2. 1.15 Variables
Variables are things that we measure, control, or
manipulate in research.
Example:
In studying a group of children, the weight of each
child is a variable – it is measurable and it varies
from child to child.
Variate: Each individual measurement of a variable
(e.g., each weight of a child)
3. Quantitative and Qualitative Variable
A Quantitative Variable: whose variates can be
ordered by the magnitude of the characteristic
such as weight, length, quantity and so on.
e.g., number of tomatoes on a plant.
A Qualitative Variable: whose variates are
different categories and cannot be ordered by
magnitude. (e.g., type of tree)
4. 1.16 Observable and Hypothetical
Variables.
Observable Variables: Directly measurable such
as height, weight.
Hypothetical Variables: Indirectly measurable
such as inherited differences between short
distance or long distance runners.
6. 1.17 Functions and Relations
If 2 variables X and Y are related that for every
specific value x of X is associated with only one
specific value y of Y, that Y is a function of X.
A domain is the set of all specific x values that X
can assume.
A range is the set of all specific y values
associated with the x values.
8. 1.17 Functions and Relations
• When an x value is selected, the y value is
determined. Therefore, the y value ‘depends’
on x value.
X is the ‘independent variable’ of the function
and Y is the ‘dependent variable.’
And Y is a function of X.
9. 1.17 Functions and Relations
Example 1.29 For the relation Y = X ± 3, what
are its domain, range, and rule of association?
There are two y values for every x.
Domain: x values, (1, 2, 3)
Range: y values. (-2 & 4, -1 & 5, 0 & 6)
10. 1.18 Functional Notation
For Y = X2, the functional notation is
y = f(x) = x2
For y = f(x) = -3 + 2x + x2 , find f(0) and f(1)
11. 1.19 Functions in Statistics
The goal of research is to study cause and effect;
to discover the factors that cause something
(the effect) to occur.
Example: a botanist want to know the soil
characteristics (causes) that influence plant
growth (effect); or an economist want to know
the advertising factors (causes) that influence
sales (effect).
12. 1.19 Functions in Statistics
Example 1.31 In the following experiment, which is
the independent variable and which is the
dependent variable?
To determine the effects of water temperature on
salmon growth, you raise 2 groups of salmon (10 in
each group) under identical conditions from
hatching, except that one group is kept in 20 C
water and the other in 24 C water. Then 200 days
after hatching, you weigh each of the 20 salmon.
13. 1.20 The real number line and rectangular
Cartesian coordinate system
Every number in the real number system can be
represented by a point on the real number line.
14. 1.20 The real number line and rectangular
Cartesian coordinate system
A rectangular Cartesian coordinate system (or
rectangular coordinate system) is constructed
by making two real number line perpendicular
to each other, such that their point of
intersection (the origin) is the zero point of both
lines.
Example 1.33 Plot the following points on a
rectangular coordinate system: A(0,0); B(-1.3);
C(1,-3); D(2,1); E(-4,-2)
15. 1.20 The real number line and rectangular
Cartesian coordinate system
A Rectangular Cartesian Coordinate System
16. 1.21 Graphing Functions
A graph is a pictorial representation of the
relationship between the variables of a function.
Example 1.34 Graph the function y=f(x)=4 + 2x
on a rectangular coordinate system.
17. 1.21 Graphing Functions
Quadratic function:
•Characteristics of Quadratic Functions
•1. Standard form is y = ax2 + bx + c, where
a≠ 0.
•2. The graph is a parabola, a u-shaped
figure.
•3. The parabola will open upward or
downward.
•4. A parabola that opens upward
contains a vertex that is a minimum
point.
A parabola that opens downward
contains a vertex that is a maximum
point.
18. 1.22 Sequences, Series and
Summation Notation
• Sequence: a function with a domain that consists
of all or some part of the consecutive positive
integers.
• Infinite Sequence: the domain is all positive
• Finite Sequence: the domain is only a part of the
consecutive positive integers.
• Term of the Sequence: Each number in the
sequence.
• f(i) = xi, for i = 1, 2, 3. the i in the xi is “subscript
or an index, and xi is read “x sub I”.
19. 1.22 Sequences, Series and
Summation Notation
Example 1.35 What are the terms of this
sequence: f(i) = i2 – 3, for i = 2, 3, 4
20. 1.22 Sequences, Series and
Summation Notation
A series is the sum of the terms of a sequence.
For the infinite sequence f(i) = I + 1, for I = 1, 2,
3, …, ∞, the series is the sum 2 + 3 + 4 + … + ∞.
For the finite sequence f(i) = xi, for i = 1, 2, 3, the
series is x1+ x2 + x3
21. 1.22 Sequences, Series and
Summation Notation
The summation notation is a symbolic
representation of the series: x1+ x2 + x3 + … + xn
23. 1.22 Sequences, Series and
Summation Notation
When it is clear that it is the entire set being
summed, the lower and upper limits of the
summation are often omitted.
Example 1.37 The height of five boys in a 3rd
grad class form the following sequence: x1 = 2.1
ft, x2 = 2.0 ft, x3 = 1.9 ft, x4 = 2.0 ft, x5 = 1.8 ft.
For this set of measurement, find sum.
24. 1.23 Inequalities
• THIS SIGN < means is less than.. This sign >
means is greater than. In each case, the sign
opens towards the larger number.
• For example, 2 < 5 ("2 is less than 5").
Equivalently, 5 > 2 ("5 is greater than 2").
• These are the two senses of an inequality: <
and > .
• the symbol ≤, "is less than or equal to;" or ≥,
"is greater than or equal to."
25. 1.23 Inequalities
Example 1.40 For the inequality 8 > 6
Multiply both sides by -3
Example 1.41 Solve the inequality: X + 7 > -3
27. Questions
For y = f(x) = 7x - 5, find
(b) f(0)
(c) f(5)
1.84 Graph the linear function y = f(x) = 3- 0.5x
on a rectangular coordinate system using its
slope and y intercept.
