartiles, minimum and maximum values
etc.) while option 2 displays statistics for both variables.
Note the LinReg(ax + b) (linear regression function) available. We will use this
function later to determine the equation of the best straight line which ﬁts the
data in the scatterplot.
investigat
ioninv
estigat
ion
2nd
STAT
2nd ENTER
ZOOM
TRACE
STAT
remember
1. A scatterplot is a graph that is used to compare two variables.
2. One variable (the independent variable) is on the horizontal axis and the other
variable (the dependent variable) is on the vertical axis.
3. Points are plotted by the pair formed by each variable.
4. A relationship between the variables exists if one increases as the other
increases or if one decreases as the other increases.
5. If the points on the scatterplot seem to approximate a straight line, the
relationship can be said to be linear.
remember
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C h a p t e r 1 1 S c a t t e r p l o t s a n d t i m e s e r i e s 457
Scatterplots
Use a graphics calculator if you have one available.
1 The table below shows the marks obtained by a group of 10 students in history and
geography.
Display this information on a scatterplot.
2 The table below shows the maximum temperature each day together with the number
of people who attend the cinema that day.
Display the information on a scatterplot.
3 The table below shows the wages of 20 people and the amount of money they spend
each week on entertainment.
Display this information on a scatterplot.
4 The table below shows the marks obtained by nine students in English and History.
a Display the information on a scatterplot.
b Is there any relationship between the marks obtained in English and in History? If
there does appear to be a relationship, is the relationship linear?
5 The table below shows the daily temperature and the number of hot pies sold at the
school canteen.
a Display the information on a scatterplot.
b Determine if there appears to be any relationship between the two variables and if
the relationship appears to be linear.
History 36 65 82 72 58 39 58 74 82 66
Geography 45 78 66 72 50 51 61 70 60 88
Temperature (°C) 25 33 30 22 15 18 27 22 28 20
No. at cinema 256 184 190 312 458 401 200 357 312 423
Wages ($) 370 380 500 510 395 430 535 490 495 550
Amount spent on
entertainment ($)
55 85 150 75 145 100 130 115 70 150
Wages ($) 810 460 475 520 530 475 610 780 350 460
Amount spent on
entertainment ($)
220 50 100 150 140 160 90 130 40 50
English 55 20 27 33 73 18 37 51 79
History 72 37 53 74 73 44 59 55 84
Temperature (°C) 24 32 28 23 16 14 26 20 29 21
No. of pies sold 56 20 24 60 84 120 70 95 36 63
11A
WORKED
Example
1
E
XCEL Spread
sheet
Scatterplot
WORKED
Example
2
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458 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d
6 Container ships arriving on a wharf are unloaded by
work teams. The table below shows the number of
people in the work team and the time taken to unload
the container ship.
a Display the information on a scatterplot.
b Determine if there appears to be a relationship
between the number of people in the work team
and the time taken to unload the container ship. If
there is a relationship, does the relationship
appear to be linear?
7
Which of the following scatterplots does not display a linear relationship?
A B
C D
8
In which of the following is no relationship evident between the variables?
A B
C D
No. in work
team
15 18 12 19 22 21 17 16 18 20
Hours taken 20 16 25 15 14 13 18 20 17 14
mmultiple choiceultiple choice
y
x
y
x
y
x
y
x
mmultiple choiceultiple choice
y
x
y
x
y
x
y
x
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C h a p t e r 1 1 S c a t t e r p l o t s a n d t i m e s e r i e s 459
9 Give an example of a situation where the scatterplot may look like the ones below.
a b
Fitting straight lines to bivariate data
The process of ‘ﬁtting’ straight lines to bivariate data enables us to analyse relation-
ships between the data and possibly make predictions based on the given data set.
Fitting a straight line by eye (line of best ﬁt)
Consider the set of bivariate data points shown at right. In this
case the x-values could be the hand lengths of a group of people,
while y-values could be the lengths of their feet. We wish to
determine a linear relationship between these two random
variables.
Of course, there is no single straight line which would go
through all the points, so we can only estimate such a line.
Furthermore, the more closely the points appear to be on or near a straight line, the
more conﬁdent we are that such a linear relationship may exist and the more accurate
our ﬁtted line should be.
Consider the estimate, drawn ‘by eye’ in the ﬁgure below. It is clear that most of the
points are on or very close to this straight line. This line was easily drawn since the
points are very much part of an apparent linear relationship.
However, note that some points are below the line and some
are above it. Furthermore, if x is the hand length and y is the
foot length, it seems that people’s feet are generally longer than
their hands.
Regression analysis is concerned with ﬁnding these straight
lines of best ﬁt using various methods so that the number of points
above and below the lines are ‘balanced’.
Collecting bivariate data
1 Choose one of the following and collect data from within your class.
a Each person’s hand span and height.
b Each person’s resting heart rate and the time it takes for them to run 400 m.
c Each person’s mark in mathematics and in science.
2 Display the results on a scatterplot.
3 Discuss any relationship that may be evident between the two variables.
Work
SHEET 11.1
0
y
x 0
y
x
inv
estigat
ioninv
estigat
ion
y
x
y
x
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460 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d
Methods of drawing lines of best ﬁt
There are many different methods of ﬁtting a straight line by eye. They may appear
logical or even obvious but ﬁtting by eye involves a considerable margin of error. We
are going to consider only one method: ﬁtting the line by balancing the number of
points
The technique of balancing the number of points involves ﬁtting a line so that there
is an equal number of points above and below the line. For example, if there are 12
points in the data set, 6 should be above the line and 6 below it.
As mentioned above, estimating the position for the line of best ﬁt by eye involves a
considerable margin of error. For this reason, when making predictions of one variable
from another using this graphical technique, the forecasts obtained can be considered
rough estimates at best. A graphics calculator allows much more reliable predictions to
be made.
Draw the line of best ﬁt for the data in the ﬁgure using
the equal-number-of-points method.
THINK DRAW
Note that the number of points (n) is 8.
Fit a line where 4 points are above the line. Using a
clear plastic ruler, try to ﬁt the best line.
The ﬁrst attempt has only 3 points below the line
where there should be 4. Make reﬁnements.
The second attempt is an improvement, but the line is
too close to the points above it. Improve the position
of the line until a better ‘balance’ between upper and
lower points is achieved.
y
x
1
2 y
x
3 y
x
4 y
x
3WORKEDExample
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C h a p t e r 1 1 S c a t t e r p l o t s a n d t i m e s e r i e s 461
Regression lines
After drawing a scatterplot, we look for a relationship between the variables. If there is
a relationship, in many cases it appears to be linear. Once it has been established that a
linear relationship exists, we can introduce a regression line that will enable us to make
predictions about the two variables.
Consider the example at the beginning of the
chapter where the manager of the ski resort wants
to predict the number of skiers at the resort on the
weekend. Based on past data, we drew a
scatterplot.
On the scatterplot, we can draw the line of best ﬁt.
This line of best ﬁt can be extended and then used
to make predictions about the data. When this is
done, the line is called a regression line. In the
above example the depth of snow can then be used
to predict the number of visitors to the resort.
0
200
400
600
800
0 2 31 4
Numberofskiers
Depth of snow (m)
0
200
400
600
800
0 2 31 4
Numberofskiers Depth of snow (m)
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462 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d
Making predictions
If a linear relationship exists between a pair of variables, it is useful to be able to make
predictions of one variable from the other. There are two methods available:
1. Algebraically — we could ﬁnd an equation for this regression line (or line of best
ﬁt), substitute the value of the known variable and so calculate the value of the
unknown variable. A graphics calculator will generate the equation of the regression
line from data entered in lists.
2. Graphically — we can draw lines horizontally or vertically from the known value
to the line, then read off the corresponding value of the unknown variable. Using a
graphics calculator, we can use the ‘trace’ function to determine variable values on
the line.
Algebraic predictions
Using a graphics calculator
If you have access to a graphics calculator, the equation of the regression line can
readily be produced from the two sets of entered data.
Manually
To calculate the line of best ﬁt, we must determine the gradient and y-intercept of the
drawn line.
Equation of a line of best ﬁt
or regression line
To have the graphics calculator automatically ﬁnd the equation of a line of best ﬁt
(also called a regression line), follow these steps. The example below uses the
skiers and snow depth data given at the beginning of this chapter.
1. Enter the depth data in L1 and number of skiers in L2 (ﬁrst press , and
select EDIT and 1:Edit...).
2. Press , select CALC and choose 4:LinReg(ax + b).
3. Enter L1, L2, Y1 (that is, press [L1] [L2] ; Y1 is found
under VARS/Y-VARS/Function).
4. Press .
5. Ensure a STATPLOT is set up by entering [STATPLOT], setting Plot 1 to ON,
setting the Type for a scatterplot, Xlist as L1 and Ylist as L2.
6. Adjust WINDOW settings to cope with extremes of the data, then press GRAPH.
7. Use [CALC] and select 1:Value or press to ﬁnd values on the line.
8. The graphics calculator shows the line of best ﬁt to have the equation
y = 186.4x + 28.3.
inv
estigat
ioninv
estigat
ion
STAT
STAT
2nd , 2nd ,
ENTER
2nd
2nd TRACE
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C h a p t e r 1 1 S c a t t e r p l o t s a n d t i m e s e r i e s 463
Once the regression line has been drawn, to make predictions, it is useful to ﬁnd its
equation.
The equation of a straight line has the form y = mx + c, where m is the gradient and
c is the y-intercept. The gradient of a regression line is best found with a ruler and
pencil by measuring the rise and the run from two points on the line of regression. The
gradient is then found using:
The y-intercept can then be seen by noting the point where the line crosses the y-axis.
Consider worked example 3.
From this ﬁgure we can see that the equation of
the regression line is approximately y = x + 3.
That is, chemistry = × physics + 3.
A teacher may then be able to estimate a student’s
chemistry mark by using their result in physics. For
example a student who got a mark of 5 in physics may
get:
y = × 5 + 3
= 6
The teacher would estimate a mark of 6 for this student in chemistry.
Use a graphics calculator to determine the equation of the regression line for the
relationship between the chemistry and physics marks. Your resulting equation should
be y = 0.6x + 2.8 (close to the one determined here).
The table below shows the marks of 10 students in a physics and chemistry quiz.
Draw a scatterplot and on the graph show the regression line.
Physics 2 5 6 7 7 8 8 9 9 10
Chemistry 4 7 5 8 6 6 9 7 10 9
THINK WRITE
Put physics on the x-axis and chemistry
on the y-axis.
Plot the point corresponding to each
pair of marks.
Add a line of regression.
1
0
2
4
6
8
10
0 4 62 8 10
Chemistry
Physics
2
3
4WORKEDExample
SkillS
HEET 11.1Gradient m( )
rise
run
--------=
5
8
---
0
2
4
6
8
10
0 4 62 8
run = 8
rise = 5
y-intercept = approx. 3
10
Chemistry
Physics
5
8
---
Cabri Geo
metry
Gradient
Cabri Geo
metry
Graphs
and
y-intercept
5
8
---
1
8
---
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464 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d
The following table shows the bus fare charged by a bus company for different distances.
a Represent the data using a scatterplot and add the regression line.
b Find the gradient and y-intercept of the regression line; hence, ﬁnd its equation,
manually.
c Enter the data as lists into a graphics calculator and determine the equation of the
regression line.
Distance (km) Fare ($)
1.5 2.10
0.5 2.00
7.5 4.50
6 4.00
6 4.50
2.5 2.60
0.5 2.10
8 4.50
4 3.50
3 3.00
THINK WRITE
a Show the distance on the x-axis and
the fare on the y-axis.
a
Plot the points given by each pair.
Add the regression line to the graph.
b Take two points on or closest to the
regression line and ﬁnd the gradient
between them by substituting in the
formula and simplifying.
b
rise = 2, run = 5.5
m =
≈ 0.36
1
0.00
1.00
2.00
3.00
4.00
5.00
0 81 2 3 4 5 6 7
Fare($)
Distance (km)
2
3
1
m
rise
run
--------=
0.00
1.00
2.00
3.00
4.00
5.00
0 8
run = 5.5
rise = 2
1 2 3 4 5 6 7
Fare($)
Distance (km)
2
5.5
-------
5WORKEDExample
EXCE
L Spreadshe
et
Gradient
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C h a p t e r 1 1 S c a t t e r p l o t s a n d t i m e s e r i e s 465
Once the equation of the regression line has been found, it is possible to use this equation
to make predictions by substituting the value of the given variable into it.
Graphical predictions
As mentioned earlier, once the regression line has been drawn, horizontal or vertical
lines can be added from the axes to the line to read off the value of the unknown vari-
able. Remember that the values so determined are only estimates, and not exact values.
THINK WRITE
Note the y-intercept. y-intercept = 1.8
Substitute into the formula y = mx + c
to ﬁnd the equation.
y = 0.36x + 1.8
c Enter distances into L1 and fares into
L2.
c The equation of the regression line is
y = 0.37x + 1.8
Follow the procedures outlined
previously to determine the equation
of the regression line.
2
3
1
2
A casino records the number of people, N, playing a jackpot game and the prize money, p,
for that game and plots the results on a scatterplot. The regression line is found to have the
equation N = 0.07p + 220.
a Find the number of people playing when the prize money is $2500.
b Find the likely prize on offer when there are 500 people playing.
THINK WRITE
a Write the equation of the regression
line.
a N = 0.07p + 220
Substitute 2500 for p. N = 0.07 × 2500 + 220
Calculate N. = 395
Give a written answer. There would be approximately 395 people
playing.
b Write the equation of the regression line. b N = 0.07p + 220
Substitute 500 for N. 500 = 0.07p + 220
Solve the equation. 280 = 0.07p
p =
p = 4000
Give a written answer. The prize would be approximately $4000.
1
2
3
4
1
2
3
280
0.07
----------
4
6WORKEDExample
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466 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d
Causality
The term causality refers to one variable
causing another. For example, there is a strong
relationship between a person’s shoe size and
shirt size (as one tends to increase, the other
also tends to increase). However, this does
not mean that one causes the other.
There also appears to be a strong
relationship between the number of cigarettes
smoked and the chance of contracting lung
cancer. In this case, we are advised that
smoking causes lung cancer.
We cannot draw the conclusion that one
variable in a relationship causes another just
because there is a strong relationship between
the two. In fact, sometimes the strong
relationship between two variables is caused
by their relationship to a third variable. In the example of shoe and shirt sizes,
above, it is more likely that they are both dependent on age.
Consider the graph drawn in worked example 5 representing bus fares for travelling
various distances. Graphically estimate:
a the fare for a journey of 5 km
b how far you could expect to travel for a cost of $3.25.
THINK WRITE
a Find the value of 5 km on the x-axis. a
Draw a vertical line from this point
to meet the regression line.
Draw a horizontal line from this
point to the y-axis.
Read this value on the y-axis. A distance of 5 km of bus travel would cost
about $3.60.
b Locate $3.25 on the y-axis. b
Draw a horizontal line from this
point to the regression line.
From this point, draw a vertical line
to the x-axis.
Read off the x-value at this point. For $3.25 you could expect to travel about
4 km.
1
0.00
1.00
2.00
3.00
4.00
5.00
0 1 2 3 4 5 6 7
Fare($)
Distance (km)
$3.60
$3.25
5 km
4 km
2
3
4
1
2
3
4
7WORKEDExampleinv
estigat
ioninv
estigat
ion
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17.
C h a p t e r 1 1 S c a t t e r p l o t s a n d t i m e s e r i e s 467
Interpolation and extrapolation
We use the term interpolation for the making of predictions from a graph’s regression
line from within the bounds of the original experimental data.
Data may be interpolated either algebraically or graphically as shown by the pre-
vious examples.
We use the term extrapolation for the making of predictions from a graph’s
regression line from outside the bounds of the original experimental data.
Suppose, in the ski resort problem,
that we wished to ﬁnd the number of
skiers if the depth of snow was 4.5 m.
The data could be extrapolated
graphically by ﬁrst extending the trend
line to make the prediction possible.
The graph shows that about 900
skiers would be attracted to the resort
if the snow depth was 4.5 m. How
reliable is this prediction?
Reliability of predictions
Results predicted (whether algebraically or graphically) from the line of best ﬁt of a
scatterplot can be considered reliable only if:
1. a reasonably large number of points are used to draw the scatterplot,
2. the plotted points lie reasonably close to the line of best ﬁt,
3. the predictions are made using interpolation and not extrapolation.
Extrapolated results can never
be considered to be reliable because
when extrapolation is used we are
assuming that the relationship
holds true for untested values.
In the case of the ski resort data
we would be assuming that the
relationship continues to hold true
even when the snow depth is
extreme. This may or may not
be the case. It might be that there
In a small group, for each of the following topics, discuss:
a whether a positive or negative relationship exists between the variables
b how strong the relationship might be
c whether one variable causes the other.
1 Hours of study and exam marks.
2 Hours of exercise and resting pulse rate.
3 Weight and shirt size.
4 The number of hotels and churches in country towns.
5 The number of motels in a town and the number of ﬂights landing at the nearest
airport.
6 Age and the time taken to run 100 metres.
0
0
200
400
600
800
1000
1 2 3 4 5
Snow depth (m)
Numberofskiers
0
0
200
400
600
800
1000
1 2 3 4 5
Snow depth (m)
Numberofskiers
MQ Maths A Yr 11 - 11 Page 467 Thursday, July 5, 2001 11:03 AM
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468 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d
are only a certain number of skiers in
the population — which places a limit
on the maximum number visiting the
resort. In this situation the trend line,
if continued, may look like that in the
graph above.
Alternatively, it might be that when
the snow depth is extreme then skiers
cannot even get to the slopes.
So — take care when extrapolating data!
Crayﬁsh
A graphics calculator is recommended for this activity.
Rock lobsters (crayﬁsh) are sized according to the length of their carapace (main
body shell).
The table below gives the age and carapace length of 15 male rock lobsters.
inv
estigat
ioninv
estigat
ion
Age (years)
Length of
carapace (mm) Age (years)
Length of
carapace (mm)
3 65 14 210
2.5 59 4.5 82
4.5 80 3.5 74
3.25 68 2.25 51
7.75 130 1.76 48
8 150 10 171
6.5 112 9.5 160
12 200
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19.
C h a p t e r 1 1 S c a t t e r p l o t s a n d t i m e s e r i e s 469
1 Draw a scatterplot of the data.
2 Draw in the line of best ﬁt.
3 Suggest reasons why the points on the scatterplot are not perfectly linear.
4 Find an equation that represents the relationship between the length of the rock
lobster, L, and its age, a.
5 Find the likely size of a 5-year-old male rock lobster.
6 Find the likely size of a 16-year-old male rock lobster.
7 Rock lobsters reach sexual maturity when their carapace length is
approximately 65 mm. Find the age of the rock lobster at this stage.
8 The Fisheries Department wishes to set minimum size restrictions so that the
rock lobsters have three full years from the time of sexual maturity in which to
breed before they can be legally caught. What size should govern the taking of
male crayﬁsh?
9 Which of your answers to parts 5 to 9 do you consider reliable? Why?
World population
The table below shows world population during the years 1955 to 1985.
1 Use a graphics calculator to create a
scatterplot of the data.
2 Does the scatterplot of the Population-versus-
Year data appear to be linear?
3 Use the graphics calculator to ﬁt a regression
line to the data; hence ﬁnd the equation of the
line and graph it on the scatterplot.
4 Use the equation to predict the world
population in 1950.
5 Use the equation to predict the world
population in 1982.
6 Use the equation to predict the world
population in 1997.
7 Which of your answers to parts 4 to 6 do you consider reliable? Why?
8 In fact the world population in 1997 was 5840 million. Account for the
discrepancy between this and your answer to part 6.
9 What would you estimate the world population to be in 2010?
inv
estigat
ioninv
estigat
ion
Year 1955 1960 1965 1970 1975 1980 1985
World population
(millions)
2757 3037 3354 3696 4066 4432 4828
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470 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d
Regression lines
Note: For many of these questions your answers may differ somewhat from those in
the back of the book. The answers are provided as a guide but there are likely to be
individual differences when ﬁtting straight lines by eye.
Where appropriate, the use of a graphics calculator is recommended if one is
available.
1 Fit a straight line to the data in the scatterplots using the equal-number-of-points method.
a b c
d e
remember
1. To ﬁt a straight line to bivariate data by eye, the line is drawn by balancing the
number of points, ensuring an equal number of points above and below the
ﬁtted line. This line is called the line of best ﬁt or regression line.
2. The equation of the regression line can be found by measuring the
gradient and the y-intercept of the regression line and using the formula
y = mx + c. Sometimes the gradient of the regression line will be negative. This
occurs if one variable increases while the other variable decreases.
3. The equation of the regression line can also be determined using a graphics
calculator.
4. Once the equation of the regression line has been found, it can then be used to
make predictions about the variables.
5. Predictions can also be made graphically by drawing lines from the axes to the
regression line.
6. Interpolation involves making predictions within the bounds of the original
data.
7. Extrapolation involves making predictions beyond the bounds of the original
data.
8. Predictions can be considered reliable if they are obtained using interpolation
from a large number of points which lie reasonably close to the line of best ﬁt.
remember
11B
WORKED
Example
3 y
x
y
x
y
x
y
x
y
x
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C h a p t e r 1 1 S c a t t e r p l o t s a n d t i m e s e r i e s 471
2 The table below shows the marks achieved by a class of students in English and
Mathematics.
Show these data on a scatterplot and on the graph show the regression line.
3 Position the straight line of best ﬁt through each of the following graphs and ﬁnd the
equation of each.
a b
c
4 In an experiment, a student measures the length of a spring (L) when different
masses (M) are attached to it. Her results are shown below.
a Draw a scatterplot of the data and on it draw
the line of regression.
b Find the gradient and y-intercept of the
regression line and, hence, ﬁnd the equation
of the regression line. Write your equation in terms of the variables L and M.
English 64 75 81 63 32 56 47 59 73 64
Maths 76 62 89 56 49 57 53 72 80 50
Mass (g) Length of spring (mm)
0 220
100 225
200 231
300 235
400 242
500 246
600 250
700 254
800 259
900 264
WORKED
Example
4
Math
cad
Gradient and
y-intercept
0
10
20
30
40
50
60
70
y
x0 4 62 8 10
0
10
20
30
40
50
60
70
0 40 6020 80 100 120 x
y
0
500
1000
1500
2000
2500
3000
0 10 155 20 25
y
x
WORKED
Example
5
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472 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d
5 A scientist who measures the volume of a gas at different temperatures provides the
following table of values.
a Draw a scatterplot of the data and on it draw the line of regression.
b Give the equation of the line of best ﬁt. Write your equation in terms of the
variables: volume of gas, V, and its temperature, T.
6 A sports scientist is interested in the importance of muscle bulk to strength. He
measures the biceps circumference of ten people and tests their strength by asking
them to complete a lift test. His results are given in the following table.
a Draw a scatterplot of the data and draw the line of regression.
b Find a rule for determining the ability of a person to complete a lift test, S, from
the circumference of their biceps, B.
Temperature (°C) Volume (L)
−40 1.2
−30 1.9
−20 2.4
0 3.1
10 3.6
20 4.1
30 4.8
40 5.3
50 6.1
60 6.7
Circumference of biceps (cm) Lift test (kg)
25 50
25 52
27 58
28 51
30 60
30 62
31 53
33 62
34 61
36 66
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C h a p t e r 1 1 S c a t t e r p l o t s a n d t i m e s e r i e s 473
7 A taxi company adjusts its meters so that the fare is charged according to the
following equation: F = 1.2d + 3 where F is the fare, in dollars, and d is the distance
travelled, in km.
a Find the fare charged for a distance of 12 km.
b Find the fare charged for a distance of 4.5 km.
c Find the distance that could be covered on a fare of $27.
d Find the distance that could be covered on a fare of $13.20.
8 Detectives can use the equation H = 6.1f − 5 to estimate the height of a burglar who
leaves footprints behind. (H is the height of the burglar, in cm, and f is the length of
the footprint.)
a Find the height of a burglar whose footprint is 27 cm in length.
b Find the height of a burglar whose footprint is 30 cm in length.
c Find the footprint length of a burglar of height 185 cm. (Give your answer correct
to 2 decimal places.)
d Find the footprint length of a burglar of height 152 cm. (Give your answer correct
to 2 decimal places.)
9 A football match pie seller ﬁnds that the number of pies sold is related to the
temperature of the day. The situation could be modelled by the equation
N = 870 − 23t, where N is the number of pies sold and t is the temperature of the day.
a Find the number of pies sold if the temperature was 5 degrees.
b Find the number of pies sold if the temperature was 25 degrees.
c Find the likely temperature if 400 pies were sold.
d How hot would the day have to be before the pie seller sold no pies at all?
10 The following table shows the average annual costs of running a car. It includes all
ﬁxed costs (registration, insurance etc.) as well as running costs (petrol, repairs etc.).
a Draw a scatterplot of the data.
b Draw in the line of best ﬁt.
c Find an equation which represents the relationship between the cost of running a
vehicle, C, and the distance travelled, d.
d Use your graph and its equation to ﬁnd:
i the annual cost of running a car if it is driven 15 000 km
ii the annual cost of running a car if it is driven 1000 km
iii the likely number of kilometres driven if the annual costs were $8000
iv the likely number of kilometres driven if the annual costs were $16 000.
Distance (km) Annual cost ($)
5 000 4 000
10 000 6 400
15 000 8 400
20 000 10 400
25 000 12 400
30 000 14 400
WORKED
Example
6
SkillS
HEET 11.4
SkillS
HEET 11.3
SkillS
HEET 11.2
WORKED
Example
7
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474 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d
11 A market researcher ﬁnds that the number of people who would purchase ‘Wise-up’
(the thinking man’s deodorant) is related to its price. He provides the following table
of values.
a Draw a scatterplot of the data.
b Draw in the line of best ﬁt.
c Find an equation that represents the relationship between the number of cans of
‘Wise-up’ sold, N (in thousands), and its price, p.
d Use the equation to predict the number of cans sold each week if:
i the price was $3.10 ii the price was $4.60.
e At what price should ‘Wise-up’be sold if the manufacturers wished to sell 80 000 cans?
f Given that the manufacturers of ‘Wise-up’ can produce only 100 000 cans each
week, at what price should it be sold to maximise production?
12 The following table gives the adult return air fares between some Australian cities.
a Draw a scatterplot of the data and on it draw the regression line.
b Find an equation that represents the relationship between the air fare, A, and the
distance travelled, d.
c Use the equation to predict the likely air fare (to the nearest dollar) from:
i Sydney to the Gold Coast (671 km)
ii Perth to Adelaide (2125 km)
iii Hobart to Sydney (1024 km)
iv Perth to Sydney (3295 km).
Price ($) Weekly sales (× 1000)
1.40 105
1.60 101
1.80 97
2.00 93
2.20 89
2.40 85
2.60 81
2.80 77
3.00 73
3.20 69
3.40 65
City Distance (km) Price ($)
Melbourne–Sydney 713 580
Perth–Melbourne 2728 1490
Adelaide–Sydney 1172 790
Brisbane–Melbourne 1370 890
Hobart–Melbourne 559 520
Hobart–Adelaide 1144 820
Adelaide–Melbourne 669 570
EXCE
L Spreadshe
et
Interpolation/
extrapolation
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C h a p t e r 1 1 S c a t t e r p l o t s a n d t i m e s e r i e s 475
An electrical repair business charges its customers using the formula C = 40h + 35, where
C is the cost of the repairs and h is the time taken for the repairs, in hours.
Find the cost of a repair job that took:
1 2 hours 2 5 hours 3 1 hour and 15 minutes.
Estimate the time taken (in hours and minutes) for repairs if the cost of the repairs was:
4 $175 5 $275 6 $145.
The information below is to be used for questions 7–10.
A survey relating exam marks to the amount of television watched ﬁnds that the
regression line has the equation M = 95 − 15t, where M is the mark obtained and t is the
average number of hours of television watched each night by the students.
7 Estimate the mark of a person who averages one hour of television per night.
8 Estimate the mark of a person who watches an average 4 hours of television per night.
9 Estimate the amount of television watched per night by a person who scores a mark of
65.
10 Jodie scored 27.5 on the exam. Estimate the average amount of television that Jodie
watches each night.
Time series and trend lines
We have looked at bivariate, or (x, y), data where both x and y could vary independ-
ently. We shall now consider cases where the x-variable is time and, generally, where
time goes up in even increments such as hours, days, weeks or even years. In these
cases we have what is called a time series. The main purpose of a time series is to see
how some quantity varies with time. For example, a company may wish to record its
daily sales ﬁgures over a 10-day period.
We could also make a graph of this
time series as shown in the ﬁgure.
As can be seen from this graph, there
seems to be a trend upwards — clearly,
this company is increasing its revenues!
Time Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7 Day 8 Day 9 Day 10
Sales ($) 5200 5600 6100 6200 7000 7100 7500 7700 7700 8000
1
Days
Sales($)
4000
5000
6000
7000
8000
9000
10000
0 1 2 3 4 5 6 7 8 9 10 t
MQ Maths A Yr 11 - 11 Page 475 Friday, July 6, 2001 2:37 PM
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476 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d
Types of trend
Many types of trend exist. They can be classiﬁed as secular, seasonal, cyclic and
random.
Secular trends
If over a reasonably long period of time a trend appears to be either increasing or
decreasing steadily, with no major changes of direction, then it is called a secular
trend. It is important to look at the data over a long period. If the trend in the ﬁgure on
the previous page continued for, say, 30 days, then we could safely conclude that the
company was indeed becoming more proﬁtable. What appears to be a steady increase
over a short term — say, stock market share prices — can turn out to be something
quite different over the long run.
Seasonal trends
Certain data seem to ﬂuctuate during the year, as the seasons change. Consequently,
this is termed a seasonal trend. The most obvious example of a seasonal trend would
be the average monthly maximum temperatures over a year.
A sample of this type of trend is
shown in the ﬁgure.
Note that the months have been
coded, so that 1 = Jan., 2 = Feb., and
so on. From which hemisphere of the
world would these data come?
Cyclic trends
Like seasonal trends, cyclic trends show ﬂuctuations upwards and downwards, but not
according to season. Businesses often have cycles where at times proﬁts increase, then
decline, then increase again. A good example of this would be the sales of a new major
software product, such as a word processor. At ﬁrst, sales are slow; then they pick up as
the product becomes popular. When enough people have bought the product, sales may
fall off until a new version of the product comes on the market, causing sales to
increase again. This cycle can be repeated many times, which is why there are many
versions of some software products.
Random trends
Trends may seem to occur at random. This can be caused by external events such as
ﬂoods, wars, new technologies or inventions, or anything else that results from random
causes. There is no obvious way to predict the direction of the trend or even when it
changes direction.
Months
Temp.(°C)
14
18
22
26
30
0 1 2 3 4 5 6 7 8 9 10 11 12 t
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27.
C h a p t e r 1 1 S c a t t e r p l o t s a n d t i m e s e r i e s 477
In the ﬁgure, there are a couple of minor
ﬂuctuations at t = 4 and t = 8, and a major one at
t = 13. The major ﬂuctuation could have been
caused by a change in government which positively
affected sales.
The trend line
If we want to predict the future values of a trend, it is important to be able to ﬁt a
straight line to the data that we already have. There are a number of techniques which
can be used to determine the trend line. We will use the method of ﬁtting the line ‘by
eye’ employing the ‘equal number of points’ technique that was used previously.
Trend lines in ABS census data
This activity can be undertaken using a spreadsheet or graphics calculator.
The Australian Bureau of Statistics has published data from the 1996 Australian
Census on their web site, http://www.abs.gov.au. (It will take some time for the
2001 Census material to be available. Download the updated data when it is
published.) An extract from their material shows the age breakdown year by year
for males and females to age 79 (groupings occur after this date).
Profits
14
18
22
26
30
0 2 4 6 8 10 12 14 16 t
Fit a straight line to the following time series data,
which represent the body temperature of a patient
with appendicitis, taken every hour.
THINK WRITE
Attempt to ﬁt a line using your eye.
By trial and error, a line such as the one
at right could be the trend line. Try to
balance the number of points either side
of the line.
Evaluate the trend. It is unlikely that the temperature will continue
to rise indeﬁnitely, but the line may be
signiﬁcant over the short term.
Hours
Temp.(°C)
30.0
30.5
31.0
31.5
32.0
32.5
33.0
0 1 2 3 4 5 6 7 8 9 10 t
1
Hours
Temp.(°C)
30.0
30.5
31.0
31.5
32.0
32.5
33.0
0 1 2 3 4 5 6 7 8 9 10 t
2
8WORKEDExample
inv
estigat
ioninv
estigat
ion
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28.
478 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d
AUSTRALIANBUREAUOFSTATISTICS1996CensusofPopulationandHousing
Australia7688965.464sqkms
B03AGEBYSEX
AllPersons
MaleFemalePersons
0years123101116506239607
1126232119980246212
2133747126933260680
3132414126111258525
4133393126489259882
0–46488876160191264906
5133908127701261609
6133759127144260903
7130505124238254743
8129595123650253245
9129424123389252813
5–96571916261221283313
10131934125566257500
11131046126007257053
12132320126563258883
13134123127098261221
14131117123939255056
10–146605406291731289713
15130655123635254290
16127911121142249053
17126772120884247656
18126441120841247282
19127615123792251407
15–196393946102941249688
20127481124020251501
21130978128427259405
22131793129213261006
23135793133924269717
24140685140143280828
20–246667306557271322457
25144157145410289567
26134645136527271172
27133443136049269492
28130488132804263292
29128474131264259738
25–296712076820541353261
MaleFemalePersons
30132867136256269123
31130269133606263875
32139492142313281805
33142209145143287352
34142150145761287911
30–346869877030791390066
35145277148560293837
36141012146205287217
37137969141896279865
38137076141660278736
39134507137849272356
35–396958417161701412011
40137481140360277841
41127118130373257491
42131417134898266315
43130649132966263615
44124503125515250018
40–446511686641121315280
45127342128336255678
46124358125646250004
47120734121165241899
48125216122796248012
49132224127795260019
45–496298746257381255612
50111979109136221115
5110151998391200082
5210248999391201880
539209589216181311
549028887136177424
50–54498370483442981812
558662483913170537
568389280821164713
578022578436158661
587796777312155279
597617974680150859
55–59404887395162800049
MaleFemalePersons
607332974559147888
616614366340132483
626786868018135886
636656068291134851
646628866370132658
60–64340188343578683766
656894772040140987
666598268925134907
676550568270133775
686373768776132513
696163966933128572
65–69325810344944670754
706062168794129415
715506062832117892
725412164498118619
735071762169112886
744818860441108629
70–74268707318734587441
754552358811104334
76405645390894472
77315584329874856
78297024288472586
79281224134569467
75–79175469240246415715
80–84103256174476277732
85–894371694151137867
90–94122863561747903
95–982331840810739
99yearsandover58721572744
Overseasvisitor6579873796139594
Total8849224904319917892423
MQ Maths A Yr 11 - 11 Page 478 Thursday, July 5, 2001 11:03 AM
29.
C h a p t e r 1 1 S c a t t e r p l o t s a n d t i m e s e r i e s 479
Time series and trend lines
Graphics calculators or spreadsheets can be used for the following where appropriate.
1 Identify whether the following trends are likely to be secular, seasonal, cyclic or
random:
a the amount of rainfall, per month, in
north Queensland
b the number of soldiers in the United
States army, measured annually
c the number of people living in Australia,
measured annually
d the share price of BHP, measured monthly
e the number of seats held by the Liberal
Party in Federal Parliament.
2 Fit a trend line to the data in the graph at right.
Consider this data set as a time series of age groupings 0–4, 5–9 etc. (Note that the
totals for these groupings occur within the data.)
1 Graph the trend of male numbers over time.
2 On the same graph, plot the female numbers over time.
3 Examine the differences in these two trends.
a Compare the numbers of males and females in Australia in each age
category.
b What has happened to the birth rate over time?
c Compare the age of death of males and females.
d Which age group represents the greatest proportion of the population?
4 Write a report of the age composition of Australians. Present your ﬁndings,
backed by reference to tables and graphs.
remember
1. Time series are a set of measurements taken over (usually) equally spaced time
intervals, such as hourly, daily, weekly, monthly or annually.
2. There are 4 basic types of trend:
(a) secular: increasing or decreasing steadily
(b) seasonal: varying from season to season
(c) cyclic: similar to seasonal but not tied to a calendar cycle
(d) random: varying from external causes happening at random.
remember
11C
Days
Temp.(°C)
10
15
20
25
30
35
40
0 1 2 3 4 5 6 7 8 9 10 t
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480 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d
3 Fit a trend line to the following data. What type of trend is best reﬂected by these data?
4 The monthly share prices of a recently privatised telephone company were recorded as
follows.
Graph the data (let 1 = Jan. 2001, 2 = Feb. 2001 . . . and so on) and ﬁt a trend line to the
data. Use this line to predict the share price in January 2002. Comment on the feas-
ibility of the predicted share price.
5 Plot the following monthly sales data for umbrellas. Fit a trend line. Discuss the type of
trend best reﬂected by the data and the limitations of your trend line.
6 Consider the data in the ﬁgure, which
represent the price of oranges over a
19-week period.
a Fit a straight trend line to the data.
b Predict the price in week 25.
7 The following table represents the quarterly sales ﬁgures (in thousands) of a popular
software product. Plot the data and ﬁt a trend line using the ‘equal number of points’
method. Discuss the type of trend best reﬂected by these data.
8 The number of employees at the Comnatpac Bank was recorded over a 10-month
period. Plot and ﬁt a trend line to the data. What would you say about the trend?
t 1 2 3 4 5 6 7 8 9 10 11 12
y 6 9 13 8 9 14 15 17 14 11 15 19
Date Jan. 01 Feb. 01 Mar. 01 Apr. 01 May 01 Jun. 01 Jul. 01 Aug. 01
Price ($) 2.50 2.70 3.00 3.20 3.60 3.70 3.90 4.20
Month Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov. Dec.
Sales 5 10 15 40 70 95 100 90 60 35 20 10
Quarter Q1-99 Q2-99 Q3-99 Q4-99 Q1-00 Q2-00 Q3-00 Q4-00 Q1-01 Q2-01 Q3-01 Q4-01
Sales 120 135 150 145 140 120 100 110 120 140 190 220
Month Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov. Dec.
Employees 6100 5700 5400 5200 4800 4400 4200 4000 3700 3300
WORKED
Example
8
Weeks
Price(cents)
20
40
60
80
100
0 5 10 15 20 25 t
Work
SHEET 11.2
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31.
C h a p t e r 1 1 S c a t t e r p l o t s a n d t i m e s e r i e s 481
Weather report for snowﬁelds
The Bureau of Meteorology publishes statistical data on their web site
(http://www.bom.gov.au). A visit to this site reveals climate averages for many
places throughout Australia. The following data set is the result of measurements
collected at Mount Buffalo Chalet over a number of years.
inv
estigat
ioninv
estigat
ion
CLIMATEAVERAGES—longtermmeanvaluesofweatherdata
083073MOUNTBUFFALOCHALETCommenced:1910Lastrecord:2000
Latitude:36.72SLongitude:146.82EElevation:1372.0mState:VIC
JANFEBMARAPRMAYJUNJULAUGSEPOCTNOVDECANNNo.%age
Yrscomp
MeanDailyMaxTemp(degC)19.518.916.411.67.94.93.74.68.010.814.217.511.643.368
Meanno.Days,Max>=40.0degC0.00.00.00.00.00.00.00.00.00.00.00.00.02.892
Meanno.Days,Max>=35.0degC0.00.00.00.00.00.00.00.00.00.00.00.00.02.892
Meanno.Days,Max>=30.0degC0.00.00.00.00.00.00.00.00.00.00.00.00.02.892
HighestMaxTemp(degC)29.023.921.724.418.912.214.012.814.818.521.529.529.52.894
MeanDailyMinTemp(degC)10.811.09.05.32.60.2–0.7–0.31.83.86.49.25.043.669
Meanno.Days,Min=<2.0degC0.70.00.74.317.322.327.326.015.511.34.71.7131.82.997
Meanno.Days,Min=<0.0degC0.00.00.31.37.013.720.019.09.55.71.30.378.22.997
LowestMinTemp(degC)1.73.30.0–0.6–1.9–4.5–5.3–4.4–4.4–3.6–1.1–0.6–5.32.986
Mean9amAirTemp(degC)14.915.012.98.75.62.71.72.44.97.410.313.28.440.889
Mean9amWet-bulbTemp(degC)10.611.29.76.43.81.40.61.23.05.07.29.65.940.688
Mean9amRelativeHumidity(%)5660647073788178716863616840.588
Mean9amWindSpeed(km/hr)6.96.48.17.79.08.410.19.39.87.97.27.58.210.165
Mean3pmWindSpeed(km/hr)4.93.44.35.36.67.98.98.17.96.86.36.36.52.175
MeanRainfall(mm)88.483.2100.8131.0197.3212.7230.8227.0196.8191.4130.1112.61902.085.596
Median(Decile5)Rainfall(mm)64.257.981.8108.3168.2181.4214.3225.1190.5159.5112.794.81884.675
Decile9Rainfall(mm)189.6184.2256.9278.5375.9392.2386.2356.2332.1399.3253.8256.82493.875
Decile1Rainfall(mm)12.19.215.122.456.656.385.186.465.545.149.727.61209.975
Meanno.ofRaindays6.66.57.49.012.413.414.615.513.512.810.38.9131.084.395
HighestMonthlyRainfall(mm)364.7412.9359.2764.1614.0607.4717.4557.7496.0525.3323.1374.685.596
LowestMonthlyRainfall(mm)1.30.04.80.00.032.02.321.431.74.010.45.185.596
HighestRecordedDailyRain(mm)193.0134.4174.0189.7173.0216.7157.0145.4195.0212.0108.6109.7216.785.196
Meanno.ofClearDays0.00.00.00.00.00.00.00.00.00.00.00.00.03.0100
Meanno.ofCloudyDays0.00.00.00.00.00.00.00.00.00.00.00.00.03.0100
MQ Maths A Yr 11 - 11 Page 481 Thursday, July 5, 2001 11:03 AM
32.
482 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d
The following article explains the ideal conditions for heavy snowfalls, and
graphically displays the recorded snow depth in the Snowy Mountains over the
years 1963 to 1995.
1 Carefully examine all the information presented.
2 Visit the web site for more information, if necessary.
3 Your school is planning a visit for your year level to the snowﬁelds some time
during the year. Analyse the data presented (together with any other you have
collected) and write a report to the organisers identifying:
a the general weather conditions in the snowﬁelds throughout the year (refer
to as many factors as possible)
b the best time of the year for the excursion (in your opinion)
c the type of weather you would hope for, prior to your visit, to produce good
snowfalls in the area.
Supply tables and graphs to support your recommendations.
Abundant snow seasons
The prime conditions for heavy snowfalls in theAustralianAlps are persistent strong westerlies
through the winter, which produce abundant precipitation and are generally accompanied by
relatively low temperatures. There is no particular relationship with El Nino or La Nina, though
the tendency for less precipitation in El Nino years usually leads to a poor season. The favoured
conditions are generally different to those which produce snow at low levels, for which a strong
cold outbreak (with generally south to southwesterly winds) is required.
On the Snowy Mountains the greatest snow depth in the 46 years from 1954 to 1999 was just
over 3.5 metres, achieved in both 1961 and 1981. The winter of 1981 was also exceptionally
wet in SouthAustralia, Victoria and southern New South Wales: a severe storm struckAdelaide
on 1 June, and this heralded three months of almost uninterrupted inﬂuence by depressions in
the westerlies, which brought copious rain and snow. At Perisher Valley the snow depth rose
steadily to 3.6 metres in early September before beginning to decline. Snow didn’t disappear
completely until mid-November. More recently, 1992 was a very long season with deep snow.
At Spencer Creek and Perisher Valley in NSW the snow depth reached the high level of three
metres at regular four-year intervals: 1956, 1960, 1964 and 1968. These were also years of
heavy snow cover on the Victorian Alps. In each case weather patterns were characterised by
strong, persistent westerlies and abundant rain at lower levels. In each case the snow cover
persisted until late Novem-
ber, and in 1956, into De-
cember. In 1964 a
particularly stormy third
week of July dumped pro-
digious quantities of snow,
stranding people at Falls
Creek and Mt Hotham.
In Australia, the annual
spring melting of the
mountain snow is not in it-
self sufﬁcient to cause
ﬂooding. However, in
years when the snow depth
is great, a combination of
heavy rain and mild to
warm conditions can aug-
ment spring ﬂoods.
Winter snow depth (cm) at Spencer Creek in the Snowy
Mountains, 1964 through 1995 (data courtesy of Snowy
Mountains Hydro-electric Authority
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33.
C h a p t e r 1 1 S c a t t e r p l o t s a n d t i m e s e r i e s 483
Scatterplots
• When looking for a relationship between two variables, data can be represented on
a scatterplot.
• One variable (the independent variable) is on the x-axis and the other variable (the
dependent variable) is on the y-axis.
• Points are plotted by the coordinates formed by each piece of data.
• If the dependent variable consistently increases or decreases as the independent
variable increases, a relationship exists.
• If all points on the scatterplot form a straight line, the relationship is said to be
linear.
• The pattern of the scatterplot gives an indication of the strength of the relationship
or level of association between the variables.
• A strong relationship between variables does not imply that one variable causes the
other to occur.
Regression lines
• The regression line is the line of best ﬁt on a scatterplot.
• By measuring the gradient and the y-intercept on the regression line, we can use the
formula y = mx + c to ﬁnd the equation. A graphics calculator also can be used to
determine the equation of the regression line.
• When the equation of the regression line has been found, it can then be used to
make predictions about the data.
Predictions
• Predictions can be made algebraically or graphically.
• Reliable predictions are produced by interpolation using a large number of points
that lie reasonably close to the regression line.
• Be careful about extrapolating data.
Time series
• A time series is a set of measurements taken over (usually) equally spaced time
intervals, such as hourly, daily, weekly, monthly or annually.
Trend lines
• There are 4 basic types of trend:
1. secular: increasing or decreasing steadily
2. seasonal: varying from season to season
3. cyclic: similar to seasonal but not tied to a calendar cycle
4. random: variations caused by external triggers happening at random.
Fitting trend lines
• The trend line is a straight line that can be used to represent the entire time series.
Trend lines can be used for predicting the future values of the time series. The line
can be found by eye using the ‘equal number of points’ technique.
summary
MQ Maths A Yr 11 - 11 Page 483 Thursday, July 5, 2001 11:03 AM
34.
484 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d
Answers to some of these questions may vary due to different positions of the regression line.
The use of a graphics calculator is recommended, where appropriate.
1 The table below shows the maximum and minimum temperature on 10 days chosen at
random throughout the year.
Display this information on a scatterplot.
2 The table below shows the number of sick days taken by ten employees and relates this to
the number of children that they have.
a Show this information on a scatterplot.
b Does a relationship appear to exist between the number of sick days taken and the
number of children they have? If so, is the relationship linear?
3 The table below shows the number of cars and number of televisions in each household.
a Show this information on a scatterplot.
b Does a relationship appear to exist between the number of televisions in each household
and the number of cars they have? If so, is the relationship linear?
4
A researcher administers different amounts of fertiliser to a number of trial plots of potato
crop. She then measures the total mass of potatoes harvested from each plot. When drawing
the scatterplot, the researcher should graph:
A mass of harvest on the x-axis because it is the independent variable, and amount of ferti-
liser on the y-axis because it is the dependent variable
B mass of harvest on the y-axis because it is the independent variable, and amount of ferti-
liser on the x-axis because it is the dependent variable
Maximum temperature (°C) 25 36 21 40 24 26 30 18 20 25
Minimum temperature (°C) 12 21 11 23 12 15 19 10 8 13
No. of children 1 0 3 2 2 4 6 0 1 2
No. of sick days 5 3 10 8 4 12 12 0 1 5
No. of cars 1 1 2 2 2 3 1 0 1 2
No. of televisions 2 1 1 2 0 1 4 3 1 1
11A
CHAPTER
review
11A
11A
11A mmultiple choiceultiple choice
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35.
C h a p t e r 1 1 S c a t t e r p l o t s a n d t i m e s e r i e s 485
C mass of harvest on the x-axis because it is the dependent variable, and amount of ferti-
liser on the y-axis because it is the independent variable
D mass of harvest on the y-axis because it is the dependent variable, and amount of ferti-
liser on the x-axis because it is the independent variable.
5
Which of the following graphs best depicts a strong negative relationship between the two
variables?
A B C D
6
What type of relationship is shown by the graph on the right?
A Strong positive relationship
B Moderate positive relationship
C Moderate negative relationship
D Strong negative relationship
7 The table below shows the relationship between two variables, x and y.
a Prepare a scatterplot of the data.
b On the scatterplot, draw the regression line.
c By measuring the gradient and the y-intercept of the regression line, ﬁnd its equation.
8 A survey is conducted comparing household income, I, with house value, V. A scatterplot is
drawn and the regression line is found to have the equation V = 3.7I + 50 000. Use the
equation to ﬁnd:
a the likely value of a house owned by a family with an income of $52 000
b the likely income (to the nearest $1000) of a family living in a house valued at $320 000.
9 An entomologist conducted an experiment in which small amounts of insecticide were
introduced to a container of 100 blowﬂies. The results are detailed below.
a Display the above information on a scatterplot and draw the line of regression.
b Find the equation of the regression line.
c Use the equation to predict the number of blowﬂies that would remain after two hours if
4.25 micrograms of insecticide was introduced.
d Estimate the amount of insecticide needed to remove all blowﬂies.
x 2 4 18 7 9 12 2 7 11 10 16
y 103 75 20 66 70 50 95 40 27 42 30
Insecticide (I) (micrograms) 1 2 3 4 5 6 7 8 9 10
No. remaining after 2 h (F) 99 92 81 74 62 68 52 45 38 24
11Ammultiple choiceultiple choice
y
x
y
x
y
x
y
x
11Ammultiple choiceultiple choice
y
x
11B
11B
11B
MQ Maths A Yr 11 - 11 Page 485 Thursday, July 5, 2001 11:03 AM
36.
486 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d
10
The price of oranges over a 16-month period is
recorded in the ﬁgure. The trend can be described
as:
A Cyclic B Seasonal C Random
D Secular E There is no trend.
11 The number of uniforms sold in a school uniform shop is reported in the table.
Fit a trend line to these data. What type of trend is best reﬂected by these data? Can you
explain these trends?
Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov. Dec.
118 92 53 20 47 102 90 42 35 26 12 58
11C mmultiple choiceultiple choice
testtest
CHAPTER
yyourselfourself
testyyourselfourself
11
t
50
40
30
20
10
0
0 2 4 6 8
Months
10 12 14
Priceoforanges
16
11C
MQ Maths A Yr 11 - 11 Page 486 Thursday, July 5, 2001 11:03 AM
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