Download Grade 12 (advanced: quantitative methods)

Mathematics standards
Summary of students’ performance by the end of Grade 12
Grade 12
Advanced
(quantitative
methods)
Reasoning and problem solving
Students analyse problems in a range of mathematical and statistical
contexts. They break problems into smaller tasks, and set up and perform
relevant manipulations, calculations and tests. They identify and use
connections between mathematical topics and appropriate statistical
techniques. They develop and explain chains of logical reasoning, using
correct mathematical notation and terms, and generalise when possible.
They approach problems systematically, knowing when and how to
enumerate all outcomes. They identify exceptional cases and statistical
outliers, and conjecture alternative possibilities with ‘What if …?’ and ‘What
if not …?’ questions. They collect, organise, analyse and interpret relevant
and realistic data, using statistical functions on a calculator and ICT. They
work to expected degrees of accuracy.
Number, algebra and calculus
Students use number, algebra and calculus to further their understanding of
statistics. They use the series expansion of ex and the rules of logarithms.
They use the remainder theorem and the factor theorem and find
permutations and combinations. They sketch and interpret the graphs of
linear, quadratic, cubic, reciprocal, exponential and logarithm functions, the
sine and cosine and tangent functions (using radian measure for angles), and
the modulus function. They solve related equations (except cubic equations)
on specified domains. They recognise when functions are increasing,
decreasing or stationary. They calculate and interpret the derivative of powers
of x, of polynomial functions, and of the exponential function, including second
and higher order derivatives. They calculate the derivative of the sum,
difference and product of any two of these functions. They know that
integration is the inverse of differentiation. They use integration to calculate
areas under curves.
Geometry and measures
Students apply and use the work on geometry and measures learned in
earlier grades to solve problems.
Probability and statistics
Students collect data, organise data and make inferences from data. They
plan questionnaires and surveys and design experiments to collect primary
data from samples, distinguishing a sample from its parent population. They
know the significance of a simple random sample and the effect of bias in a
sample. They understand the importance of a random variable. They
distinguish a parameter for a population from a statistic for a sample. They
formulate problems based on primary data, or on secondary data from
published sources, including government statistics and the Internet. They
calculate measures of central tendency and of spread. They construct
histograms and frequency distributions, using box-and-whisker plots and
associated vocabulary in presenting their findings and conclusions. They
distinguish between nominal, ordinal and interval or ratio scales. They look
for correlation between two random variables and calculate the rank order
281 | Qatar mathematics standards | Grade 12 advanced | Quantitative methods
© Supreme Education Council 2004
correlation coefficient and the product moment coefficient of correlation and
interpret their meaning. They draw lines of best fit where linear correlation is
exhibited. They calculate probabilities of single and combined events, and
use and understand vocabulary associated with the probabilities of
occurrence of different events. They use tree diagrams to represent and
calculate the probabilities of compound events when the events are
independent and when one event is conditional on another. They compare
probabilities derived from sampling with theoretical models of probability,
including both discrete and continuous models. They use the chi-squared
test to compare observation with theoretical expectation. They use
significance levels to accept or reject a null hypothesis and to make
inferences from data. They use random numbers to choose random
samples, to assign individual units to samples, and to simulate a range of
statistical situations. They present findings and conclusions using a range of
graphs, charts and tables. They use relevant statistical functions on a
calculator and ICT.
Content and assessment weightings for Grade 12
The advanced mathematics standards for Grade 12 have two pathways:
mathematics for science and quantitative mathematics, to support the social
sciences and economics. Each pathway includes reasoning and problem
solving, and number, algebra and calculus. The mathematics for science
standards include substantial work on calculus but no new work on
probability and statistics, whereas the quantitative methods standards
include substantial work on probability and statistics, less calculus, and no
new work on geometry and measures.
The reasoning and problem solving strand cuts across the other strands.
Reasoning, generalisation and problem solving should be an integral part of
the teaching and learning of mathematics in all lessons.
The weightings of the content strands relative to each other are as follows:
Number, algebra
Geometry, measures
Probability and
and calculus
and trigonometry
statistics
Grade 10
55%
30%
15%
Grade 11
55%
30%
15%
40%
–
60%
75%
25%
–
Advanced
Grade 12
(quantitative)
Grade 12
(for science)
The standards are numbered for easy reference. Those in shaded rectangles,
e.g. 1.2, are the performance standards for all advanced quantitative
mathematics students. The national tests for advanced quantitative
mathematics will be based on these standards.
Grade 12 teachers should consolidate earlier standards as necessary.
282 | Qatar mathematics standards | Grade 12 advanced | Quantitative methods
© Supreme Education Council 2004
Mathematics standards
Reasoning and problem solving
Grade 12
Advanced
(quantitative
methods)
Key standards
By the end of Grade 12, students analyse problems in a range of
mathematical and statistical contexts. They break problems into smaller tasks,
and set up and perform relevant manipulations, calculations and tests. They
identify and use connections between mathematical topics and appropriate
statistical techniques. They develop and explain chains of logical reasoning,
using correct mathematical notation and terms, and generalise when possible.
They approach problems systematically, knowing when and how to
enumerate all outcomes. They identify exceptional cases and statistical
outliers, and conjecture alternative possibilities with ‘What if …?’ and ‘What if
not …?’ questions. They collect, organise, analyse and interpret relevant and
realistic data, using statistical functions on a calculator and ICT. They work
to expected degrees of accuracy.
Key performance standards
are shown in shaded
rectangles, e.g. 1.2.
Cross-references
Standards are referred to
using the notation RP for
reasoning and problem
solving, NAC for number,
algebra and calculus, GM
for geometry and measures
and PS for probability and
statistics, e.g. standard
NAC 2.3.
Students should:
Examples of problems
1
1.1
1.2
Use mathematical reasoning to solve problems
Examples of problems in
italics are intended to clarify
Solve routine and non-routine problems in a range of mathematical and
statistical contexts, including open-ended and closed problems.
the standards, not to
represent the full range of
Use statistical techniques to model and predict the outcomes of statistical
situations, including real-world applications; work to definitions and
perform appropriate tests.
Reasoning and problem
solving
1.3
Identify and use interconnections between mathematical topics.
1.4
Break down complex problems into smaller tasks.
1.5
Use a range of strategies to solve problems, including working the problem
backwards and redirecting the logic forwards; set up and solve relevant
equations and perform appropriate calculations and manipulations; change
the viewpoint or mathematical representation; and introduce numerical,
algebraic, graphical, geometrical or statistical reasoning as necessary.
1.6
Develop chains of logical reasoning, using correct terminology and
mathematical notation.
1.7
Explain their reasoning, both orally and in writing.
1.8
Identify exceptional cases and statistical outliers.
1.9
Generalise whenever possible.
Reasoning, generalisation
and problem solving should
1.10
Approach complex problems systematically, recognising when and how it
is important to enumerate all outcomes.
1.11
Conjecture alternative possibilities with ‘What if …?’ and ‘What if not …?’
questions.
1.12
Synthesise, present, discuss, interpret and criticise mathematical
information presented in various mathematical forms.
283 | Qatar mathematics standards | Grade 12 advanced | Quantitative methods
possible problems.
be an integral part of the
teaching and learning of
mathematics in all lessons.
© Supreme Education Council 2004
1.13
Work to expected degrees of accuracy, and know when an exact solution is
appropriate.
1.14
Check statistical data for reliability, internal consistency, arithmetic errors
and plausibility.
There is an error in the following table, which appeared in a report called ‘Smoking and
Health Now’ published by the Royal College of Physicians in the United Kingdom. The
table shows the number of men aged 35 or more who died due to smoking related
diseases. Locate the error and correct it.
Lung cancer
Chronic
bronchitis
Coronary heart
disease
All causes
Number of deaths
26 973
24 976
85 892
312 537
Percentage
8.6%
8.0%
2.75%
100%
Find an article of a statistical nature in a newspaper.
Check the article for lack of consistency, faulty arithmetic, implausible numbers and
omission of relevant information.
1.15
Use statistical functions on a calculator to analyse real data sets.
1.16
Recognise when to use ICT and when not to, and use it efficiently; use ICT
to present findings and conclusions.
Number, algebra and calculus
By the end of Grade 12, students use number, algebra and calculus to further
their understanding of statistics. They use the series expansion of ex and the
rules of logarithms. They use the remainder theorem and the factor theorem
and find permutations and combinations. They sketch and interpret the
graphs of linear, quadratic, cubic, reciprocal, exponential and logarithm
functions, the sine and cosine and tangent functions (using radian measure
for angles), and the modulus function. They solve related equations (except
cubic equations) on specified domains. They recognise when functions are
increasing, decreasing or stationary. They calculate and interpret the
derivative of powers of x, of polynomial functions, and of the exponential
function, including second and higher order derivatives. They calculate the
derivative of the sum, difference and product of any two of these functions.
They know that integration is the inverse of differentiation. They use
integration to calculate areas under curves.
Number, algebra and
calculus
The standards for number,
algebra and calculus are
designed, in the main, to
support the statistical
techniques developed in
the probability and statistics
strand.
Students should:
2
2.1
Manipulate algebraic expressions
Multiply, factorise and simplify expressions and divide a polynomial by a
linear or quadratic expression.
Write down in ascending powers of x the expansion of (2 – 3x)3.
Factorisation
Factorise 27x3 + 8y3. Use your factorisation to write an identity for 27x3 – 8y3.
Include the sum and
difference of two cubes.
Simplify
x3 + 1
.
x2 − 1
Solve the equation (x + 3)(3x –1) – (x + 3)(5x + 2) = 0.
284 | Qatar mathematics standards | Grade 12 advanced | Quantitative methods
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2.2
Combine and simplify rational algebraic fractions; decompose a rational
algebraic fraction into partial fractions (with denominators not more
complicated than repeated linear terms).
Simplify
5
3x − 2
.
−
x + 1 x2 − 1
Show that
1
1
1
≡ −
.
r ( r + 1) r r + 1
Find the values of A, B and C in the identity
2.3
1
A
B
C
≡
+
+
.
(3r + 1)( r + 1) 2 3r + 1 r + 1 (r + 1) 2
Understand and use the remainder theorem.
3x3 – 2x2 – ax – 28 has a remainder of 5 when divided by x – 3. What is the value of a?
2.4
Understand and use the factor theorem.
Show that 2x3 + x2 – 13x + 6 is divisible by x – 2 and find the other factors. Hence find
the solution set in \ of the equation 2x3 + x2 – 13x + 6 = 0. What is the solution set in `?
3
3.1
Use index notation and logarithms to solve numerical
problems
Understand exponents and nth roots, and apply the laws of indices to
simplify expressions involving exponents; use the xy key and its inverse on
a calculator.
Without using a calculator, simplify
8n × 25 n
.
42 n
Plot a graph of y = 1000 × 2–t for values of t from 0 to 1, increasing in steps of 0.2.
3.2
Know the definition of a logarithm in number base a (a > 0), and the rules
of combination of logarithms, including change of base.
Logarithms
In addition to formulae for
the sum and difference of
Give the value of log10 1000.
two logarithms, include:
loga a x = x
Evaluate log2 64.
aloga x = x
Express log 27 – 2 log 3 as a single logarithm.
loga a = 1
log c b
1
Prove that log a b =
and use this result to show that log a b =
.
log b a
log c a
loga 1 = 0 for any a > 0
loga x n = n loga x
Given logb 2 = 1/3, logb 32 is equal to
A. 2
B. 5
C.
–3
/5
D. 5/3
E.
3
log 2 32
TIMSS Grade 12
Explain why the number base of a logarithm must be positive, but why the logarithm
itself may take any value.
3.3
Use the ln and log keys on a calculator and the corresponding inverse
function keys.
Use a calculator to evaluate log5 4 correct to three decimal places.
In 1916, two scientists each named du Bois derived a formula to estimate the surface
area S m2 of human beings in terms of their mass M kg and their height H cm.
The formula is S = 0.000 718 4 × M0.425 × H0.725.
A boy has a mass of 60 kg and a height of 160 cm. Calculate the boy’s surface area to
three significant figures.
The intensity of sound, N, is measured in decibels (dB) and is defined by the formula
N = 10 log (I / 10–16), where I is the power of sound measured in watts.
Find N for normal speech with a power of 10–10 watts.
Find the power of a jet aircraft with a sound intensity of 150 dB.
285 | Qatar mathematics standards | Grade 12 advanced | Quantitative methods
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In chemistry, the pH value of a solution is defined as pH = – log [H], where [H] is the
concentration of hydrogen ions in the solution. Solutions with pH < 7 are acidic and
solutions with pH > 7 are alkaline. Solutions with pH = 7 are neutral.
A solution has [H] = 5 × 10–7. Find its pH value and state whether it is acid, neutral or
alkaline.
4
4.1
Work with sequences, series, recurrence relations and
arrangements
Understand that nCr is the number of combinations of r different objects
from n different objects and that the number of permutations of r different
objects from n different objects is r! nCr, which is denoted by nPr.
How many different committees of four people can be chosen from a group of nine
people?
Permutations
A committee is to be chosen from five men and three women. The committee will have
two men and two women. In how many ways may the committee be selected?
distinguishes a permutation
from a combination.
The order of selection
An examination consists of 13 questions. A student must answer only one of the first two
questions and only nine of the remaining ones. How many choices of question does the
student have?
A. 13C10 = 286
B. 11C8 = 165
C. 2 × 11C9 = 110
D. 2 × 11P2 = 220
E. some other number
TIMSS Grade 12
Evaluate 9C6 and 10P5.
Prove that nCr = nCn–r and that nCr + nCr+1 = n+1Cr+1.
In how many ways can 5 thick books, 4 medium sized books and 3 thin books be
arranged on a bookshelf so that the books of the same size remain together?
A.
B.
C.
D.
E.
5! 4! 3! 3! = 103 680
5! 4! 3! = 17 280
(5! 4! 3!) × 3 = 51 840
5 × 4 × 3 × 3 = 180
212 × 3 = 12 288
TIMSS Grade 12
4.2
Investigate properties of nCr; expand the binomial series (1 + x)n for any
rational value of n.
Binomial coefficients
The properties of these
coefficients can form
interesting extension work.
Show that nC0 + nC1 + nC2 + …+ nCn = 2n.
5
5.1
5.2
Work with functions and their graphs
Use a graphics calculator to plot exponential functions of the form y = ekx;
describe these functions, distinguishing between cases when k is positive or
negative, and the special case when k is zero.
Plot and describe the features of the natural logarithm function y = ln x;
understand that the natural logarithm function is inverse to the exponential
function.
Functions
Include use of the notation
f : x 6 f( x ) , as well as
y = f(x) or f(x) = …
A radioactive element decomposes according to the formula y = y0e–kt, where y is the
mass of the element remaining after t days and y0 is the value of y for t = 0. Find the
value of the constant k for an element whose half-life (i.e. the time taken to decompose
half of the material) is 4 days.
1
A. 14 log e 2
B. log e 12
C. log 2 e
D. (log e 2) 4
E. 2e 4
TIMSS Grade 12
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5.3
Understand the modulus function y = | x | and sketch its graph.
Sketch the graph of the function f(x) = | x – 2 | + | x | for –3 ≤ x ≤ 3.
5.4
Form composite functions and use the notation y = g(f(x)).
Composite functions
Study of the functions in
2
The function f(x) = 4x – 7 is defined on \. A second function g(x) = (x + 1) is also
defined on \. Find f(g(x)) and g(f(x)). Comment on why these functions are not the
same.
5.5
standards NAC 5.4−5.6 can
form extension work for the
most able students.
Form inverse functions (on a restricted domain, if necessary) and use the
notation y = f −1 ( x) .
Show that the function f ( x) =
x
, where x ≠ 1, is its own inverse.
x −1
Find the inverse of the function f : t 6 t + 2 , defined for t ≥ 0.
5.6
Know that f −1 (f ( x)) = x and that the graph of y = f −1 ( x) is the reflection of
the graph of y = f ( x) about the straight line y = x.
Show that eln x = x .
Sketch the graph of y = x for non-negative values of x.
6
6.1
Solve equations associated with functions
Given a quadratic equation of the form ax2 + bx + c = 0, know that if the
discriminant ∆ = b2 − 4ac is negative, there are two complex roots, which
are conjugate to each other.
6.2
Solve exponential and logarithmic equations of the form ekx = A, where A is
a positive constant, and ln kx = B, where B is constant.
6.3
Solve trigonometric equations of the form:
sin (ax + b) = A, where –1 ≤ A ≤ 1;
cos (ax + b) = A, where –1 ≤ A ≤ 1;
tan (ax + b) = A, where A is constant;
and find solutions in the interval 0 ≤ x ≤ 2π.
6.4
Find approximate solutions for the intersection of any two functions from
the intersection points of their graphs, and interpret this as the solution set
of pairs of simultaneous equations.
Complex numbers
Complex numbers could
form extension work for the
most able students.
An approximate solution of the equation x = ex can be found by plotting on the same axes
the curves y = x and y = ex and finding the x-coordinate of their point of intersection.
7
7.1
Calculate the derivative of a function
Know that the derivative of f ′(x) is called the second derivative of the
Reading the second
2
function y = f(x) and that this can also be written in the forms f′′(x) or
d y
;
dx 2
know that higher derivatives may be taken in the same way.
7.2
derivative
f′′(x) is read as ‘f-double2
2
dash of x’ and d y/dx is
read as ‘dee-two-y by dee-
For the function f(x) = x3 – 6x, show that f′′(x) = 6x.
What is the value of f′′(x) at the points where f′(x) is zero?
x-squared’.
Interpret the numerical value of the derivative at a point on the curve of the
function; know that:
when the derivative is positive the function is increasing at the point;
when the derivative is negative the function is decreasing at the point;
when the derivative is zero the function is stationary at the point.
Higher derivatives
287 | Qatar mathematics standards | Grade 12 advanced | Quantitative methods
In general an nth order
derivative is denoted by
dn y
(n)
or f (x).
dx n
© Supreme Education Council 2004
Discuss how knowledge of when a function increases, when it decreases and when it is
stationary gives important information about the function as a whole and helps to
analyse what it looks like.
Discuss the derivative function associated with the function f : x 6 | x | .
Is there anywhere on this function where the derivative does not exist?
Justify your answer.
7.3
7.4
Understand that stationary points of any function may correspond to a local
maximum or minimum of the function, or may be a point of inflexion;
understand how the derivative changes as the point at which the derivative
is calculated moves through the local maximum or minimum, or through an
inflexion; understand that not all points of inflexion are stationary points.
Maxima and minima
These are sometimes
referred to as turning
points.
Understand and use the second derivative to test whether a stationary point
is a local maximum, or a local minimum, or a point of inflexion.
Which of the following graphs has these features:
f′(0) > 0, f′(1) < 0, and f′′(x) is always negative?
TIMSS Grade 12
7.5
Know that of all exponential functions, the exponential function y = ex is
defined as the one in which the slope at the y-intercept point (0, 1) has the
value 1.
7.6
Know that the function f(x) = ex is the only non-zero function in
mathematics for which the derivative f′(x) = ex gives back the original
x 2 x3
xn
function, and that e x = 1 + x + + + ... + + ... (ad infinitum).
2! 3!
n!
The function e
x
x
This expansion of e is
important for the Poisson
distribution in PS 12.2.
x 2 x3 x 4
+ + + ... represents the number ex.
2! 3! 4!
Show that if you differentiate the series term by term and add all these terms together,
you get back to what you started with.
It can be shown that the infinite series 1 + x +
7.7
Know that the derivative of the natural logarithm function ln x is 1/x.
Derivative of combinations of functions
7.8
Understand that given any function f(x) = f1(x) + f2(x) then the derivative of
this sum of two functions is f′(x) = f1′(x) + f2′(x).
Find the derivative of the function given by y = 5x + ex.
7.9
Understand that given a function f(x) = u(x) v(x) then the derivative of this
product of two functions is given by f′ = uv′ + vu′; use this result in
calculating the derivative of the product of two functions; know the special
case of this result that if y = a f(x), where a is constant, then y′ = a f′.
Find
7.10
dy
when y = 10x ex.
dx
Understand that given a function f(x) = u(x) / v(x) then the derivative of this
quotient of two functions is given by f′ = (uv′ – vu′) / v2; use this result in
calculating the derivative of the quotient of two functions.
288 | Qatar mathematics standards | Grade 12 advanced | Quantitative methods
© Supreme Education Council 2004
7.11
Understand that given a composite function h(x) = g(f(x)) then the
derivative of this composite function is given by h′(x) = g′(f(x))f′(x); use
this result in calculating the derivative of the composite of two functions.
Use the chain rule to show that if y = 10 e–2t then
dy
= −20e −2t .
dx
acceptable way of writing
dy dy dz
this is
,
=
×
dx dz dx
function is formed by first
mapping x to z and then
d(ln x) 1
= .
dx
x
mapping z to y. This rule is
often called the chain rule
dx
1
=
.
Show that
dy dy
dx
because it extends for
composite functions formed
in more than two stages.
7.12
Recognise that the derivative of f(x) = A ekx, where A and k are constants, is
f′(x) = kA ekx.
7.13
Find the derivative of a function defined implicitly.
Find
An alternative and equally
where the composite
Find the derivative with respect to x of y = (3x + 1)2 (2 – x).
Prove that
Composite functions
dy
for the implicit function x2 + y2 = 25 for y ≥ 0.
dx
Applications using derivatives
7.14
Use the first and second derivatives of functions to analyse the behaviour of
functions and to sketch curves.
Sketch the curve y =
1
, showing clearly its turning points and asymptotes.
x( x − 2)
Sketch the curve y = xe–x for x ≥ 0, showing clearly its turning points and any points of
inflexion.
7.15
8
Use the derivative to explore a range of problems in which a function is
maximised or minimised.
Reconstruct a function from its derivative
The indefinite integral
8.1
Understand integration as the inverse process to differentiation.
8.2
Understand and use the notation for indefinite integrals, knowing that
∫ f ′( x) dx = f ( x) + c , where c is any constant, and that there is a whole family
of curves y = f(x) + c, each member of which has derivative function f′(x).
8.3
Integration
The word integration comes
from integrating, i.e. adding,
the contributions of many
Discuss how the individual members of the family of curves represented by y = f(x) + c
are related to each other.
small parts. The symbol
∫ ...dx denotes summation
The graph of the function g passes through the point (1, 2). The slope of the tangent to
the graph at any point (x, y) is given by g′(x) = 6x – 12. What is g(x)? Show all your work.
with respect to x. In the
TIMSS Grade 12
are called the limits of the
integral, or the limits of
integration.
Know the integrals of the functions:
xn, where n ≠ –1
1/x, with x ≠ 0
ekx;
write the integrals of multiples of these functions and of linear
combinations of these functions.
289 | Qatar mathematics standards | Grade 12 advanced | Quantitative methods
definite integral
∫
b
a
, a and b
© Supreme Education Council 2004
The definite integral
8.4
Use the definition of the definite integral:
∫
b
a
f ′( x) dx = f (b) − f (a) , where f(x) is a function of x and a ≤ x ≤ b;
interpret this as ‘the integral of a rate of change of a function is the total
change of that function’; understand the effect of interchanging the limits of
integration; know that
8.5
b
c
b
a
a
c
∫ =∫ +∫
.
Use summation of areas of rectangles to calculate lower and upper bounds
for the area between the x-axis and a curve y = f(x) with y > 0, bounded on
either side by lines x = constant; understand that as the width δx of each of
the rectangles tends to zero the sums Σ f(x) δx for the lower and upper
bounds on the area under the curve tend to the same value, and that this
value is called the area under the curve.
8.6
Understand that the area bounded by a positive function y = f(x), the x-axis
and the lines x = a and x = b, with a ≤ x ≤ b, is the definite integral
∫
b
a
f ( x) dx .
The line l in the figure is the graph of y = f(x).
∫
3
−2
f ( x) dx is equal to
A. 3
B. 4
C. 4.5
D. 5
E. 5.5
TIMSS Grade 12
8.7
Use the trapezium rule to find an approximation to the area represented by
the definite integral of a particular function when it is not easy or possible
to integrate the function.
8.8
Understand that if a curve y = f(x) lies entirely below the x-axis, so that its
y-value is always negative, then the definite integral
Areas under curves
Students should work with
‘area-so-far’ for the area
under a curve y = f(x),
using definite integrals as in
NAC 8.6, or the trapezium
rule as in NAC 8.7, to
reinforce work in the
probability and statistics
strand.
b
∫a f ( x)dx over the
interval a ≤ x ≤ b has a negative value.
Calculate the area between the curve y = x2 + 5, the x-axis and the lines x = –2 and x = 3.
Find the area between the curves y = x2 – 4 and y = 4 – x2.
Find the area between the curves y = x3 and y = x.
This figure shows the graph of y = f(x).
S1 is the area enclosed by the x-axis,
x = a and y = f(x);
S2 is the area enclosed by the x-axis,
x = b and y = f(x);
where a < b and 0 < S2 < S1.
The value of
A.
B.
C.
D.
E.
∫
b
a
f ( x) dx is
S1 + S2
S1 – S2
S2 – S1
| S1 – S2 |
1
S + S2 )
2( 1
TIMSS Grade 12
290 | Qatar mathematics standards | Grade 12 advanced | Quantitative methods
© Supreme Education Council 2004
8.9
Understand that the integration by parts formula
∫ uv′ dx = uv − ∫ vu ′ dx
Integration by parts
Extension work on
probability distributions
could make use of
standards 8.9, 8.10 and
8.11.
reverses the derivative of the product of two functions.
Find
8.10
∫ xe
x
dx .
Understand that ∫ g ′(f ( x)) f ′( x) dx = g(f ( x)) + c reverses the derivative of a
composite function; recognise ‘simple’ functions for which this formula can
be instantly applied.
Explain why
8.11
∫x
2x
dx = ln ( x 2 + 1) + c .
+1
2
Use partial fractions to integrate.
Find
1
∫ ( x + 1)( x + 2) dx .
291 | Qatar mathematics standards | Grade 12 advanced | Quantitative methods
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Probability and statistics
By the end of Grade 12, students collect data, organise data and make
inferences from data. They plan questionnaires and surveys and design
experiments to collect primary data from samples, distinguishing a sample
from its parent population. They know the significance of a simple random
sample and the effect of bias in a sample. They understand the importance
of a random variable. They distinguish a parameter for a population from a
statistic for a sample. They formulate problems based on primary data, or
on secondary data from published sources, including government statistics
and the Internet. They calculate measures of central tendency and of
spread. They construct histograms and frequency distributions, using boxand-whisker plots and associated vocabulary in presenting their findings
and conclusions. They distinguish between nominal, ordinal and interval or
ratio scales. They look for correlation between two random variables and
calculate the rank order correlation coefficient and the product moment
coefficient of correlation and interpret their meaning. They draw lines of best
fit where linear correlation is exhibited. They calculate probabilities of single
and combined events, and use and understand vocabulary associated with
the probabilities of occurrence of different events. They use tree diagrams
to represent and calculate the probabilities of compound events when the
events are independent and when one event is conditional on another. They
compare probabilities derived from sampling with theoretical models of
probability, including both discrete and continuous models. They use the
chi-squared test to compare observation with theoretical expectation. They
use significance levels to accept or reject a null hypothesis and to make
inferences from data. They use random numbers to choose random
samples, to assign individual units to samples, and to simulate a range of
statistical situations. They present findings and conclusions using a range of
graphs, charts and tables. They use relevant statistical functions on a
calculator and ICT.
Probability and statistics
Students should know that
probability lies at the heart
of statistics. They should be
aware of the uses of
statistics in society and
recognise when statistics
are used sensibly and when
they are misused or likely to
be misunderstood.
Students should:
9
Collect, organise and analyse data, and make inferences
from data
Measuring and sampling
9.1
Know the difference between categorical data, discrete data and continuous
data.
9.2
Understand what it means to measure a property of a person or thing in a
statistical sense.
Explain the difference between a scientific measurement (e.g. the measurement of the
amount of heat generated in a chemical reaction) and a statistical measurement (e.g. the
number of homeless people in a big city like London).
Why is measuring ‘authoritarian personality’ different from measuring height?
9.3
Distinguish between nominal, ordinal, interval and ratio scales.
Explain the difference between nominal, ordinal, interval and ratio scales, giving an
example of each type.
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What sort of measurement scale is used for categorical data? Give an example of such a
scale.
A badly designed opinion poll poses the proposition: ‘The price of oil should be linked
to the political situation in the Middle East.’ Subjects are asked whether they
• strongly agree
• agree
• are undecided
• disagree
• strongly disagree
with the proposition.
Identify the type of scale used for this variable response.
An oil company keeps records on its employees. The database includes the age of the
employees, the date on which they were hired, their gender, their grouping by country of
origin (Qatari, Palestinian, Syrian, Indian, etc.), their type of job, their salary. Name the
scale on which each of these variables is measured
9.4
Distinguish between population, sample and census; know the importance
of choosing a representative sample; locate obvious sources of bias within a
sample.
Explain, with examples, the important differences between a sample from a population
and a census of the population.
Describe the effect of bias in a sample.
Explain why it is often in the interest of politicians, those that espouse good causes,
telephone interviewers, advertisers and others to use questionable statistics and biased
sampling methods.
Vocabulary of sampling
Know the meanings of
vocabulary associated with
sampling, including
population, census, sample,
unit, sampling frame,
variable, parameter and
statistic.
Two common types of sampling are convenience sampling (when the easiest option is
chosen to select units for sampling, e.g. stopping any shopper in a Suq) and voluntary
response samples (where people choose themselves, e.g. to respond to a television poll).
Explain why both convenience sampling and voluntary response sampling are likely to
be biased.
9.5
Know the distinction between bias and precision.
Give examples of samplings with:
high bias and high precision;
low bias and low precision;
high bias and low precision;
low bias and high precision.
9.6
Understand and use the concept of a random variable; understand the
meaning and properties of random numbers; know how to generate random
numbers using the random number function(s) on a calculator; know how to
assign random numbers in a variety of situations; use tables of random
numbers.
Explain to someone who knows no statistics what a random variable is.
Collection and organisation of data
9.7
Random numbers
Random numbers can be
used in a variety of ways,
including choosing a simple
random sample, assigning
subjects to different
treatments in a randomised
trial and in carrying out
simulations.
Understand how to collect a simple random sample using random numbers.
A class wishes to make a complaint to the school principal. The girls decide to select a
simple random sample of five girls from the class to go to the principal with the
complaint. Show how to use a random number table to select the five girls from the class
of the class of 28 whose names are given below:
Aida
Nada
Naima
Haifa
Maram
Farha
Zahra
Muna
Inas
Roza
Safa
Majda
Sharifa
Farida
Badriya
Halima
Hayat
Deena
Huwaida Haya
Mariam
Zalikha
Amina
Anissa
Noor
Fathiya
Habiba
Salma
293 | Qatar mathematics standards | Grade 12 advanced | Quantitative methods
© Supreme Education Council 2004
9.8
Distinguish different types of sampling: simple random sampling, stratified
sampling and cluster sampling.
Write a short essay comparing different types of sampling techniques.
9.9
Plan surveys and design questionnaires to collect meaningful primary data
from samples in order to test hypotheses about, or estimate, characteristics
of the population as a whole.
9.10
Formulate problems based on secondary data from published sources,
including the Internet.
9.11
Display categorical data with a pie chart or bar graph.
Data collection and
analysis
Use primary data collected
in other subjects, such as
science or social science,
and also secondary
sources of data, e.g. from
government statistics and
from the Internet.
What is the essential difference between a pie chart and a bar graph?
The table below, from the 1995 Statistical Abstract of the United States, shows the
percentage of females who were awarded doctorates in a number of subjects
Subject
Percentage of doctorates
obtained by females
Education
59.5
Psychology
59.7
Life sciences
38.3
Computer science
13.3
Engineering
9.6
Explain why a pie chart cannot be used to represent this information.
Display the information in a bar chart.
9.12
9.13
Construct and interpret frequency tables and histograms for continuous
grouped data, using equal and unequal class intervals; know that the
frequency of occurrence in each class interval of a histogram is represented
by the area of the rectangle constructed on that class interval; display
discrete data in a vertical line chart; comment on how outliers might affect
these distributions.
Calculate measures of central tendency: the arithmetic mean, the median
and the mode; distinguish between these measures.
Relative frequency
Include the terms frequency
and frequency distribution,
relative frequency and
relative frequency
distribution.
Mean
Include the terms mode,
modal class, modal
frequency.
Why is the median often a more useful statistic than the mean?
A frequency diagram for a set of data is shown in the diagram below. No scale is given
on the frequency axis, but summary statistics are given for the distribution:
The sample mean x is
defined as
n
Σ f = 50, Σ fx = 100, Σ fx2 = 344
x=
n
1
xi =
n∑
1
∑f x
i
i
1
n
∑f
i
1
The population mean is
usually denoted by µ.
a. State the mode and the mid-range value of the data.
b. Identify two features of the distribution.
c. Calculate the mean and standard deviation of the data and explain why the value 8,
which occurs just once, may be regarded as an outlier.
d. Explain how you would regard the outlier if the diagram represents:
i.
the difference of the scores obtained when throwing a pair of ordinary dice,
ii. the number of children per household in a neighbourhood survey.
e. Find new values for the mean and standard deviation if the single outlier is removed.
MEI
294 | Qatar mathematics standards | Grade 12 advanced | Quantitative methods
© Supreme Education Council 2004
A group of seven professional tennis players compare their financial winnings in one
year. The amounts they win in US dollars, to the nearest thousand dollars, are
138 000 2 597 000 155 000 146 000 369 000 199 000 283 00
Find the mean and median earnings of these players. Explain which is the better statistic
to use as an average measure of their winnings? What would be the new mean and
median if an eighth player, with winnings of US$ 538 000, joins the group?
A scientist took a sample of 80 eggs. She measured the length of each one and grouped
the data as follows and then plotted frequency density against length:
Length (l) in cm
Frequency
4.4 ≤ l < 5.0
5.0 ≤ l < 5.4
5.4 ≤ l < 5.8
5.8 ≤ l < 6.3
6.3 ≤ l < 6.5
4
20
36
16
4
In grouped data, one way to estimate the mode is to use similar triangles.
a. Explain why AB = 40 and DE = 58.
Use similar triangles to calculate an estimate of the mode.
b. The best estimate of the median is 5.6 (to one decimal place). The estimate is
calculated like this:
16.5
5.4 +
(0.4)
35
5.4 is the initial value of the class containing the median.
Explain what the other three numbers in the formula stand for.
QCA, modified
9.14
Calculate measures of spread, including the variance and standard
deviation; know the distinction between population and sample variance,
and the corresponding standard deviations.
9.15
Use calculator function keys for mean, standard deviation and variance.
9.16
Understand how mean, variance and standard deviation are affected by the
linear coding yi = a + bxi .
Sample variance
2
This is denoted by s . The
2
definition of s is
n
1
s2 =
( xi − x )2
n − 1∑
1
The sample variance is s,
2
the square root of s .
The population variance is
2
denoted by σ , and the
standard deviation is given
by its square root, σ.
Find the mean and standard deviation of the set of numbers 5, 7, 4, 3, 8, 7, 8, 6.
Add 3 to each number. Repeat the calculation of the mean and standard deviation.
Explain how these values relate to the original values for the mean and standard
deviation?
A golf tournament is taking place. For each round, the players’ scores are recorded
relative to a fixed score of 72. [For example, a true score of 69 would be recorded as
–3.] The recorded scores, x, for the ten players to complete the first round were:
–4 –3 –7 6 2 0 0 3 5 7
a. Calculate the mean and standard deviation of the values of x.
b. Deduce the mean and standard deviation of the true scores.
In the second round of the tournament, the recorded scores, x, for the same ten golfers
produced a mean of –0.3 and standard deviation 2.9.
c. Comment on how the performance of the golfers has changed from the first to the
second round.
d. Calculate the mean and the standard deviation of the twenty true scores for the two
rounds.
MEI
295 | Qatar mathematics standards | Grade 12 advanced | Quantitative methods
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9.17
Plot and interpret cumulative frequency distributions and box-and-whisker
plots, using grouped continuous data as necessary; use the vocabulary
range, percentile, interquartile range, semi-interquartile range.
Explain why making a comparison of the performance ratings of two different types of
motor vehicle is easily made using box-and-whisker plots.
9.18
Draw stem-and-leaf diagrams.
Compare the advantages and disadvantages of histograms and stem-and-leaf diagrams.
9.19
10
10.1
Interpret, in qualitative terms, the skewness of a frequency distribution and
understand the importance of a symmetric distribution.
Understand random variables and calculate probability
Know that all probability values lie between 0 and 1, and that the extreme
values correspond respectively to impossibility and certainty of occurrence;
calculate probabilities.
The eighteenth-century French scientist Count Buffon did many experiments on
probability. One of these was to estimate the value of π by dropping a needle between
two parallel lines drawn exactly one needle length apart. Buffon showed that the
probability that the needle lands on or across one of the parallel lines is 2/π. Estimate
the probability of needles falling on a line by throwing similar length needles in the air
and recording how they land. Use this estimate to calculate π.
Can you prove Buffon’s result?
In a group of 36 blood donors, 16 are male and 20 are female. Four of these people are
chosen for an interview.
a.
b.
In how many ways can they be chosen?
Find the probability that they are all of the same sex.
MEI
10.2
Know that all possible outcomes for an experiment form the sample space
for that experiment; use the sample space to calculate probabilities for each
outcome.
Two fair six-sided dice are thrown and their total is recorded. Give a diagrammatic
representation of the sample space. Calculate the probability that when the dice are
thrown the total is at least 8.
10.3
Understand that a random variable has a range of values that cannot be
predicted with certainty; investigate common examples of random
variables; measure the empirical probability (relative frequency) of
obtaining a particular value of a random variable.
Discrete random
variables
The probability that a
discrete random variable X
has the value x is denoted
by P(X = x).
Comment on this statement: ‘The behaviour of a random variable is not randomly
chaotic, but represents a kind of order that emerges only in the long run’.
Repeatedly throw an ordinary six-sided dice. Investigate the number of throws ‘on
average’ to first record a five on the uppermost face.
Here are two statements:
An unbiased coin is tossed many times; the probability of heads will be close to one half.
An unbiased coin is tossed very many times; the number of heads will be close to half
the number of tosses.
Are both statements true? Is only one of the statements true? Are both statements false?
Explain the reasoning behind your answer.
Use computer generation of random numbers to simulate the tossing of a coin. What is
the long-term probability of tossing heads?
What does ‘the law of averages’ really mean?
296 | Qatar mathematics standards | Grade 12 advanced | Quantitative methods
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10.4
Know that a probability distribution for a random variable assigns the
probabilities of all the possible values of the variable and that these values
total to 1; use a simple mathematical probability distribution to calculate,
for a particular set of events, the theoretical probability of obtaining a
particular outcome for a random variable associated with those events.
In a ‘heads or tails’ experiment a coin that is believed to be unbiased is repeatedly
tossed in the air and the uppermost face on landing (H or T) is recorded. The first six
throws of the experiment give TTTTTT. Is this evidence that the coin is biased? If not,
explain how this can happen. Is the next throw likely to be a head?
Probability models
Include as mathematical
models for discrete random
variables the discrete
uniform distribution and the
binomial distribution, and
simple applications of
these.
Use a random number table to simulate the tossing of a fair coin. Estimate the
probability of getting a run of four heads or four tails.
Assume that it is equally likely for a woman to give birth to a girl as it is to give birth to
a boy. What is the probability that a woman with six children has four girls and two
boys? What is the probability that if another woman has four children they are all boys?
A woman with three daughters is going to have a fourth child. What is the probability
that the fourth child will be boy?
A couple wants to have children. They would like to stop having children once they have
a girl, but they do not want to have more than four children. Use a random number table
to simulate the birth of children using the couple’s strategy. Assume that boys and girls
are equally likely to be born. Estimate the probability that the couple will have a girl.
Carry out your own simulation for this model. Use the data collected to estimate the
mean number of children for families using this model of child bearing.
A quiz consists of six multiple choice questions, each with four possible answers. The
questions are very unusual and no one is expected to know any of the answers. So
everyone has to guess the answers. What is the probability that someone guesses four
correctly? What is the probability that someone guesses at least two correctly?
10.5
Understand risk as the probability of occurrence of an adverse event;
investigate some instances of risk in everyday situations, including in
insurance and in medical and genetic matters.
Huntingdon’s disease is a serious condition that may be passed on by women to their
children. Waafaa has a probability of 0.5 of passing on the disease to her daughter
Moza, but her daughter has been killed in a car crash and it is not known whether or not
she was a carrier for the disease. What is the probability that Moza’s daughter will
inherit the disease?
A person who is a carrier for cystic fibrosis has a 0.5 probability of passing on the gene
to his or her child. Since cystic fibrosis is a recessive disorder, any children that inherit
the disease must inherit the gene for cystic fibrosis from both their parents. If both
parents are carriers, what is the probability that their child will have cystic fibrosis?
Further properties of discrete random variables
10.6
Use and understand expected value, or expectation, of a quantified random
variable as the sum of the products of each possible value and the
probability of obtaining that value, and that this is the mean value.
Expectation and variance
A state lottery offers the following 100 prizes for every 100 000 tickets sold:
1 prize of US $5000, 9 prizes of US $500, and 90 prizes of US $50.
A man buys one ticket for US $1.What is his probability of winning nothing? What is the
expectation for his winnings? Is it worth the man’s trouble?
Var( X ) = E( X 2 ) − µ 2
n
E( X ) = ∑ xi P( xi )
1
or
Var( X ) = E[( X − µ )2 ]
The number, X, of occupants of cars coming into a city centre is modelled by the
k
probability distribution P( X = r ) = for r = 1, 2, 3, 4.
r
a. Tabulate the probability distribution and determine the value of k.
b. Calculate E(X) and Var(X).
MEI
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10.7
Use alternative formulae for the variance of a discrete random variable.
Prove that Var( X ) = E( X 2 ) − µ 2 is equivalent to Var( X ) = E[( X − µ ) 2 ] .
Probability of combined events
10.8
Understand when two events are mutually exclusive, and when a set of
events is exhaustive; know that the sum of probabilities for all outcomes of
a set of mutually exclusive and exhaustive events is 1, and use this in
probability calculations.
10.9
Know that:
•
when two events A and B are mutually exclusive the probability of A or
B, denoted by P(A ∪ B), is P(A) + P(B), where P(A) is the probability
of event A alone and P(B) is the probability of event B alone;
•
two events A and B are independent if the probability of A and B
occurring together, denoted by P(A ∩ B), is the product P(A) × P(B);
•
when two events A and B are not mutually exclusive the probability of
A or B, denoted by P(A ∪ B), is
P(A ∪ B) = P(A) + P(B) – P(A ∩ B), where P(A) is the probability of
event A alone, P(B) is the probability of event B alone and P(A ∩ B) is
the probability of both A and B occurring together.
The probability that A occurs is 0.5.
The probability that B occurs is 0.35.
The probability that neither A nor B occurs is 0.3.
Find the probability that both A and B occur.
MEI, modified
10.10
Use tree diagrams to represent and calculate the probabilities of compound
events when the events are independent and when one event is conditional
on another.
An unbiased dice is thrown until a five is recorded. Calculate the probability of winning
after one throw, after two throws, after three throws, and so on. Imagine this process
keeps on building up for ever-increasing numbers of throws to first record a five. Now
calculate the expectation of the number of throws needed to throw a five. If you have set
this up correctly the probability distribution that you obtain is called a geometric
distribution. Can you see why?
45 per cent of the population of a country has a particular disease. A screening test can
be given to help determine whether or not people have the disease. The probability that
the test is positive for those that have the disease is 0.7. But there is a 0.1 chance that a
patient who does not have the disease registers positive on the test. Find the probability
that an individual selected at random tests positive, but does not have the disease.
Another person is chosen at random. Calculate the probability that the test result for
this person is positive.
10.11
Know that in general if event B is dependent on event A, then the
probability of A and B both occurring is P(A ∩ B) = P(A) × P(B|A), where
P(B|A) is the conditional probability of B given that A has occurred.
In a set of 28 dominoes each domino has from 0 to 6 spots at each end. Each domino is
different from every other and the ends are indistinguishable so that, for example, the
two diagrams below represent the same domino.
298 | Qatar mathematics standards | Grade 12 advanced | Quantitative methods
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A domino which has the same number of
spots at each end, or no spots at all, is
called a ‘double’. A domino is drawn at
random from the set. The sample space
diagram on the right represents the
complete set of outcomes, each of which is
equally likely.
6
5
4
3
2
1
0
0
1
2
3
4
5
6
Let the event A be ‘the domino is a
double’, event B be ‘the total number of
spots on the domino is 6’ and event C be
‘at least one end of the domino has 5
spots’.
The diagram on the right shows the
sample space with event A marked.
a. Write down the probability that event A occurs.
b. Find the probability that either B or C or both occur.
c. Determine whether or not events A and B are independent.
d. Find the conditional probability P(A|C). Explain why events A and C are not
independent.
e. After the first domino has been drawn, a second domino is chosen at random from
the remainder. Find the probability that at least one end of the first domino has the
same number of spots as at least one end of the second domino. [Hint: Consider
separately the cases where the first domino is a double and where it is not.]
MEI
In the United Kingdom, pregnant women are screened to see if there is a high risk that
their baby has Down syndrome. The screening test indicates if the risk is high enough to
warrant the woman having further investigations. As with all screening tests, some
women with Down syndrome pregnancies will fail to be detected as being in the high
risk group while a number of normal pregnancies will be identified as high risk. The
result is false positive if the baby tests positive but does not have the syndrome, and a
false negative if the baby has the syndrome but the test result is negative. The true
incidence of the syndrome can be found from other tests and after the babies are born.
The table below gives the result of the screening test on 1400 babies.
Down syndrome
Not Down syndrome
Number of
babies
Number of
positive results
20
14
1380
Number of
negative results
1310
Complete the table.
What percentage of babies had false positive results?
What is the probability that a baby selected at random will have Down syndrome and
give a positive test result?
In the fictitious country of Virtualia there are three prisoners who cannot communicate
with each other. They have been told that next day two of them will be executed, but that
the choice made by random selection will be revealed next morning. Each prisoner
calculates his probability of being executed. One of the prisoners begs the jailor to
reveal the name of one of the other prisoners that will be executed and the jailor finally
agrees. The jailer thinks he has given nothing away. The prisoner thinks his chances of
survival have increased from 1/3 to 1/2. Which of them is correct?
299 | Qatar mathematics standards | Grade 12 advanced | Quantitative methods
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11
11.1
Understand and use the binomial distribution to make
inferences from data
Recognise when to use the binomial distribution and know how to identify
the probability of success, p, and the probability of failure, (1 – p); know
the notation X ~ B(n, p) for a random variable X modelled by the binomial
distribution.
State the conditions to assume that a random variable has a binomial distribution.
The binomial distribution
The binomial distribution is
the distribution in which the
probability of r successes in
n trials is n Cr p r (1 − p )n − r
for r = 0, 1, 2, …, n.
A binomial distribution has n = 5 and p = 0.1. Plot a vertical line graph of the
probability of success against the number of successes. Comment on the skewness of this
distribution.
11.2
Know that the sum of all probabilities in a binomial distribution totals to 1.
Calculate binomial probabilities and expected frequencies for different
numbers of successes.
1 in 20 people are believed to be left-handed. What size sample is needed so that the
expected number of left-handers in the sample is 3?
Over the years two football teams play each other six times. Calculate the probability
that one team wins the toss 4 times. Calculate the probability that one team wins the toss
at least 3 times.
11.3
Calculate the mean and variance of the binomial distribution as µ = np and
σ2 = np(1 – p); use the mean and variance to model sample data expected to
have a binomial distribution.
Mean and variance
No proofs of these results
will be required.
A random variable is X ~ B(n, p). Find the expected value of X and its variance.
Calculate P(µ – σ < X < µ + σ).
[Extension example] Consider the random variable X ~ B(n, p). Let the random variable
Yi (i = 1, 2, …, n) represent the number of successes on the ith trial. Find E(Yi) and
Var(Yi). Use the fact that E( X ) = ∑1 E(Yi ) and that Var( X ) = ∑1 Var(Yi ) to show
n
n
that µ = np and σ2 = np(1 – p).
11.4
Understand the principle of a hypothesis test involving a null hypothesis or
alternative hypothesis, and use the related vocabulary of significance level,
one-tail or two-tail test, critical value, critical region, acceptance region.
Explain the meaning of a significance level to someone who knows no statistics.
11.5
Hypothesis testing
Use the notation H0 for the
null hypothesis and H1 for
the alternative hypothesis.
Set up and perform a hypothesis test on a binomial probability distribution
model, identifying the null hypothesis and the alternative hypothesis, and
make correct inferences from the test.
A road safety team examines the tyres of a large number of commercial vehicles. They
find that 20% of vans have defective tyres. Following a campaign to reduce the
proportion of vehicles with defective tyres, 18 vans are stopped at random and their
tyres are inspected. Just one of the vans has defective tyres. Carry out a suitable
hypothesis test to examine whether the campaign appears to have been successful.
a. State your hypotheses clearly, justifying the form of the alternative hypothesis.
b. Carry out the test at the 5% significance level, stating your conclusions clearly.
c. State, with reason, the critical value for the test.
d. Give a level of significance such that you would come to the opposite conclusion for
your test. Explain your reasoning.
MEI
300 | Qatar mathematics standards | Grade 12 advanced | Quantitative methods
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12
12.1
12.2
Understand and use the Poisson distribution to make
inferences from data
Recognise that the Poisson distribution is used when single events in space
or time occur independently of each other at a constant rate.
P( X = r ) =
e λ
r!
This is named after the
French mathematician
Use the probability distribution
−λ
The Poisson distribution
Simeon Denis Poisson
(1781–1840).
r
to calculate values of P(X) when r = 0, 1, 2, 3, … (ad infinitum), and know
that the sum to infinity of all these probabilities is 1; know that λ is the
parameter of the distribution.
12.3
Know the notation X ~ P(λ) describes the Poisson distribution for a discrete
random variable representing the number of events that occur at random in
a certain interval of space or time, where λ is the mean number of events
that occur in the interval.
If X ~ P(4), calculate P(X = 0), P(X = 1) and P(X ≤ 2).
The number of goals per game scored by football teams playing at home or away in a
football competition are modelled by independent Poisson distributions with means 1.63
and 1.17 respectively.
a. Find the probability that in a game chosen at random:
i.
the home team scores at least 2 goals;
ii.
the result is a 1–1 draw;
iii. the teams score 5 goals between them.
b. Give two reasons why the proposed model might not be suitable.
c. The number of goals scored per game by the Alpha team is modelled by the Poisson
distribution with mean 1.63. In a season they play 19 home games. Use a suitable
approximating distribution to find the probability that Alpha will score more than 35
goals in their home games.
MEI
12.4
Know that both the mean and variance of X ~ P(λ) are equal to λ.
12.5
Know without proof that the Poisson distribution X ~ P(λ) can be used as an
approximation to the binomial distribution X ~ B(n, p) when n tends to
infinity and the mean, np, is kept constant.
13
13.1
Understand and use the normal distribution to make
inferences from data
Understand and describe the main features of a normal distribution for a
continuous random variable.
Normal distribution
The normal distribution is
Explain the main features of the normal distribution to someone who knows no statistics.
13.2
2
Use the notation X ~ N(µ, σ ) for a continuous random variable modelled
by a normal distribution with mean µ and variance σ2.
13.3
Standardise a normally distributed continuous random variable.
13.4
Use statistical tables to read off probabilities for a standardised normal
distribution; know that the total area under the standardised normal
distribution curve is 1; know probabilities for obtaining a result 1, 2 or 3
standard deviation units either side of the mean.
the distribution of many
naturally occurring
variables, such as the
heights of adult men in a
city, the masses of carrots
in a field, and so on.
Explain what you think the ‘68–95–99.7’ rule means in relation to the standard normal
distribution.
301 | Qatar mathematics standards | Grade 12 advanced | Quantitative methods
© Supreme Education Council 2004
13.5
Use the normal distribution as an approximation for the binomial
distribution X ~ N(np, npq) ≈ X ~ B(n, p) when n is large and where
q = 1 – p.
13.6
Use the normal distribution as an approximation for the Poisson distribution
X ~ N(λ, λ) ≈ X ~ P(λ).
13.7
Know that if a population distribution is normal the sampling distribution of
the mean is also normal; know that if the population distribution is not
normal the sampling distribution of the mean is approximately normal for
large samples; know the mean and variance of the sampling distribution of
means in terms of the mean and variance (or estimated variance) of the
population distribution.
13.8
Perform a hypothesis test on a population mean using the standardised
normal distribution in situations where the population variance is known or
where the population variance is unknown but the sample size is large.
Standard error
If samples of size n are
taken from a population
2
with distribution N(µ, σ )
then the distribution of the
sample mean is N(µ,
σ
n
σ2
standard error of the mean.
a. Illustrate this information on a sketch.
b. Show that σ =396 and find the value of µ.
In the remainder of this question take µ to be 4650 and σ to be 400.
c. Find the probability that a bulb chosen at random has a lifetime between 4250 and
4750 hours.
d. Find the probability that a bulb has a lifetime of over 4500 hours.
e. Extralite wish to quote a lifetime which will be exceeded by 99% of bulbs. What
time, correct to the nearest 100 hours, should they quote?
f. A new school classroom has 6 light fittings, each fitted with an Extralite long-life
bulb. Find the probability that no more than one bulb needs to be replaced within
the first 4250 hours of use.
MEI
The lengths of metal rods used in an engineering structure is specified as being 40 cm. It
does not matter if they are slightly longer, but they should not be any shorter. These
rods are made by a machine in such a way that their lengths are normally distributed
with standard deviation 0.2 cm. The mean, µ cm, of the lengths is set to a value slightly
above 40 cm to give a margin for error.
To examine whether the specification is being met, a random sample of 12 rods is taken.
Their lengths, in cm, are found to be
40.43 40.49 40.19 40.36 40.81 40.47
40.46 40.63 40.41 40.27 40.34 40.54
It is desired to test whether µ = 40.5.
a. State a suitable alternative hypothesis for the test.
b. Carry out the test at the 5% level of significance, stating your conclusions carefully.
MEI
Calculate the standard error for a population mean and give a confidence
interval for the mean after applying the confidence test described in PS 13.8
above.
302 | Qatar mathematics standards | Grade 12 advanced | Quantitative methods
).
is often called the
Extralite are testing a new long-life light bulb. The lifetimes, in hours, are assumed to be
normally distributed with mean µ and standard deviation σ. After extensive tests, they
find that 19% of bulbs have a lifetime exceeding 5000 hours, while 5% have a lifetime
under 4000 hours.
13.9
n
© Supreme Education Council 2004
14
14.1
Test for association in bivariate data
Understand the distinction between an independent variable and a
dependent variable.
Variables
In some statistical texts an
14.2
Draw a scatter diagram to suggest strength of relationship between two
variables with interval or ratio scales of measurement and with each
measured on the same subject; know that two variables have a positive
association if larger values of one seem to link to larger values of the other,
and a negative association if larger values of one seem to link with smaller
values of the other.
14.3
Know and use the term linear correlation to indicate if scatter points on a
scatter diagram are clustered around a straight line.
14.4
Calculate the product moment correlation coefficient from bivariate data,
and know that the value lies between –1 and 1; understand the distinction
between positive and negative correlation, and the special cases when
r = –1, 0 or 1.
14.5
Understand that the value of r may be severely affected by outliers.
14.6
Test for evidence for a null hypothesis of no correlation using the calculated
value of r from data and from tables of critical values.
independent variable is
called an explanatory
variable and a dependent
variable is called a
response variable.
Product moment
correlation coefficient
This is usually denoted by r.
A medical statistician wishes to carry out a hypothesis test to see if there is any
correlation between the head circumference and body length of newly born babies.
a. State appropriate null and alternative hypotheses for the test.
A random sample of 20 newly born babies have had their head circumference, x cm, and
body length, y cm, measured. This bivariate sample is illustrated below.
Summary statistics for this data set are as follows.
n = 20
Σ x = 691
Σ y = 1018
Σ x2 = 23 917
Σ y2 = 51 904
Σ xy = 35 212.5
b. Calculate the product-moment correlation coefficient for the data.
Carry out the hypothesis test at the 1% significance level, stating the conclusion
carefully. What assumption is necessary for the test to be valid?
Originally, the point x = 34, y = 51 had been recorded incorrectly as x = 51, y = 34.
c. Calculate the values of the summary statistics if this error had gone undetected.
Use the uncorrected summary statistics to show that the value of the productmoment correlation coefficient would be negative.
d. How is it that this one error produces such a large change in the value of the
correlation coefficient and also changes its sign?
MEI
14.7
Calculate the line of best fit for linear correlation using least squares
regression of dependent variable y on independent variable x.
14.8
Calculate and use Spearman’s coefficient of rank correlation.
303 | Qatar mathematics standards | Grade 12 advanced | Quantitative methods
© Supreme Education Council 2004
14.9
14.10
15
15.1
15.2
Test for evidence of the null hypothesis no association using the rank
correlation coefficient and from tables of critical values.
Appreciate the difference between association and causation.
Understand and apply the chi-squared test
Know that the chi-squared (χ2) test is used to test whether two or more
population proportions are independent of each other and that this is done
using observed frequencies from a sample and expected frequencies from a
probability model to calculate the value of χ2 statistic.
Chi-squared test
This is denoted as the
2
χ test.
Interpret the result of a χ2 test applied to a contingency table in which one
basis for classification is across the columns and the other basis for
classification is along the rows.
Data are extracted from medical records of a random sample of patients of a large
clinic, showing for part of a particular year the frequencies of contracting or not
contracting influenza for patients who had not had influenza inoculations.
Influenza
Inoculated
Yes
No
Yes
8
18
No
35
17
State the null hypothesis for a suitable test of independence of inoculation and
occurrences of influenza. Carry out the test at the 5% level of significance.
MEI
16
16.1
Simulation
Use coins, dice or random numbers to generate models of events described
by random variables and to calculate probabilities and frequencies.
Do an investigation using random numbers to investigate the building up of a queue of
vehicles at a set of traffic lights.
Scientists have invented a fictitious beetle, called the stochastic beetle, that reproduces
in the following manner:
• Different females reproduce independently.
• 50% of the females have two offspring.
• 30% of the females have one offspring.
• The remaining females die out.
Use random numbers to simulate the growth of the population of stochastic beetles,
stating any assumptions that are made in carrying out the simulation and stating clearly
how the random numbers are used in the simulation. What conclusions can be made
about the population of stochastic beetles? How do these conclusions change if you vary
the percentages of the female beetles in the above categories?
You wish to find the least number of people in a gathering so that the probability that
two of them have the same birthday (date of month only, not year of birth) exceeds 0.5.
Plan and carry out a simulation to do this.
Random numbers
These can be generated on
scientific or graphics
calculators using the RND
and RAN function keys.
Large-scale simulations are
best done with computer
software. Random number
tables are also very useful.
ICT opportunity
A range of ICT applications
can support data handling.
Random numbers can be
rapidly generated on a
computer and programs
developed to simulate
particular situations.
Secondary data sets are
readily available on the
17
17.1
Internet. Statistical
Use of ICT
calculations are rapidly
Use a calculator with statistical functions to aid the analysis of large data
sets, and ICT packages to present statistical tables and graphs.
carried out using statistical
software packages or
statistical functions on a
calculator. Statistical charts
and graphs can be drawn
using appropriate software
and graphic calculators.
304 | Qatar mathematics standards | Grade 12 advanced | Quantitative methods
© Supreme Education Council 2004