Statistics

Lesson 02

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)

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)

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.

1.17 Functions and Relations

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.

1.17 Functions and Relations

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.

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)

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)

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).

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.

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.

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)

1.20 The real number line and rectangular Cartesian coordinate system

A Rectangular Cartesian Coordinate System

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.

1.21 Graphing FunctionsQuadratic 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.

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”.

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

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

1.22 Sequences, Series and Summation Notation

The summation notation is a symbolic representation of the series: x1+ x2 + x3 + … + xn

1.22 Sequences, Series and Summation Notation

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.

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."

1.23 Inequalities

Example 1.40 For the inequality 8 > 6Multiply both sides by -3

Example 1.41 Solve the inequality: X + 7 > -3

Questions

1.80 Using the quadratic formula to solve 4X2 = 1

Questions

For y = f(x) = 7x - 5, find (a) f(0)(b) 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.

Questions