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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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slide1
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
slide3
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)

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?

slide7
^

Y = a + bx

0

X-axis

Equation of a straight line.

b =

rise

run

“y” intercept

a

slide8
^

Y = a + bx

0

X-axis

Equation of a straight line.

a

“y” intercept

b =

rise

run

slide9
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.

slide10
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?

slide12
·

.

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

slide13
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

slide16
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!

slide19
^

Y = a + bX

where

Prediction equation

1

=

b

Estimated Slope

2

=

a

y

b

x

Estimated Y - intercept

Least Squares Regression Line

slide20
1

=

b

2

555

=

77.2

Example 1

Slope, for Weight vs. Height

= 7.189

slide21
=

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

slide22
^

Y = a + b X

^

^

Y = – 332.73 + 7.189 X

^

Wt = – 332.73 + 7.189 Ht

Example 1

Prediction equation

slide23
^

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

slide24
^

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

slide25
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.
slide27
^

Y = – 332.7 + 7.189(65) =

Example 2

220

·

200

·

180

·

160

WEIGHT

·

140

·

120

100

60

64

68

72

76

HEIGHT

slide29
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:

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:
slide34
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 = _____________

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, . . . .
slide42
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

slide44
Real estate data,previous section

Example 6

Positive Linear Relationship

r =

slide45
AL school data,previous section

Example 7

r =

Negative Linear Relationship

slide46
Rainfalldata ,previous section

Example 9

r =

No linear Relationship

slide47
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:

Inches to centimeters

Pounds to kilograms

Celsius to Fahrenheit

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 ___?

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