Chapter 10: Inferences Involving
Two Populations
Chapter Goals
Independent versus dependent samples.
Compare two populations using:
(i) the mean of the paired differences,
(ii) the difference between two means,
(iii) the ratio of two variances.
1
10.1:
MA2182 Take Home Assignment 1 Note: (a) Deadline: 8pm on Wednesday, 22 Oct, 08. Normally, late submission is NOT accepted. (b) Please submit the assignment by depositing it at Y6514. (c) Please make your own photocopies of your assignment before submittin
MA2182 Notes 6
Trigonometry
Trigonometric Functions 1. In elementary trigonometry, the trigonometric functions are defined as the ratios of sides of a right-angled triangle and the angles are restricted to acute angles. 2. The six trigonometric functions
MA2182 Practice Exercise 4 Algebraic Functions, Factor and Remainder Theorems, Partial Fractions 1. (a) (b) (c) (d) It is given that g ( x ) = -3 x 2 + 24 x - 36 . Find g (2 x ) and g (- x ) . Determine whether g ( x ) is odd or even or neither of them. D
MA2182 Practice Exercise 5
1.
Exponential and Logarithmic Functions
Determine whether each of the following is true or false. If it is false, give a counterexample.
i. If N is positive, then as N increases, log5 N increases. (T / F)
ii. If a, b are positi
MA2182 Practice Exercise 3 1.
Set Notation and Bounded Intervals, More on Functions
Let A = cfw_x R | -3 < x 8, B = cfw_x R | -11 x < -3 , C = cfw_x Z | -11 x < -3 and D = cfw_x R | x > 5. (a) Use bounded intervals to represent the above sets, if it is po
MA2182 Practice Exercise 4 Algebraic Functions, Factor and Remainder Theorems, Partial Fractions 1. (a) (b) (c) (d) It is given that g ( x ) = -3 x 2 + 24 x - 36 . Find g (2 x ) and g (- x ) . Determine whether g ( x ) is odd or even or neither of them. D
MA2182 Practice Exercise 1
Coordinate Geometry and Conic Section
Completing the square 2 2 (With reference to the identity (a + b ) a 2 + 2ab + b 2 and (a - b ) a 2 - 2ab + b 2 .)
Express each of the following expression in the form ( x + a ) + b or ( y +
MA2182 Notes 1
Coordinate Geometry and Conic Section
Review In the rectangular/Cartesian coordinates system, we describe the location of points using coordinates.
y
P2(x2, y2)
P(x, y)
P1(x1, y1)
x
The distance d between two points P (x1 , y1 ) and P2 ( x2
MA2182 Notes 2 Binomial Coefficient
Binomial Theorem, Arithmetic and Geometric Progressions
If n and m are integers with 0 m n , then the binomial coefficient n C r , is defined by n! , n Cr = r!(n - r )! where r!= r (r - 1)(r - 2 )K3 2 1 for r > 0 (calle
MA2182 Notes 3 Set Notation
Set Notation and Bounded Intervals, More on Functions
A set is a collection of distinct objects. e.g. Let V be the set of all vowels of the English alphabets, then V = cfw_a, e, i, o, u e.g. Let M be the set of all months in a
MA2182 Practice Exercise 1
Answers (Questions 8-11)
8.
(a)
(b)
(c)
1
2
9.
(a)
(b)
(c)
(d)
10. (a) (b) (c) (d) 11. (a) No intersection (b)
It represents a circle. It represents a parabola. It represents a hyperbola. It represents an ellipse.
(0,-2) and (1,
MA2182 Practice Exercise 2 1.
Binomial Theorem, Arithmetic and Geometric Progressions
Evaluate each of the following. (a) n C n -3 (b) n C n- 2 + n C n -1 Write out each of the sums below. (a)
2.
(i
i =1
6
2
+1
)
(b)
[(- 2) - 5]
r r =4
7
(c)
r -1 r r =7
MA2182 Notes 4
Algebraic Functions, Factor and Remainder Theorems, Partial Fractions
A function f ( x ) = an x n + an -1 x n -1 + L + a0 (where the a ' i s are real numbers and n is a non-negative integer) is called a polynomial function. If a n 0 , n is
MA2182 Practice Exercise 1
Coordinate Geometry and Conic Section
Completing the square 2 2 (With reference to the identity (a + b ) a 2 + 2ab + b 2 and (a - b ) a 2 - 2ab + b 2 .)
Express each of the following expression in the form ( x + a ) + b or ( y +
MA2182 Practice Exercise 6
1.
Trigonometry
(a) Convert the following angles to radians.
(i) 48
(ii)
120
(b) Convert the following angles to degree.
123
(i) rad
(ii)
rad
6
180
(iii)
315
(iii)
2
rad
5
to x = 2 ; (ii) find its domain
2
and range; (iii) deter
MA2182 Notes 10
Matrix, System of Linear Equations and Determinant
Introduction of Matrix
A matrix of order m n or an m n matrix is a rectangular array of numbers having m rows and n columns.
a11 a12 . a1n
a
21 a22 . a2 n
It can be written as A = .
.
MA2182 Notes 11
Matrix, System of Linear Equations and Determinant (Contd)
Inverse of a Matrix
Definition:
The inverse of a square n n matrix A (denoted by A 1 ) is an n n matrix such that
AA1 = A1 A = I n ,
where I n is the n n identity matrix.
Illustrat
MA2182 Practice Exercise 9
Complex Numbers
1.
Determine whether each of the following is true or false.
(g)
5 cos 390 o + i sin 390 o = 5 cos 30 o + i sin 30 o
Solutions
(a)
F
5 = 5 + 0i is a complex number.
(b)
F
8i 2 = 8( 1) = 8 is a real number.
(c)
T
MA2182 Practice Exercise 10 and 11
Matrix, System of Linear Equations and Determinant
1.
2.
3.
2 3 0
2 0 0
Given A = 1 3 4 and D = 0 4 0 .
1 2 1
0 0 1
(a)
Compute A2 and A3.
(b)
Compute D2 and D3.
(c)
What can you observe from the results of (a) and (b)