1.15 VariablesVariables are things that we measure, control, ormanipulate in research.Example:In studying a group of children, the weight of eachchild is a variable – it is measurable and it variesfrom child to child.Variate: Each individual measurement of a variable(e.g., each weight of a child)
Quantitative and Qualitative VariableA Quantitative Variable: whose variates can beordered by the magnitude of the characteristicsuch as weight, length, quantity and so on.e.g., number of tomatoes on a plant.A Qualitative Variable: whose variates aredifferent categories and cannot be ordered bymagnitude. (e.g., type of tree)
1.16 Observable and Hypothetical Variables.Observable Variables: Directly measurable suchas height, weight.Hypothetical Variables: Indirectly measurablesuch as inherited differences between shortdistance or long distance runners.
1.17 Functions and RelationsIf 2 variables X and Y are related that for everyspecific value x of X is associated with only onespecific value y of Y, that Y is a function of X.A domain is the set of all specific x values that Xcan assume.A range is the set of all specific y valuesassociated with the x values.
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 RelationsExample 1.29 For the relation Y = X ± 3, whatare 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 NotationFor Y = X2, the functional notation is y = f(x) = x2For y = f(x) = -3 + 2x + x2 , find f(0) and f(1)
1.19 Functions in StatisticsThe 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 soilcharacteristics (causes) that influence plantgrowth (effect); or an economist want to knowthe advertising factors (causes) that influencesales (effect).
1.19 Functions in StatisticsExample 1.31 In the following experiment, which isthe independent variable and which is thedependent variable?To determine the effects of water temperature onsalmon growth, you raise 2 groups of salmon (10 ineach group) under identical conditions fromhatching, except that one group is kept in 20 Cwater and the other in 24 C water. Then 200 daysafter hatching, you weigh each of the 20 salmon.
1.20 The real number line and rectangular Cartesian coordinate systemEvery number in the real number system can berepresented by a point on the real number line.
1.20 The real number line and rectangular Cartesian coordinate systemA rectangular Cartesian coordinate system (orrectangular coordinate system) is constructedby making two real number line perpendicularto each other, such that their point ofintersection (the origin) is the zero point of bothlines.Example 1.33 Plot the following points on arectangular 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 FunctionsA graph is a pictorial representation of therelationship between the variables of a function.Example 1.34 Graph the function y=f(x)=4 + 2xon a rectangular coordinate system.
1.21 Graphing FunctionsQuadratic function:•Characteristics of Quadratic Functions•1. Standard form is y = ax2 + bx + c, wherea≠ 0.•2. The graph is a parabola, a u-shapedfigure.•3. The parabola will open upward ordownward.•4. A parabola that opens upwardcontains a vertex that is a minimumpoint.A parabola that opens downwardcontains a vertex that is a maximumpoint.
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 NotationExample 1.35 What are the terms of thissequence: f(i) = i2 – 3, for i = 2, 3, 4
1.22 Sequences, Series and Summation NotationA 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, theseries is x1+ x2 + x3
1.22 Sequences, Series and Summation NotationThe summation notation is a symbolicrepresentation of the series: x1+ x2 + x3 + … + xn
1.22 Sequences, Series and Summation NotationWhen it is clear that it is the entire set beingsummed, the lower and upper limits of thesummation are often omitted.Example 1.37 The height of five boys in a 3rdgrad class form the following sequence: x1 = 2.1ft, 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 InequalitiesExample 1.40 For the inequality 8 > 6Multiply both sides by -3Example 1.41 Solve the inequality: X + 7 > -3
Questions1.80 Using the quadratic formula to solve 4X2 = 1
QuestionsFor y = f(x) = 7x - 5, find(b) f(0)(c) f(5)1.84 Graph the linear function y = f(x) = 3- 0.5xon a rectangular coordinate system using itsslope and y intercept.