kut86916_ch01

kut86916_ch01 - Part Simple Linear Regression I Chapter 1...

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Simple Linear Regression Part I
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Chapter 1 Linear Regression with One Predictor Variable Regression analysis is a statistical methodology that utilizes the relation between two or more quantitative variables so that a response or outcome variable can be predicted from the other, or others. This methodology is widely used in business, the social and behavioral sciences, the biological sciences, and many other disciplines. A few examples of applications are: 1. Sales of a product can be predicted by utilizing the relationship between sales and amount of advertising expenditures. 2. The performance of an employee on a job can be predicted by utilizing the relationship between performance and a battery of aptitude tests. 3. The size of the vocabulary of a child can be predicted by utilizing the relationship between size of vocabulary and age of the child and amount of education of the parents. 4. The length of hospital stay of a surgical patient can be predicted by utilizing the rela- tionship between the time in the hospital and the severity of the operation. In Part I we take up regression analysis when a single predictor variable is used for predicting the response or outcome variable of interest. In Parts II and III, we consider regression analysis when two or more variables are used for making predictions. In this chapter, we consider the basic ideas of regression analysis and discuss the estimation of the parameters of regression models containing a single predictor variable. 1.1 Relations between Variables The concept of a relation between two variables, such as between family income and family expenditures for housing, is a familiar one. We distinguish between a functional relation and a statistical relation, and consider each of these in turn. Functional Relation between Two Variables A functional relation between two variables is expressed by a mathematical formula. If X denotes the independent variable and Y the dependent variable, a functional relation is 2
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Chapter 1 Linear Regression with One Predictor Variable 3 FIGURE 1.1 Example of Functional Relation. Dollar Sales 0 50 100 100 150 Y 5 2 X Units Sold X Y 200 300 of the form: Y = f ( X ) Given a particular value of X , the function f indicates the corresponding value of Y . Example Consider the relation between dollar sales ( Y ) of a product sold at a Fxed price and number of units sold ( X ) .If the selling price is $2 per unit, the relation is expressed by the equation: Y = 2 X This functional relation is shown in ±igure 1.1. Number of units sold and dollar sales during three recent periods (while the unit price remained constant at $2) were as follows: Number of Dollar Period Units Sold Sales 17 5 $150 22 5 5 0 3 130 260 These observations are plotted also in ±igure 1.1. Note that all fall directly on the line of functional relationship. This is characteristic of all functional relations.
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kut86916_ch01 - Part Simple Linear Regression I Chapter 1...

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