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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Lesson Objectives
Best graphical tool for “seeing”the relationship between two quantitative variables.
Use to identify:
• Patterns (relationships)
• Unusual data (outliers)
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)
b = slope
a = the “y” intercept.
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.
How are the slope and yinterceptdetermined?
^
Y = a + bx
0
Xaxis
Equation of a straight line.
b =
rise
run
“y” intercept
a
^
Y = a + bx
0
Xaxis
Equation of a straight line.
a
“y” intercept
b =
rise
run
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.
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?
DTDP
·
.
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
S
(xi  x)(yi  y)
S
(xi  x)2
S
(yi  y)2
= 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
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
=

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
=
=
=
= 555.0
1
= 77.2
2
= 4858.0
3
Example 1
Once 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
1
=
b
2
555
=
77.2
Example 1
Slope, for Weight vs. Height
= 7.189
=
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 = a + b X
^
^
Y = – 332.73 + 7.189 X
^
Wt = – 332.73 + 7.189 Ht
Example 1
Prediction equation
^
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:
Example 2:Estimate the weight of a student 5’ 5” tall.
^
Y = a + bX = – 332.73 + 7.189 X
^
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 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.
r= sample correlation.
r= population correlation,“rho”.
ris an “estimator” ofr.
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.
Y
Y
Xaxis
Xaxis
Slope and correlationmust have the same sign.
r > 0.0
r < 0.0
Y
Y
Xaxis
Xaxis
Perfect positiverelationship
Perfect negativerelationship
r = _____
r = _____
Y
Y
Xaxis
Xaxis
r = _____________
r = _____________
rclose to1 or +1 means_________________________ linear relation.
rclose to0 means_________________________ linear relation.
Y
Y
Xaxis
Xaxis
r = ________________
r = ________________
Y
Y
Xaxis
X axis
Strong parabolic pattern! We can fix it.
r = ________________
r = ________________
Sxy
1
r
=
=
Sxx
Syy
2
3
Formula for Sample Correlation(Page 627)
Look at your formula sheet
1
2
3
Example 1; Weight versus Height
r=
=
“Go to Slide 18 for values.”
Look at your formula sheet
Real estate data,previous section
Example 6
Positive Linear Relationship
r =
AL school data,previous section
Example 7
r =
Negative Linear Relationship
Rainfalldata ,previous section
Example 9
r =
No linear Relationship
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:
Inchestocentimeters
Poundstokilograms
CelsiustoFahrenheit
X to Z (standardized)
Example:X = dryer temperatureY = drying time for clothes
High correlation does not always imply 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?
Comment 5:
The end