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Today’s Topics:
Simple Linear Regression (SLR)
What is it? Why is it used?
The link with correlation.
Estimation (using Excel and not hand calculation).
Interpretation of Excel analysis results.
ECON1310
Introductory Statistics for Social Sciences
Statistics
LECTURE 11
Simple Linear Regression– Part 1
1 hour of online YouTube videos on Blackboard
to be viewed to complete Lecture 11.
Sample slope coefficient, b1
Coefficient of Determination, r2
Standard error of the estimate, se
Confidence interval for the slope coefficient.
1
What is Simple Linear Regression (SLR)?
2
Why use Simple Linear Regression?
used to predict the value of one variable
(dependent variable) based on a given value of
another variable (independent variable).
a procedure to find the equation of the line of
best fit between two variables of the form:
Y = mX + c
used to explain the impact of a change in the
independent variable on the dependent variable.
SLR is an inferential statistics technique allowing
conclusions to be made about a population
parameter based on a sample statistic.
where
m = the slope of the line
c = the yaxis intercept
3
4
Questions.
1. What is the impact on a retail store’s sales revenue
with an increase in advertising expenditure?
2. Do house prices decrease with an increase in
distance from the Central Business District (CBD)?
How can these questions be analysed?
Firstly, collect observed data on the two
variables of interest.
Plot the data on a graph with one variable on
the X axis and the other on the Y axis.
3. Do home values increase as land size increases?
4. Is there a relationship between gross box office movie
revenue and the number of videos sold once the
video is released a short time later?
5
Observe the scatter plot (graph).
Note: a scatter plot can be drawn using Excel by
setting up a table of XY coordinates, then using
insert/scatter.
6
1
2. Scatter plot
2000
1800
1600
1400
1200
1000
800
600
400
200
0
House price ($’000)
sales ($000)
1. Scatter plot
0
20
40
60
1000
900
800
700
600
500
400
300
200
100
0
80
0
20
TV advertising ($000)
40
60
Distance from CBD (km)
3. Scatter plot
4. Scatter plot
number of videos sold
1000
900
800
appraised value ($000)
700
600
500
400
300
70
60
50
40
30
20
10
0
0
100
200
300
400
200
Box Office Gross ($m)
100
0
0
500
1000
1500
2000
2)
Land (m
2500
3000
3500
10
What can scatter plots reveal?
Correlation Coefficient, r
Is there a linear (straight line) relationship?
It has no units (it is just a number)
Is there a curvilinear (eg: parabolic shape)
Y
relationship? Y
Can only have a value between –1 and 1
X
1 r 1
r close to 1 implies a strong positive linear relationship
X
r close to –1 implies a strong negative linear relationship
If the relationship looks linear, is the line
sloping upward (positive relationship) or
downward sloping (negative relationship)?
r closer to 0 implies a weaker linear relationship
r = 0 implies no linear relationship exists
To calculate r using Excel, use:
Data/Data Analysis/Correlation or Insert/ fx /correl
Is the linear relationship weak or strong?
11
12
2
Correlation coefficients and the strength
of linear relationship.
Line of best fit in scatter plot
House prices ($’000)
sales ($000)
rTV advertising & sales = 0.83
2000
1800
1600
1400
1200
1000
800
600
400
200
0
0
20
40
60
80
rdistance and price = 0.90
1000
900
800
700
600
500
400
300
200
100
0
0
Television advertising ($000)
r
number of videos sold
appraised value ($000)
20
40
60
Distance from CBD (km)
rland and value = 0.61 (weak)
1000
Y
Y
900
800
700
600
500
400
300
200
X
= 0.85
Box office & videos sold
70
60
50
40
30
20
10
0
100
200
300
10 0
400
X
r = 1
Perfectly negatively
correlated = a very
strong relationship
r = +1
Perfectly positively
correlated = a very
strong relationship
Box Office Gross ($m)
100
land (m2)
0
0
500
1000
1500
2000
2500
14
3000
3500
Correlation coefficients and strength of
linear relationship.
Y
Correlation coefficients and strength of
linear relationship.
X
r = +0.2
A weak, positive (upward
sloping) linear relationship.
Y
Y
Y
X
r = 0.6
A negative (downward
sloping) linear relationship
that is neither strong nor
weak.
X
X
r=0
r=0
NO linear relationship exists.
The slope of the regression line = 0
15
Correlation summary.
16
Simple Linear Regression
A scatter diagram can indicate if some kind of
relationship might exist between two variables.
(line of best fit using sample data)
Ŷi b 0 b1X i
Correlation analysis is used to measure the
strength of the linear relationship between two
variables.
the relationship between the two variables X
and Y is described as a linear function.
Note:
Correlation does NOT imply a causal effect
(ie: a change in X does not cause a change in Y)
17
changes in Y are assumed to be caused by
changes in X.
18
3
Sample Linear Regression Equation
Ŷi b 0 b1X i
Y
Observed value
Yi
of Yi for Xi
Predicted
value of Y for Xi
(estimated Y
value = Ŷi )
Defining and Interpreting the terms
Ŷi
(Xi ,Yi)
ei Yi Ŷi
n = 12 (number
of observed data
points in
scatter plot)
1 unit
(0 , b0)
0
(Xi ,Yi) is a measured (observed) data point in the
sample (of size n) used to help estimate the
sample linear regression equation.
(Xi ,Ŷi )
b1 units
X
Ŷi is the predicted value (estimated value) of Y
for a particular chosen value of Xi
X
Xi
b0 = the value of Ŷ when the value of X is zero.
It is the “estimate sample regression line” y
intercept value.
b1 = the slope of the estimated regression line.
rise
change in y
run
change in x
If b1 is positive, for every one unit increase in X
there is b1 units increase in Y.
If b1 is negative, for every one unit increase in X
there is b1 units decrease in Y.
21
Population Simple Linear Regression Equation
(for the true, but unknown, relationship)
Y
Y β 0 β1X i ε i
error
εi= random
for Xi value
Expected value
of Y for Xi
(0 , β0) X
1 unit
0
(Xi , Y)
Population SLR Equation (for the true,
but unknown, relationship)
Population Population slope Independent Random
error
Y intercept coefficient
variable
term
Dependent
variable
Yi β 0 β1X i ε i
where β0 and β1 are the parameters respectively
for the y intercept value and slope coefficient of X.
ε i = an error term to allow for a range of values of Y to
occur for any given Xi . In the population, there may be
many different Y values for the same X value.
22
Finding the equation to the Simple Linear
Regression Equation (based on a sample):
The error at any value of Xi is defined as the
difference between the observed data value of
Yi and the predicted value Ŷi
β1 units
N = very large for
population model
Xi
20
Note: some data points are above the estimated
regression line (a positive error) and some
below the line (a negative error).
(Xi ,Yi)
Sample SLR
equation
Yi is called the dependent variable (or response
variable). The value of Y changes when X
changes. (ie: Y depends on the value of X).
(units)
Defining and Interpreting the terms.
Observed value
Yi
of Yi for Xi
Xi is called the independent variable (or
explanatory variable).
X
(units)
ei Yi Ŷi
24
4
Observations of the Simple Linear
Regression Equation (based on a sample).
the error (ei) at any of Xi is always measured in the
vertical direction.
So far, what do we know?
Simple Linear Regression aims to find a straight line
relationship between two variables, X and Y, using
sample data.
The estimated equation is the line of best fit (the
Sample Linear Regression Model) is given by the
equation:
Ŷ b 0 b1X
the slope of the SLR equation has the same sign
as r.
if b1 positive, r > 0, the line slopes upwards
if b1 negative, r < 0, the line slopes downwards
If b1 = 0, the line is horizontal and there is no linear
relationship between X and Y
ie: +ve if slopes upward to the right,
 ve if slopes downward to the right.
25
So far, what do we know?
Finding the values of b0 and b1
The simple linear regression equation (using sample data)
provides an estimate of the population regression line.
estimated sample linear regression equation is given by:
Ŷ b 0 b1X
where bo and b1 are statistics.
The unknown population linear regression equation is
given by:
Yi β0 β1Xi ε i
26
Least Squares Method.
b0 and b1 are obtained by finding the values of b0
and b1 that minimises the sum of the squared
differences between all pairs of Yi and Ŷi values.
min (Yi Ŷi )2 min (Yi (b0 b1Xi ))2
where β0 and β1 are parameters.
27
28
Finding the Least Squares Equation.
Using Excel and Kaddstat for Regression
the value of the coefficients b0 and b1, and
other regression results in ECON1310, are to be
found using Excel.
formulae to find b0 and b1 are shown in
textbooks (for those keen to follow the maths).
no hand calculations to find b0 and b1 from the
observed data will be required in ECON1310.
an ability to use Excel, and interpret the output
results of regression analysis, ARE required in
ECON1310.
29
Use Excel (rather than Kaddstat).
use Data (or Tools)/Data Analysis/ Regression
Output lists
Regression Statistics
ANOVA table
coefficients table
values of b0 and b1 are seen in the coefficients
column of the coefficients table
30
5
Sample Data for Fire Damage Model
Example 1.
Simple Linear Regression
Distance from Fire Station (km)
An insurance company wishes to determine the
dollar amount of fire damage a house will suffer
as the distance from the local fire station
increases.
The insurer collects a random sample of 15 fire
damage claims from historical records. The fire
damage suffered is measured in $’000 and the
distance to the fire station to the house is in
kilometers (km).
31
Example 1.
Simple Linear Regression
Fire Damage ($ ’000)
Xi
Yi
0.7
1.1
1.8
2.1
2.3
2.6
3.0
3.1
3.1
3.8
4.3
4.6
4.8
5.5
6.1
14.1
17.3
17.8
24
23.1
19.6
22.3
27.5
26.2
26.1
31.3
31.3
36.4
36
43.2
Example 1.
Simple Linear Regression
a. What are the independent and dependent
variables, as well as their units?
a. What are the independent and dependent
variables, as well as their units?
b. Using the output from Excel, write down the
estimated linear regression equation.
House fire damage depends on distance from the
fire station. So,
c. Interpret the constant and slope coefficient.
Give the units of each.
Independent variable (X) = distance (km)
Dependent variable (Y) = house fire damage ($’000)
33
Fire Damage ($'000)
Graphical Presentation of sample data
50
45
40
35
30
25
20
15
10
5
0
34
Example 1.
Simple Linear Regression
b. Using the output from Excel, write the
estimated linear regression equation.
Ŷi b 0 b1X i
0
2
4
6
8
Distance from Fire Station (km)
36
6
Excel Simple Linear Regression Output
Example 1.
Analysing Excel Regression Output
Regression Statistics
Multiple R
r2
R Square
Adjusted R Square
n = sample size
(can find from df. residual = n2, or from df total = n1)
Standard Error se
Observations
n
SS = Sum of Squares
SSR = Sum of Squares Regression
SSE = Sum of Squares Residuals (Errors)
SST = Sum of Squares Total = SSR + SSE
ANOVA
(R) Regression
df
SS
MS
1
SSR
MSR
(E) Residual
n2 +
SSE +
MSE
(T) Total
n1
SST
Coefficients
Standard
Error
F
MSE = Mean Square Error
b0
X
b1
Pvalue
Lower
95%
S
Lecture 11,b0
ECON1310
Upper
95%
37
38
Sb1
Example 1.
Excel Fire Damage Model Output
Fire Damage vs Distance to Fire Station
b. The estimated regression equation: Ŷi b 0 b1X i
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.960978
R Square
0.923478
Adjusted R Square
0.917592
Standard Error
2.316346
Observations
house fire damage 10.28 4.92 X
n =15
ANOVA
df
Regression
Residual
Total
SS
841.7664
MS
1
n2 = 13
n1 = 14
Coefficients
841.7664
69.75098
Significance F
156.8862
1.25E08
5.36546
911.5173
Standard Error
t Stat
Intercept 10.2779 =b01.420278
X
F
4.9193 =b10.392748
Pvalue
Lower 95%
Upper 95%
Lower 95.0%
Upper 95.0%
7.236562
6.59E06
7.209605
13.34625
7.209605
13.34625
12.52542
1.25E08
4.070851
5.767811
4.070851
5.767811
Fire Damage ($'000)
Intercept
t Stat
(t calc)
50
house fire damage 10.28 4.92 X
45
40
35
4.92 ($’000) = $4,920
30
1 km
25
20
15
10
(0, 10.28)
5
n = 15
0
0
2
4
6
8
Distance from Fire Station (km)
39
Next lecture…
Simple Linear Regression – Part 2
41
7
Today’s Topics:
ECON1310
Simple Linear Regression (SLR)  continued
Introductory Statistics for Social Sciences
LECTURE 12
Assumptions when using SLR
Residual plot analysis
Hypothesis testing of the slope coefficient
Simple Linear Regression– Part 2
1 hour of online YouTube videos on Blackboard
to be viewed to complete Lecture 12.
1
The Estimated Regression Line
(from Lecture 11)
2
Assumptions when using SLR
(and the Least Squares Method)
line of best fit through sample data points.
Sample data points lie above and below the
estimated line. Hence, the error (ei) at each
point Xi may be positive or negative, where
the error is given by ei Yi Ŷi
Predictions (inferences) require certain
“assumptions”, about the error terms, to
be satisfied.
To ensure predictions are valid and reliable
using SLR:
need to check four assumptions are true.
only make inferences after assumptions
are satisfied.
3
Least Squares Method Assumptions.
4
How to check the Assumptions?
Use observations (a subjective method)
1. The model is linear.
The simplest way is to observe:
Error term assumptions.
2. The error terms have constant variance.
3. The error terms are independent (ie: they are
not correlated) and occur randomly.
4. The error terms are normally distributed with
an expected value (=mean) of zero.
ie: E(ei)=0.
5
1. The scatter plot of (Xi ,Yi) sample points
to see whether a linear relationship is
appropriate (rather than curvilinear).
2. The residual plot of (Xi ,ei).
6
1
Check Assumption 1 – is the model linear?
Check Assumption 1 – is the model linear?
scatter plot
appraised value ($000)
1000
sales ($000)
scatter plot
2000
1800
1600
1400
1200
1000
800
600
400
200
0
0
20
40
60
80
900
800
700
600
500
400
300
200
100
0
0
500
1000
Television advertising ($000)
1500
2000
2500
3000
3500
land (m2)
Plot looks linear, so a linear model can be used.
The “linear assumption” is satisfied.
Does NOT look linear, so a linear model should NOT
be used. The “linear assumption” is violated.
7
8
Residual Plots to check Assumptions
Example 1:
Calculating a Residual (an error)
a residual plot is a plot of all (Xi ,ei) points from
sample data.
From lecture 11, the estimated regression
equation between distance from the fire
station and value of house damage was:
also used to identify if a linear relationship is
appropriate (Assumption 1).
house fire damage 10.2779 4.9193* X
used to verify Assumptions 2 and 3 about the
error term.
From the sample of 15 observations, the
third data point was (1.8, 17.8).
What is the value of the residual at this third
data point observation?
9
Sample Data for Fire Damage Model
Distance from Fire Station (km)
(Xi)
0.7
1.1
1.8
2.1
2.3
2.6
3.0
3.1
3.1
3.8
4.3
4.6
4.8
5.5
6.1
Fire Damage ($ ’000)
(Yi)
14.1
17.3
17.8
24
23.1
19.6
22.3
27.5
26.2
26.1
31.3
31.3
36.4
36
43.2
10
Calculating a Residual.
The estimated house damage for X3 is
house fire damage Ŷ3
10.2779 4.9193*1.8
19.13264
e 3 Y3 Ŷ3
17.8 19.13264
1.3326
1.33 (2 dec. places)
12
2
Error = Observed Value  Predicted Value
Regression and Residual Plots
Ŷi b 0 b1X i
Y
Y
Ŷ3 Predicted
Y3
e3 1.33
Y3 Observed
(1.8,17.8)
(1.8,17.8)
Observed data
+
X
Residual Plot
error
0
X
X3
Residuals ($’000)
4
3
2
1
0
3
4
1
2
3
4
X3
14
(units)
Residual Plot
House damage vs Distance to Fire Station
0
X
(1.8,1.33)

Residual Plot from Excel
2
X
X3
0
1.33
1
Ŷ
(1.8,19.13)
(1.8,19.13)
e3 1.33
(0 , b0)
Regression Plot
Ŷ3
5
6
7
(1.8,1.33)
Distance from Fire Station (km)
15
Residual Plot from Excel
Excel: Data/Data Analysis/Regression and tick the
box “Residual Plots”. X is plotted on the horizontal axis.
Kaddstat: Kadd/regression and correlation/ simple/multiple
and tick Plot the residuals.
Note: the resulting plot has the observation number, not
the X variable, on the horizontal axis (useful for time series).
Note: (X, ei) plot can be thought of as the regression
line of best fit, with the plotted points around it,
being made horizontal. (This removes what is called
the trend effect).
RESIDUAL OUTPUT (Damage v Distance)
Observation, X
Predicted, Y
1
13.72146
2
15.68919
3
19.13272
4
20.60852
5
21.59239
6
23.06819
7
25.03592
8
25.52785
9
27.00365
10
28.97139
11
31.43105
12
32.90685
13
33.89072
14
37.33425
15
40.28585
ei Yi Ŷ
Residuals
0.37854
1.610808
1.33272
3.391477
1.507611
3.46819
2.73592
1.972146
0.80365
2.87139
0.13105
1.60685
2.509284
1.33425
2.914154
Residual plot: check Assumptions 1, 2 & 3.
Look at the residual plot and determine if any
pattern exists in the residuals.
a. A pattern may indicate the linear relationship is
NOT appropriate.
b. A pattern in a residual plot is NOT good. Ideally,
the residual values should be random in value
as X increases.
c. A residual plot pattern can indicate one or more
error assumptions may have been violated.
3
Fire Damage ($'000)
Residual Plot Fire Damage v Distance
Residual plot to check Assumptions 2
50
40
(X,Y) scatter plot
looks linear, so
assumption 1
about being
linear looks OK.
30
20
10
0
0
1
2
3
4
5
6
Constant variance = homoskedasticity
(homo means the same, good for SLR)
the residual plot should show no major
changes in spread of errors (in the vertical
direction) over the range of X values.
7
Residuals ($’000)
4
3
2
1
0
0
1
2
3
4
5
1
2
3
4
6
7
Residual plot has
no pattern, and
errors are random.
Assumptions 1,
and 3 are OK.
Distance from Fire Station (km)
Residual Plot Fire Damage v Distance
if the spread of errors is not constant as X
increases, a violation (of assumption 2) has
occurred. This violation is called
heteroskedasticity.
50
40
(X,Y) scatter plot
looks linear, so
assumption 1
about being
linear looks OK.
30
20
10
0
0
1
2
3
4
5
6
7
4
Residuals ($’000)
Nonconstant variance = heteroskedasticity
(hetero means different, bad for SLR)
Fire Damage ($'000)
Residual plot to check Assumptions 2
3
2
1
0
0
2
3
4
5
6
7
2
3
4
21
1
1
Residual plot has
no pattern, and
errors are random.
Assumptions 1, 2,
and 3 are OK.
Distance from Fire Station (km)
Residual plot examples.
Residual plot to check Assumptions 3
Independent and random errors (= good).
23
House prices ($’000)
1000
900
800
700
600
500
400
300
200
100
0
(X,Y) scatter plot
looks nonlinear,
so assumption 1
about being linear
is violated.
0
10
20
30
40
50
60
Distance from CBD (km)
Residuals
the residual plot should show no pattern in the
residuals.
several consecutive positive errors followed by
several consecutive negative errors (a pattern)
as X increases can indicate a violation of the
independence of errors assumption.
If time is on the horizontal axis (or observations
are ordered as measured), and a pattern in the
residuals exists, this violation is called
autocorrelation.
200
150
100
50
0
50 0
100
150
20
40
Distance from CBD (km)
60
Residual plot has
a pattern as X
increases, and
errors are not
random. Violates
assumption 3 and
the independence
of errors.
4
Assumptions 4 – Normality of Errors
Example 2
Weekly Income and Food Expenditure
The error terms are assumed to be normally
distributed with an average, or expected value, equal
to zero ie: E(ei) = 0
Sample data on weekly income and weekly
food expenditure (both in dollars) was used
to produce a simple linear regression
equation.
The residual plot is NOT used to check the
assumption of normality of errors.
From the regression analysis, a residual plot
was produced (see next slide). Observe and
comment on the usefulness of the SLR
model using the residual plot.
A normality plot (or histogram of errors showing
the distribution) is needed and this will NOT be
covered in ECON1310.
25
26
Example 2
Weekly Income and Food Expenditure
Example 2
Weekly Income and Food Expenditure
Residual Plot
Residual Plot
100
80
80
60
60
Residuals ($)
Residuals ($)
100
40
20
0
20
20
0
20
40
40
60
60
80
80
0
200
400
600
800
1000
1200
1400
Example 2
Weekly Income and Food Expenditure
Cover the other
half of plot and
“judge” the
vertical spread
of the points
(errors)
40
20
0
20
40
400
600
800
1000
1200
1400
From the residual plot observations, is
the spread of residuals constant?
Residual Plot
60
200
Example 2 solution
100
80
0
Weekly Income ($)
Weekly Income ($)
Residuals ($)
Cover one half
of the plot and
“judge” the
vertical spread
of the points
(errors)
40
YES
Hence, the residuals satisfy the
assumption of homoskedasticity.
The errors look random (no pattern).
60
The scatter plot should look linear.
80
0
200
400
600
800
Weekly Income ($)
1000
1200
1400
Assumptions OK.
30
5
Example 4.
Constant variance of residuals.
Example 3.
Problem of nonconstant variance
Residual Plot
300
Age Residual Plot
500
200
400
300
0
0
100
200
300
400
500
100
Residuals
Residuals
100
200
100
0
0
20
40
60
80
100
120
100
200
200
300
400
300
house size
(m2)
400
31
Age (years)
Overview:
Simple Linear Regression so far.
Overview:
Simple Linear Regression so far.
1. Estimate the model (ie: the equation to the
line of best fit) using Excel assuming
certain assumptions are true.
3. Assess how “good” the model is by considering:
i) Statistics (need “r” to be close to 1 or
+1, and se to be relatively small).
ii) the confidence interval of the slope
coefficient.
2. Check the assumptions used in SLR before
going any further. If any assumptions are
violated (not true) the model will not be
reliable for making estimates.
33
4. If the model is “good”, then it can used to
predict the value of Y for a chosen value of X.
Otherwise, if the model is NOT good, any
predicted values of Y will be unreliable.
34
6
Data
MPG Horsepower Weight
43,1
48
1985
19,9
110
3365
19,2
105
3535
17,7
165
3445
18,1
139
3205
20,3
103
2830
21,5
115
3245
16,9
155
4360
15,5
142
4054
18,5
150
3940
27,2
71
3190
41,5
76
2144
46,6
65
2110
23,7
100
2420
27,2
84
2490
39,1
58
1755
28,0
88
2605
24,0
92
2865
20,2
139
3570
20,5
95
3155
28,0
90
2678
34,7
63
2215
36,1
66
1800
35,7
80
1915
20,2
85
2965
23,9
90
3420
29,9
65
2380
30,4
67
3250
36,0
74
1980
22,6
110
2800
36,4
67
2950
27,5
95
2560
33,7
75
2210
44,6
67
1850
32,9
100
2615
38,0
67
1965
24,2
120
2930
38,1
60
1968
39,4
70
2070
25,4
116
2900
31,3
75
2542
34,1
68
1985
34,0
88
2395
31,0
82
2720
27,4
80
2670
22,3
88
2890
28,0
79
2625
17,6
85
3465
34,4
65
3465
20,6
105
3380
Page 1
Appraised Value Land (sq m) House size(sq m) Age Rooms Baths
466
1044.1
204 46
7
3.5
364
996.4
161.8 51
7
2.5
429
740.9
172.7 29
5
3
548.4
2094.5
225.5 18
8
2.5
405.9
1158.6
170.1 46
7
1.5
374.1
1040.9
174 88
7
2
315
821.8
119.4 48
7
2
749.74
2279.5
249.2
7
9
2.5
217.7
1013.2
84 52
5
1
635.7
590.9
266.8 15
8
2.5
350.7
801.4
185.8 54
8
2
455
1909.1
154 48
7
2
356.2
1145.5
175 46
6
2
271.7
521.8
153.8 12
5
3
304.3
769.5
110.9 64
5
1
288.4
779.1
112 52
8
1
396.7
1749.5
151.8 44
6
2
613.5
2975.0
206.5 46
6
2.5
314.1
782.7
133.7 52
6
3
363.5
652.3
173.3 78
11
2
364.3
1252.3
200.8 71
6
1
305.1
521.8
146 97
8
2
441.7
1652.7
157 45
7
2
353.1
670.0
170.8 41
10
2
463.3
1036.8
248.1 40
6
2.5
320
2102.7
177.6 82
7
1
332.84
858.6
129.2 54
6
2
276.6
558.2
94 44
5
1
397
678.2
139.5 34
7
2
221.9
387.3
98.6 94
5
1
min
387.3
84.0 7.0
5.0
max
2975
266.8 97
11
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simple linear regression
Solar Panels
sales revenue
Slope of the line
average energy output
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