Business Mathematics and Statistics

(MATH0203)

Chapter 1:

Correlation & Regression

Dependent 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 variable.

Example:

• 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 relationship.

Definition of correlation

Correlation is concerned with describing the strength of the relationship between two variables.

Scatter Diagrams

• Visual representation can give an immediate impression of a set of data. Are these two variables having strong relationship, moderate relationship, weak relationship or no relationship?

• Independent variable?

• Dependent variable?

• Relationship?

Question 1.1

• 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 diagram.

Typist 1 2 3 4 5 6 7 8 9 10

Number of

hour of training

120 70 100 50 150 90 30 40 80 20

Speed

(word/minute)

30 18 25 14 35 21 10 15 20 10

CORRELATION

To measure how well the regression line fits the actual data

By:

i. Coefficient of determination (R2)

ii. Coefficient 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.

• Notice that:

-1 ≤ r ≤ 1

Perfect correlation Partial correlation

No correlation

11

r = 1 perfectly positive

relationship

r = -1 perfectly negative

relationship

r = 0.9 strong positive

relationship

r = -0.9 strong negative

relationship

r = 0.5 moderate positive

relationship

r= -0.5 moderate negative

relationship

r = 0.2 weak positive

relationship

r = -0.2 weak negative

relationship

r = 0 no correlation / no

relationship

Positive correlation

• Two variables x and y are moving in the same direction.

• i.e. If x increases, y will increases. If x decreases, y decreases.

Examples:

1) Numbers of calls made by salesman and number of sales obtained.

2) Age of employee and salary.

Negative correlation

• Two variables x and y are moving in the opposite direction.

• i.e. If x increases, y will decreases. If x decreases, y increases.

Example:1) Number of weeks of experience and number of

errors made.2) Grade obtained and number of hours watching

television.

• We calculate correlation coefficient by using the following formula:

Question 1.2:

• 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 cost.

Machine 1 2 3 4 5

Age 5 10 15 20 30

Cost 10 20 20 30 30

x

y

xy

x²

y²

r = = __________________

= __________________

2222 )()( yynxxn

yxxyn

Working

• An alternative method of measuring correlation is based on the ranks of the sizes of item values.

• Rank correlation coefficient:

)1(

61

2

2

nn

dr

Question 1.3:

• Find relationship between mid test and final exam using rank correlation.

Person A B C D E F G

Mid test score 50 62 85 91 74 59 84

Final Exam score 67 70 80 79 68 67 81

Solution:

Person A B C D E F G

x 50 62 85 91 74 59 84

y 67 70 80 79 68 67 81

xr

yr2)( yx rr

)1(

61

2

2

nn

dr =

The coefficient of determination, r²

• The correlation coefficient is calculated as r = A

• The coefficient of determination, r ²= A²

• In words, the B% (A² x 100) variation in variable y (specify) is due to variable x (specify). The other (A-B) % of the variation is due to other factors such as………..

Definition 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 x least squares regression line, y = a + bx, where

•

Question 1.4:

• Refer back to question 1.2, find the least square regression line of machine maintenance cost (y) on machine age (x).

• Solution:

Question 1.5:

• Suppose you obtain the least square regression line: y = 1.5x - 96.9,

• 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 °F.

Solution:• Given y = 1.5x – 96.9, • when x = 95, • y = 1.5(95) – 96.9 • = ___________ ounces