Lesson04

IBS Statistics Year 1

Dr. Ning DING [email protected]

What we are going to learn?

• Review

• Chapter 12: Simple Regression and Correlation– dependent / independent variables– scatter diagrams– regression analysis– Least-squares estimating equation– the coefficient of determination– the coefficient of correlation

• Review

• Chapter 12: Simple Regression and Correlation

• Exercises

Review

Find the interquartile range: 146014711637172117581787194020382047205420972205228723112406

Interquartile Range=Q3-Q1

=2205-1721=484

• Review


• Exercises

Review EXCEL Lesson

L=(8+1)*25%=2.25

Q1=133.5

L=(8+1)*75%=6.75

Q3=274.5

Interquartile Range=274.5-133.5=141

Review

Boxplot

12245789

12

Median1224

789

12

Quartile

Q1=2

Q3=8.5

5Interquartile

Range

Decile

1st D

9th D

Percentile

http://cnx.org/content/m11192/latest/

How to interpret?


The distribution is skewed to __________ because the mean is __________the median.

the right larger than


€ 20 € 2000Q1= € 250 Q3= € 850Median= € 350

Mean= € 450a b

• Review


• Exercises

Review



0.81.01.01.21.21.31.51.72.02.02.12.24.0

2.03.23.63.74.04.24.24.54.54.64.85.05.0

Mean > Median

Mean < Median

Positively skewed

Negatively skewedhttp://qudata.com/online/statcalc/

Review

http://qudata.com/online/statcalc/

This means that the data is symmetrically distributed.

Zero skewness

mode=median=mean

Zero skewness

mode=median=mean

Review

– scatter diagrams– dependent / independent variables– regression analysis– Least-squares estimating equation– the coefficient of determination– the coefficient of correlation

Chapter 12• Review

• Chapter 12: •scatter diagrams•dependent / independent variables•regression analysis•Least-squares estimating equation•the coefficient of determination•the coefficient of correlation

Regression and Correlation Analyses

– How to determine both the nature and the strength of a relationship between variables.

• Review



• Review


Scatter Diagram:

28

Describing Relationship between Two Variables – Scatter Diagram Examples

Positive correlationPositive correlation

28



• Review


Scatter Diagram:

Negative correlationNegative correlation

28



• Review


Scatter Diagram:

No correlationNo correlation


• Review


Scatter Diagrams:• Patterns indicating that the variables are related• If related, we can describe the relationship

Strong & Positivecorrelation

Strong & Negativecorrelation

Weak & Positivecorrelation

Weak & Negativecorrelation

Nocorrelation

28


28

Describing Relationship between Two Variables – Scatter Diagram ExamplesVariables: – Independent variables: known

– Dependent variables: to predict

Independent Variable

Dependent Variable

• Review



• Review



Correlation & Cause Effect?

• The relationships found by regression to be relationships of association

• Not necessarilly of cause and effect.

• Review


Least-squares estimating equation:• The dependent variable Y is determined by the independent

variable X

Ŷ = a + bX

Y

X

Independent Variable

Dependent Variable• Review


88 ?I

Ŷ = a + bX

• Review


Least-squares estimating equation:

xn-x

y xn-xy=b 22

Y = a + bX a = Y - bX

• Review



• Review


75.09*444

6*3*478

-

-=b

the relationship between the age of a truck and the annual repair expense?

X=3 Y=6

xn-x

y xn-xy=b

22

a = 6 - 0.75*3 = 3.75

Ŷ = 3.75 + 0.75 X

If the city has a truck that is 4 years old,

the director could use the equation to predict $675 annually in repairs.

6.75 = 3.75 + 0.75 * 4


Y = a + bX a = Y - bX

Step 1:

Step 2:

Step 4:

Step 5:

Step 6:

Step 7:

Step 8:

• Review


Example:• To find the simple/linear regression of Personal Income (X) and

Auto Sales (Y)

Count the number of values.

Step 1:

Find XY, X2 See the below tableStep 2:

N = 5N = 5

If X=64, what about Y?


Step 3:

Step 4:

Find ΣX, ΣY, ΣXY, ΣX2. ΣX = 311 Mean = 62.2 ΣY = 18.6 Mean = 3.72 ΣXY = 1159.7 ΣX2 = 19359

xn-x

y xn-xy=b

22

Substitute in the above slope formula given. Slope(b) = = 0.19 1159.7-5*62.2*3.72

19359-5*62.2*62.2


• Review


Step 5:

Then substitute these values in regression equation formula Regression Equation(Ŷ) = a + bX

Ŷ = -8.098 + 0.19X

Slope(b) = 0.19

Now, again substitute in the above intercept formula given.

Intercept(a) = Y - bX = 3.72- 0.19 * 62.2= -8.098

Suppose if we want to know the approximate y value for the variable X = 64. Then we can substitute the value in the above equation.

Regression Equation:Ŷ = a + bX = -8.098 + 0.19(64) = -8.098 + 12.16

= 4.06

Regression Equation:Ŷ = a + bX = -8.098 + 0.19(64) = -8.098 + 12.16

= 4.06


Step 6:

• Review


to minimize the sum of the squares of the errors to measure the goodness of fit of a line

ei = residuali

Least-squares estimating equation:• Review


Strongcorrelation

Weakcorrelation

SESE

to minimize the sum of the squares of the errors to measure the goodness of fit of a line

ei = residuali


• Review


Correlation Analysis:

• Review


describe the degree to which one variable is linearly related to another.

Coefficient of Determination:Measure the extent, or strength, of the association that existsbetween two variables.

Coefficient of Correlation:Square root of coefficient of determination

r 2r 2

rr

• Review


Coefficient of Determination:Measure the extent, or strength, of the association that

exists between two variables.

r 2r 2

• 0 ≤ r2 ≤ 1.• The larger r2 , the stronger the linear relationship.• The closer r2 is to 1, the more confident we are in our prediction.

Yn-YYn-XYb+Ya

=r 22

22

• Review


Coefficient of Determination: r 2r 2Yn-Y

Yn-XYb+Ya=r 22

22

Coefficient of Correlation:Square root of coefficient of determination

rr• Review


ReviewWhich value of r indicates a stronger correlation than 0.40? A. -0.30B. -0.50C. +0.38D. 0

If all the plots on a scatter diagram lie on a straight line, what is the standard error of estimate? A. -1B. +1C. 0D. Infinity

• Review


• Review


In the least squares equation, Ŷ = 10 + 20X the value of 20 indicates A. the Y intercept.B. for each unit increase in X, Y increases by 20.C. for each unit increase in Y, X increases by 20.D. none of these.

Review

A sales manager for an advertising agency believes there is a relationship between the number of contacts and the amount of the sales. To verify this belief, the following data was collected: What is the Y-intercept of the linear equation? A. -12.201B. 2.1946C. -2.1946D. 12.201

Review• Review


– scatter diagrams– dependent / independent variables– regression analysis– Least-squares estimating equation– the coefficient of determination– the coefficient of correlation

What we have learnt?

Lesson04

Technology

coefficient of correlation

coefficient of determination

simple regression

squares estimatingequation

ibs statisticsyear

ning ding