m22- regression & correlation 1 department of ism, university of alabama, 1992-2003 lesson...

M22- Regression & Correlation 1 Department of ISM, University of Alabama, 1992-2003

Lesson Objectives

Know what the equation of a straight line is, in terms of slope and y-intercept.

Learn how find the equation of the least squares regression line.

Know how to draw a regression line on a scatterplot.

Know how to use the regression equation to estimate the mean of Y for a given value of X.


Best graphical tool for “seeing”the relationship between two quantitative variables.

Use to identify:

• Patterns (relationships)

• Unusual data (outliers)

Scatterplot


Y

X

Y

X

Y

X

Y

X

Y

Positive Linear Relationship

Negative Linear Relationship

Nonlinear Relationship,need to change the model

No Relationship (X is not useful)


RegressionRegression

AnalysisAnalysis

mechanicsmechanics

RegressionRegression

AnalysisAnalysis

mechanicsmechanics


Equation of a straight line.

Y = mx + b m = slope = “rate of change”

b = the “y” intercept.

Y = a + bx

^

b = slope

a = the “y” intercept.

Days of algebra

Days of algebra

Statistics form

Statistics form

Y = estimate of the mean of Y for some X value.

^


by “eyeball”.

by using equations by hand.

by hand calculator.

by computer: Minitab, Excel, etc.

Equation of a straight line.How are the slope and y-interceptdetermined?



Y = a + bx ^

X-axis0

rise

run

a“y” intercept

b =


Population: All ST 260 students

Each value of X defines a subpopulation of “height” values.

The goal is to estimate the true mean weight for each of the infinite number of subpopulations.

Example 1:

Y = Weight in pounds,X = Height in inches.

Measure:

Is height a goodestimator of mean weight?


Sample of n = 5 studentsY = Weight in pounds,X = Height in inches.

1

2

3

4

5

Ht Wt

73 175

68 158

67 140

72 207

62 115

Case

Example 1:

Step 1?


DTDPDTDP


100

120

140

160

180

200

220

60 64 68 72 76HEIGHT

.

.

.

.

.

WE

IGH

T

Where should the line go?

Where should the line go?

X Y

73 175

68 158

67 140

72 207

62 115

X Y

73 175

68 158

67 140

72 207

62 115

Example 1


2 2

( )

( )i i

i

x y nxyb

x nx

page 615

a y bx

Equation of Least Squares Regression LineEquation of Least Squares Regression Line

y a bx Slope:Slope:

y-intercepty-intercept These are not

the preferred

computational

equations.

These are not

the preferred

computational

equations.


Basic intermediate calculations

(xi - x)(yi - y)

(xi - x)2

(yi - y)2

1

2

3

= Sxy =

= Sxx =

= Syy =

Numerator part of S2

Look at your formula sheet Look at your formula sheet


1

2

3

= Sxy = xy ( x)( y)

n

= Sxx =

= Syy = y2

ny)2 (

x2

nx)2 (

Alternate intermediate calculations


Numerator part of S2

1

2

3

4

5

Case x y

Ht Wt

73 175

68 158

67 140

72 207

62 115

342 795

x y

xy

Ht*Wt

12775

10744

.

.

__.___54933

xy

x2

Ht 2

5329

4624

.

.

_ .___23470

x2

30625

24964

.

.

_ _.___131263

y2

Wt 2

y2

Example 1


Intermediate Summary Values

xy ( x)( y) n54933 ( 342 ) ( 795 ) 5

1

=

x2 n x)2 (2

23470 (342 ) 2 5 =

y2 n y)2 (3

131263 (795 )2 5 =

Example 1


Intermediate Summary ValuesExample 1

1

2

3

= 555.0

= 77.2

= 4858.0Once these values are calculated,the rest is easy!


Least Squares Regression Line

whereY = a + b X

b

a y b x

1

2

Prediction equation Prediction equation

Estimated Slope

Estimated Slope

Estimated Y - intercept Estimated Y - intercept


Slope, for Weight vs. Height

b 1

2 77.2555

=

= 7.189

Example 1


Intercept, for Weight vs. Height

a b y x

– 332.73 =

=795 5

y = 159342

5 x = = 68.4

= 159a (+7.189) 68.4

Example 1


Prediction equation

^Y = a + b X

Wt = – 332.73 + 7.189 Ht^Y = – 332.73 + 7.189 X^

Example 1


100

120

140

160

180

200

220

60 64 68 72 76HEIGHT

Y = – 332.7 + 7.189X^

WE

IGH

TExample 1 Draw the line on the plot


100

120

140

160

180

200

220

60 64 68 72 76HEIGHT

Y = – 332.7 + 7.189 60^

Y = 98.64

X

Y = – 332.7 + 7.189 76^

Y = 213.7

XW

EIG

HT

Example 1 Draw the line on the plot


What a regression equation gives you:

The “line of means” for the Y population.

A prediction of the mean of the population of Y-values defined by a specific value of X.

Each value of X defines a subpopulation of Y-values; the value of regression equation is the

“least squares” estimate of the mean of that Y subpopulation.


Example 2: Estimate the weight of a student 5’ 5” tall.

Y = a + b X = – 332.73 + 7.189 X^


100

120

140

160

180

200

220

60 64 68 72 76HEIGHT

Y = – 332.7 + 7.189(65) =

^

WE

IGH

TExample 2


Calculate your own weight.

Why was your estimate not exact?


1. Calculate the least squares regression line.

2. Plot the data and draw theline through the data.

3. Predict Y for a given X.

4. Interpret the meaning of the regression line.

Regression: Know How To:


CorrelationCorrelation


Sample Correlation Coefficient, r

A numerical summary statistic that measures the strength of

the linear association between two quantitative variables.


Notation:

• r = sample correlation.

• = population correlation, “rho”.

r is an “estimator” of


Interpreting correlation:

-1.0 -1.0 rr +1.0 +1.0

r > 0.0 Pattern runs upward from left to right; “positive” trend.

r < 0.0 Pattern runs downward from left to right; “negative” trend.


Upward & downward trends:

r > 0.0 r < 0.0

Y

X-axis

Y

X-axisSlope and correlation

must have the same sign.

Slope and correlationmust have the same sign.


All data exactly on a straight line:

r = _____ r = _____

Perfect positive

relationship

Perfect positive

relationship Perfect negative

relationship

Perfect negative

relationship

Y

X-axis

Y

X-axis


r = _____________ r = _____________

Which has stronger correlation?

Y

X-axis

Y

X-axis


r close to -1 or +1 means _________________________ linear relation.

r close to 0 means _________________________ linear relation.

"Strength": How tightly the data follow a straight line.


r = ________________ r = ________________


Y

X-axis

Y

X-axis


Y

X-axis X -axis

Y


Strong parabolic pattern! We can fix it.

Strong parabolic pattern! We can fix it.

r = ________________ r = ________________


Computing Correlation

by hand using the formula

using a calculator (built-in)

using a computer: Excel, Minitab, . . . .


Formula for Sample Correlation (Page 627)

2 2 2 2

( ) ( )( )

( ) ( ) ( ) ( )

i i i i

i i i i

n x y x yr

n x x n y y

2 3

1r Sxy

Syy

Sxx



Calculating Correlation

2 3

1r =


Example 1; Weight versus Height

=

“Go to Slide 18 for values.”


3000250020001500

200000

150000

100000

SQFT

ECI

RPLES

Scatterplot of Selling Price vs Square Footage for 50 Houses

Positive Linear Relationship

Example 6 Real estate data,Real estate data, previous sectionprevious section

r =


1009080706050403020100

90

80

70

60

50

40

30

20

10

FRLUNCH

8TPTAS

Scatterplot of 8th Grade SAT Percentile vs Free Lunch Participationfor the 128 Public School Systems in Alabama in 1995

Negative Linear Relationship

Example 7 AL school dataAL school data ,, previous sectionprevious section

r =


6543210

6

5

4

3

2

1

M_RAIN

NIAR_T

Scatterplot of Tuscaloosa Rainfall vs Moscow Rainfall

for 60 Months

No linear Relationship

Example 9 RainfallRainfall data data ,, previous previous sectionsection

r =


Size of “r” does NOT reflect the steepness of the slope, “b”;

but “r” and “b” must have the same sign.

r = b s x s y

and = b r s y

s x

Comment 1:


Changing the units of Y and X does not affect the size of r.

Comment 2:

Inches to centimetersPounds to kilogramsCelsius to FahrenheitX to Z (standardized)


Comment 3: High correlation does not always imply causation.

Example: X = dryer temperature Y = drying time for clothes

Causation: Changes in X

actually do cause changes in Y.

Consistency, responsiveness, mechanism


Common ResponseBoth X and Y change as some unobserved third variable changes.

Comment 4:

Example:In basketball, there is a high correlation between points scored and personal fouls committed over a season. Third variable is ___?


ConfoundingThe effect of X on Y is"hopelessly" mixed up with the effects of other variables on Y.

Example:

Is adult behavior most affected

by environment or genetics?

Comment 5:


The end

m22- regression & correlation 1 department of ism, university of alabama, 1992-2003 lesson...

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