Business Mathematics and Statistics (MATH0203)

1 Business Mathematics and Statistics (MATH0203) Chapter 1: Correlation & RegressionDependent and independent variables The independent variable(x) is the one that is chosen freely or occur naturally. The dependent variable(y) occurs as a consequence of the value of the independent : Numbers of item produced (x) and total cost of production (y). The time spent on promotion (x) and the level of sales volume (y). Sometimes the relationship between a dependent and an independent variable is called a causal of correlationCorrelation is concerned with describing the strength of the relationshipbetween two variables. Scatter Diagrams Visual representation can give an immediate impression of a set of data.

2 Are these two variables having strong relationship, moderate relationship, weak relationship or no relationship? Independent variable? Dependent variable? Relationship?Question The table below presents the data concerning the number of hours of training in typewriting and the speed of typing a given text for 10 randomly selected typists. Draw a scatter of hour of training12070100501509030408020 Speed (word/minute)30182514352110152010 CORRELATIONTo measure how well the regression line fits the actual of determination (R2) of correlation (R)8 The correlation coefficient, r We need a way of measuring the value of the correlation between two variables. This is achieved through a correlation coefficient, r.

3 Notice that: -1 r 1 Perfect correlationPartial correlationNo correlation11r=1 perfectlypositiverelationshipr=-1 perfectlynegativerelationshipr= strongpositiverelationshipr= strongnegativerelationshipr= moderatepositiverelationshipr= moderatenegativerelationshipr= weakpositiverelationshipr= weaknegativerelationshipr=0 nocorrelation/norelationshipPositive correlation Two variables x and y are moving in the same direction. If x increases, y will increases. If x decreases, y :1)Numbers of calls made by salesman and number of sales )Age of employee and correlation Two variables x and y are moving in the opposite direction. If x increases, y will decreases.

4 If x decreases, y :1)Number of weeks of experience and number of errors )Grade obtained and number of hours watching television. We calculate correlation coefficient by using the following formula:Question : The data of the following table relates the weekly maintenance cost (RM) to the age (in months) of five machines of similar type in a manufacturing company. Calculate the product moment correlation coefficient between age and y r = = _____= _____ 2222)()(yynxxnyxxyn Working An alternative methodof measuring correlation is based on the ranksof the sizes of item values. Rank correlation coefficient:)1(6122 nndrQuestion : Find relationship between mid test and final exam using rank :PersonABCDEFGx50628591745984y6770807968 6781xryr2)(yxrr )1(6122 nndr=The coefficient of determination, r The correlation coefficient is calculated as r = A The coefficient of determination, r = A In words, the B% (A x 100) variation in variable y (specify) is due to variable x (specify).

5 The other (A-B) % of the variation is due to other factors such of regression Regression is concerned with obtaining a mathematical equation, which describes the relationship between two variables. The equation can be used for comparison or estimation purposes. Obtaining a regression line (least square regression line) Formula for obtaining the y on xleast squares regression line, y = a + bx, where Question : Refer back to question , find the least square regression line of machine maintenance cost (y) on machine age (x). Solution:Question : Suppose you obtain the least square regression line: y = , Where x = temperature of the weather ( F), y = water consumption (ounces) Predict the amount of water a person would drink when the temperature is 95 : Given y = , when x = 95, y = (95) = _____ ounces