STK 123 Summary of whole semester

Contents
Section A: Optimization techniques .............................................................................. 2
Syllabus theme 1: Data Transformations and relationships with economic
applications ..................................................................................................................... 2
1.7. Linear inequalities and systems thereof .............................................................. 2
1.8 Graphical solution ................................................................................................. 6
1.9 Extreme point method ........................................................................................ 10
Syllabus theme 2 ........................................................................................................... 14
2.1 LIMIT OF A FUNCTION .............................................................................................. 14
2.2 Continuity .................................................................................................................... 20
2.3 Rate of change........................................................................................................... 23
2.4 Derivative of a function....................................................................................... 26
2.5 Differentiation rules .............................................................................................. 28
2.6 Higher order derivatives ...................................................................................... 30
2.7 Maxima and minima ........................................................................................... 31
2.8 Area under the curve .......................................................................................... 37
2.9 Application of definite integrals ......................................................................... 45
Section B ........................................................................................................................ 48
Syllabus Theme 1: Probability ....................................................................................... 48
1.2 Discrete Probability distribution........................................................................... 48
1.3 Continuous probability ........................................................................................ 55
Syllabus Theme 2: Statistical Inference ........................................................................ 61
2.1 Sampling and Sampling Distribution .................................................................. 61
2.2 Interval Estimation ................................................................................................ 73
2.3 Hypothesis testing (One sample) ........................................................................ 80
2.4 Hypothesis testing (Two Samples)....................................................................... 93




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,Section A: Optimization techniques
Syllabus theme 1: Data Transformations and relationships with
economic applications
1.7. LINEAR INEQUALITIES AND SYSTEMS THEREOF
Definitions Linear function y =x (line)

Strict linear inequality y < x (region, dotted line)

Weak linear inequality y ≤ x(region, solid line)

Graphical 1. Plot the line
representation
2. Shade region of feasible solutions:

 all coordinates that satisfy the inequality



Example 1 2x + 3y  12

 weak linear inequality



1. Rewrite as y  4 – 2/3x
2. Plot line
o y = 0:
o 0 = 4 – 2/3x
o x=6

o x = 0:
o y= 4 – 2/3(0)
o y=4

3. Shade region
o if x = 0 then y  4
o if y = 0 then x  6

4. Graph




Example 2 x – y > -3


2

, 1. Rewrite as y  4 – 2/3x
2. Plot line
y = 0:
0=x+3
x = -3

x = 0:
y= 0 + 3
y=3

3. Shade region
if x = 0 then y < 4
if y = 0 then x > -3

4. Graph
Straight line is not included




System of Definition:
linear
When more than one linear inequality is represented on the same
inequalities
graph


Region of feasible solutions:

All points whose co-ordinates satisfy all the inequalities
simultaneously, thus the solution area



Example

1. 2x + 3y  12:

 (0;4)(6;0)


2. x + 2y > 4

 (0;2)(4;0)




3

, 3. x – y > -3

 (0;3)(-3;0)




Shaded region = region of feasible solutions: consisting of all
points whose co-ordinates satisfy all the inequalities
simultaneously

Practical 1. Man’s hours available:
application
24 for moulding / painting

16 for firing process



x coffee sets:
3h for moulding/ painting

2h for firing



y tea sets:

2h for moulding/ painting

1h for firing




2. Conditions:
3x + 2y  24

2x + 1y  16

x≥0



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