Lesson Objectives
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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.

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Lesson Objectives

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Lesson objectives

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.


Scatterplot

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

Use to identify:

• Patterns (relationships)

• Unusual data (outliers)

Scatterplot


Lesson objectives

Y

Y

X

X

Y

Y

Y

X

X

Positive Linear Relationship

Negative Linear Relationship

Nonlinear Relationship,need to change the model

No Relationship (X is not useful)


Regression analysis mechanics

RegressionAnalysismechanics


Equation of a straight line

b = slope

a = the “y” intercept.

Equation of a straight line.

Y = mx + b

m = slope

= “rate of change”

Days of algebra

b = the “y” intercept.

^

Y = a + bx

^

Statistics form

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


Equation of a straight line1

r by “eyeball”.

rby using equations by hand.

rby hand calculator.

r by computer: Minitab, Excel, etc.

Equation of a straight line.

How are the slope and y-interceptdetermined?


Lesson objectives

^

Y = a + bx

0

X-axis

Equation of a straight line.

b =

rise

run

“y” intercept

a


Lesson objectives

^

Y = a + bx

0

X-axis

Equation of a straight line.

a

“y” intercept

b =

rise

run


Lesson objectives

Example 1:

Is height a goodestimator of mean weight?

Population: All ST 260 students

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

Measure:

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

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


Lesson objectives

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

Example 1:

HtWt

Case

1

2

3

4

5

73 175

68 158

67 14072 20762 115

Step 1?


Lesson objectives

DTDP


Lesson objectives

·

.

XY

73 175

68 158

67 140

72 207

62 115

Example 1

220

·

.

200

Where should the line go?

180

·

160

.

WEIGHT

·

140

.

·

120

.

100

60

64

68

72

76

HEIGHT


Lesson objectives

Equation of Least Squares Regression Line

Slope:

page 615

These are notthe preferred computational equations.

y-intercept


Basic intermediate calculations

S

(xi - x)(yi - y)

S

(xi - x)2

S

(yi - y)2

Basic intermediate calculations

= Sxy =

1

= Sxx =

2

Numerator part of S2

= Syy =

3

Look at your formula sheet


Alternate intermediate calculations

Alternate intermediate calculations

å

å

(

)

(

)

x

y

= Sxy =

-

å

xy

1

n

2

x)

= Sxx =

-

å

x2

2

n

Numerator part of S2

2

y)

= Syy =

-

å

y2

3

n

Look at your formula sheet


Lesson objectives

1

2

3

4

5

S

S

S

S

x

xy

x2

y

S

y2

Example 1

Case

x y

HtWt

xy

Ht*Wt

x2

Ht 2

y2

Wt 2

30625

24964

.._ _.___

73 175

68 158

67 14072 20762 115

12775

10744

. .__.___

5329

4624

. . _ .___

342 795

54933

23470

131263


Intermediate summary values

=

-

å

å

å

(

xy

)

(

)

x

y

n

1

=

-

54933

(

342

)

(

795

)

5

=

2

-

x)

å

2

x2

n

2

=

-

(

)

342

23470

5

=

2

-

y)

å

y2

3

n

2

=

-

(

)

795

131263

5

Example 1

Intermediate Summary Values

=

=

=


Intermediate summary values1

= 555.0

1

= 77.2

2

= 4858.0

3

Example 1

Intermediate Summary Values

Once these values are calculated,

the rest is easy!


Lesson objectives

^

Y = a + bX

where

Prediction equation

1

=

b

Estimated Slope

2

=

a

y

b

x

Estimated Y - intercept

Least Squares Regression Line


Lesson objectives

1

=

b

2

555

=

77.2

Example 1

Slope, for Weight vs. Height

= 7.189


Lesson objectives

=

a

b

x

y

795

342

y

=

= 159

x =

= 68.4

5

5

-

= 159

a

(+7.189) 68.4

– 332.73

=

Example 1

Intercept, for Weight vs. Height


Lesson objectives

^

Y = a + b X

^

^

Y = – 332.73 + 7.189 X

^

Wt = – 332.73 + 7.189 Ht

Example 1

Prediction equation


Lesson objectives

^

Y = – 332.7 + 7.189X

Example 1

Draw the line on the plot

220

·

200

·

180

·

160

WEIGHT

·

140

·

120

100

60

64

68

72

76

HEIGHT


Lesson objectives

^

Y = – 332.7 + 7.189 60

^

Y = 98.64

^

Y = – 332.7 + 7.189 76

^

Y = 213.7

Example 1

Draw the line on the plot

220

X

·

200

·

180

·

160

WEIGHT

·

140

·

120

100

X

60

64

68

72

76

HEIGHT


Lesson objectives

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”estimateof the mean of that Y subpopulation.


Lesson objectives

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

^

Y = a + bX = – 332.73 + 7.189 X


Lesson objectives

^

Y = – 332.7 + 7.189(65) =

Example 2

220

·

200

·

180

·

160

WEIGHT

·

140

·

120

100

60

64

68

72

76

HEIGHT


Calculate your own weight

Why was your estimate not exact?

Calculate your own weight.


Lesson objectives

Calculate the least squares regression line.

Plot the data and draw theline through the data.

Predict Y for a given X.

Interpret the meaning of the regression line.

Regression: Know How To:


Correlation

Correlation


Sample correlation coefficient r

A numerical summary statistic that measures the strength of the linear association between two quantitative variables.

Sample Correlation Coefficient, r


Notation

r= sample correlation.

r= population correlation,“rho”.

ris an “estimator” ofr.

Notation:


Lesson objectives

Interpreting correlation:

-1.0£r£ +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

Y

Y

X-axis

X-axis

Upward & downward trends:

Slope and correlationmust have the same sign.

r > 0.0

r < 0.0


All data exactly on a straight line

Y

Y

X-axis

X-axis

All data exactly on a straight line:

Perfect positiverelationship

Perfect negativerelationship

r = _____

r = _____


Which has stronger correlation

Y

Y

X-axis

X-axis

Which has stronger correlation?

r = _____________

r = _____________


Strength how tightly the data follow a straight line

rclose to-1 or +1 means_________________________ linear relation.

rclose to0 means_________________________ linear relation.

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


Which has stronger correlation1

Y

Y

X-axis

X-axis

Which has stronger correlation?

r = ________________

r = ________________


Which has stronger correlation2

Y

Y

X-axis

X -axis

Which has stronger correlation?

Strong parabolic pattern! We can fix it.

r = ________________

r = ________________


Computing correlation

Computing Correlation

  • by hand using the formula

  • using a calculator (built-in)

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


Lesson objectives

Sxy

1

r

=

=

Sxx

Syy

2

3

Formula for Sample Correlation(Page 627)

Look at your formula sheet


Calculating correlation

1

2

3

Example 1; Weight versus Height

Calculating Correlation

r=

=

“Go to Slide 18 for values.”

Look at your formula sheet


Lesson objectives

Real estate data,previous section

Example 6

Positive Linear Relationship

r =


Lesson objectives

AL school data,previous section

Example 7

r =

Negative Linear Relationship


Lesson objectives

Rainfalldata ,previous section

Example 9

r =

No linear Relationship


Lesson objectives

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

but“r” and “b” must have the samesign.

.

s

s

×

y

x

r

b

b

r

=

=

s

s

x

y

Comment 1:

and


Changing the units of y and x does not affect the size of r

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

Comment 2:

Inchestocentimeters

Poundstokilograms

CelsiustoFahrenheit

X to Z (standardized)


Causation

Example:X = dryer temperatureY = drying time for clothes

High correlation does not always imply causation.

Causation:

Comment 3:

Changes in X

actually docause changes in Y.

Consistency, responsiveness, mechanism


Common response both x and y change as some unobserved third variable changes

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 ___?


Confounding the 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?

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

Comment 5:


Lesson objectives

The end


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