Lesson Objectives. Know what the equation of a straight line is, in terms of slope and yintercept . 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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Best graphical tool for “seeing”the relationship between two quantitative variables.
Use to identify:
• Patterns (relationships)
• Unusual data (outliers)
ScatterplotY
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)
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.
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 yinterceptdetermined?
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.
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?
.
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
Equation of Least Squares Regression Line
Slope:
page 615
These are notthe preferred computational equations.
yintercept
(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
å
å
(
)
(
)
x
y
= Sxy =

å
xy
1
n
2
(å
x)
= Sxx =

å
x2
2
n
Numerator part of S2
2
(å
y)
= Syy =

å
y2
3
n
Look at your formula sheet
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

å
å
å
(
xy
)
(
)
x
y
n
1
=

54933
(
342
)
(
795
)
5
=
2
(å

x)
å
2
x2
n
2
=

(
)
342
23470
5
=
2
(å

y)
å
y2
3
n
2
=

(
)
795
131263
5
Example 1
Intermediate Summary Values=
=
=
1
= 77.2
2
= 4858.0
3
Example 1
Intermediate Summary ValuesOnce these values are calculated,
the rest is easy!
Y = a + bX
where
Prediction equation
1
=
b
Estimated Slope
2
=
a
y
b
x
Estimated Y  intercept
Least Squares Regression Line
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
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
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
What a regression equation gives you:
Y = – 332.7 + 7.189(65) =
Example 2
220
·
200
·
180
·
160
WEIGHT
·
140
·
120
100
60
64
68
72
76
HEIGHT
Why was your estimate not exact?
Calculate your own weight.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:
A numerical summary statistic that measures the strength of the linear association between two quantitative variables.
Sample Correlation Coefficient, r1.0£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.
Y
Xaxis
Xaxis
Upward & downward trends:Slope and correlationmust have the same sign.
r > 0.0
r < 0.0
Y
Xaxis
Xaxis
All data exactly on a straight line:Perfect positiverelationship
Perfect negativerelationship
r = _____
r = _____
rclose to1 or +1 means_________________________ linear relation.
rclose to0 means_________________________ linear relation.
"Strength":How tightly the data follow a straight line.Y
Xaxis
X axis
Which has stronger correlation?Strong parabolic pattern! We can fix it.
r = ________________
r = ________________
2
3
Example 1; Weight versus Height
Calculating Correlationr=
=
“Go to Slide 18 for values.”
Look at your formula sheet
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
Comment 2:
Inches to centimeters
Pounds to kilograms
Celsius to Fahrenheit
X to Z (standardized)
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
Comment 4:
Example:
In basketball, there is a high correlation between points scored and personal fouls committed over a season. Third variable is ___?
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: