1. Chapter 5
STATISTICS (PART 4)

2. introduction
Many problems in science and engineering involve exploring the relationship between two or more variables.
Two statistical techniques:
Regression Analysis
Computing the Correlation Coefficient (r).
The correlation coefficient (r )
Tells us how strongly two variables are related.

3. introduction
In simple correlation and regression studies, the researcher collects data on two numerical or quantitative variables to see whether a relationship exists between the variables.
For example, if a researcher wishes to see whether there is a relationship between number of hours of study and test scores on an exam, she must select a random sample of students, determine the hours each studied, and obtain their grades on the exam. A table can be made for the data, as shown here.
Student A B C D E F
Hour of study, x 6 2 1 5 3
Grade, y (%) 82 63 57 88 68 75

4. Regression and Correlation
Regression is a statistical procedure for establishing the relationship between 2 or more variables
Linear regression - study on the linear relationship between two or more variables. This is done by fitting a linear equation to the observed data.
2 types of relationship:
1) Simple ( 2 variables)
2) Multiple (more than 2 variables)

5. In simple linear regression only two variables are involved:
X is the independent variable.
Y is dependent variable.
Independent variable:
Variable in regression that can be manipulated.
Also known as explanatory variable.
Dependent variable:
Variable in regression that cannot be manipulated.
Also known as response variable.
Correlation describes the strength of a linear relationship between two variables
Linear means “straight line”

6. Example 6.1:
A nutritionist studying weight loss programs
might wants to find out if reducing intake of
carbohydrate can help a person reduce weight.
a) X is the carbohydrate intake (independent
variable).
b) Y is the weight (dependent variable).
An entrepreneur might want to know whether increasing the cost of packaging his new product will have an effect on the sales volume.
a) X is cost
b) Y is sales volume

7. Regression Uses a variable (x) to predict some outcome variable (y)
Tells you how values in y change as a function of changes in values of x

8. Regression Calculates the “best-fit” line for a certain set of data.
Best fit means that the sum of the squares of the vertical distances from each point to the line is at a minimum.
The reason you need a line of best fit is that the values of y will be predicted from the values of x; hence, the closer the points are to the line, the better the fit and the prediction will be

9. The Simple Linear Regression Model
is an equation that describes a dependent variable (Y) in terms of an independent variable (X) plus random error,
where,
= intercept of the line with the Y-axis
= slope of the line
= random error
Random error, is the difference of data point from the deterministic value.
This regression line is estimated from the data collected by fitting a straight line to the data set and getting the equation of the straight line,

10. Regression Equation
Regression equation describes the regression line mathematically
Intercept,
Slope,
By using the least squares method (a procedure that minimizes the vertical deviations of plotted points surrounding a straight line) we are able to construct a best fitting straight line to the scatter diagram points and then formulate a regression equation in the form of:

11. Regression Equation Positive Linear Relationship E(y) Regression linex
Regression line
Intercept b0
Slope b1
is positive

12. Negative Linear Relationship
E(y) x
Intercept b0
Regression line
Slope b1
is negative

13. No Relationship
E(y) x
Intercept b0
Regression line
Slope b1
is 0

14. INFERENCES ABOUT ESTIMATED PARAMETERS: Least Squares Method
The least squares method is commonly used to determine values for and that ensure a best fit for the estimated regression line to the sample data points.
Rounding rule for intercept and slope: 3 decimal places.
The straight line fitted to the data set is the line:

15. LEAST SQUARES METHOD Theorem:
Given the sample data , the coefficients of the least squares line are:
i) y-intercept for the Estimated Regression
Equation:
and are the mean of x and y respectively.

16. ii) Slope for the Estimated Regression Equation,
where

17. LEAST SQUARES METHOD
Given any value of the predicted value of the dependent variable , can be found by substituting into the equation

18. Exercise 6.1:
A sample of 6 persons was selected the value of their age ( x variable) and their weight is demonstrated in the following table. Find the regression equation and what is the predicted weight when age is 8.5 years.
Serial no.
Age (x)
Weight (y) 1 2 3 4 5 6 7 8 9 12 10 11 13

19. Answer
Serial no.
Age (x)
Weight (y) xy X2 Y2 1 2 3 4 5 6 7 8 9 12 10 11 13 84 48 96 50 66
117 49 36 64 25 81
144
100
121
169
Total 41
461
291
742

20. Compute
Compute

21. Compute
Thus, regression model is
let x = 8.5:

22. let x = 7.5 :
we create a regression line by plotting two estimated values for y against their X component, then extending the line right and left.

23. Exercise 6.2:
The data below represent scores obtained by ten primary school students before and after they were taken on a tour to the museum (which is supposed to increase their interest in history)
Before,x 65 63 76 46 68 72 57 36 96
After, y 66 86 48 71 42 87
Find a linear regression model with “before” as the explanatory variable and “after” as the dependent variable.
Predict the score a student would obtain “after” if he scored 60 marks “before”.

26. Correlation
Finding the relationship between two quantitative variables without being able to infer causal relationships
Correlation is a statistical technique used to determine the degree to which two variables are related

27. Scatter plots The pattern of data is indicative of the
type of relationship between your two
variables:
positive relationship
negative relationship
no relationship

28. Positive relationship

30. Negative relationship
Reliability
Age of Car

31. No relation

32. Simple Correlation coefficient (r)
Correlation coefficient – statistic showing the degree of relation between two variables
It is also called Pearson's correlation or product moment correlation coefficient.
It measures the nature and strength between two variables of the quantitative type.
The symbol for the sample coefficient of correlation is r, population ρ.
Formula :

33. The sign of r denotes the nature of association
while the value of r denotes the strength of association.
If the sign is +ve this means the relation is direct (an increase in one variable is associated with an increase in the other variable and a decrease in one variable is associated with a decrease in the other variable).
While if the sign is –ve this means an inverse or indirect relationship (which means an increase in one variable is associated with a decrease in the other).

34. Properties of :
Values of close to 1 implies there is a strong positive linear relationship between x and y.
Values of close to -1 implies there is a strong negative linear relationship between x and y.
Values of close to 0 implies little or no linear relationship between x and y.

35. The value of r ranges between ( -1) and ( +1)
The value of r denotes the strength of the association as illustrated by the following diagram.
strong
intermediate
weak -1 1
-0.25
-0.75
0.75
0.25
no relation
perfect correlation
Direct
indirect

36. If r = Zero this means no association or correlation between the two variables.
If 0 < r < 0.25 = weak correlation.
If 0.25 ≤ r < 0.75 = intermediate correlation.
If 0.75 ≤ r < 1 = strong correlation.
If r = l = perfect correlation.

38. Refer Exercise 6.2: Students score in history c) Calculate the value of r and interpret its meaning Solution: Thus, there is a strong positive linear relationship between score obtain before (x) and after (y).

39. Exercise 6.3:
A sample of 6 children was selected, data about their age in years and weight in kilograms was recorded as shown in the following table . It is required to find the correlation between age and weight.
serial No
Age (years)
Weight (Kg) 1 7 12 2 6 8 3 4 5 10 11 9 13

40. Serial n.
Age (years)
(x)
Weight (Kg)
(y) xy x2 y2 1 7 12 84 49
144 2 6 8 48 36 64 3 96 4 5 10 50 25
100 11 66
121 9 13
117 81
169
Total
∑x= 41
∑y=
∑xy= 461
∑x2=
291
∑y2=
742

41. r = 0.759
Since the r value close to 1 implies that there is strong positive linear relationship between age (x) and weight (y).

42. Exercise 6.4:
1. Hospital Beds Licensed beds and staffed beds data follow:
Find y when x 44.
2. Commercial Movie Releases New movie releases per studio
and gross receipts are as follows:
i) Find y when x = 200 new releases
ii) Is there a significant relationship between these two
variables?
Licensed beds, x
144 32
175
185
208
100
169
Staffed beds, y
112
162
141
103 80
118
No. of releases
361
270
306 22 35 10 8 12 21
Gross receipts
(million $)
3844
1962
1371
1064
334
241
188
154
125

43. Answer: 1. 2.
positive linear relationship between x and y