ECON 1310 Havard University Simple Linear Regression Exercises

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16 statistics problems

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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 y-axis 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 = n-2, or from df total = n-1) 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 n-2 + SSE + MSE (T) Total n-1 SST Coefficients Standard Error F MSE = Mean Square Error b0 X b1 P-value 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 n-2 = 13 n-1 = 14 Coefficients 841.7664 69.75098 Significance F 156.8862 1.25E-08 5.36546 911.5173 Standard Error t Stat Intercept 10.2779 =b01.420278 X F 4.9193 =b10.392748 P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% 7.236562 6.59E-06 7.209605 13.34625 7.209605 13.34625 12.52542 1.25E-08 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) Non-constant 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 non-linear, 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 non-constant 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 Purchase answer to see full attachment Tags: simple linear regression Solar Panels sales revenue Slope of the line average energy output User generated content is uploaded by users for the purposes of learning and should be used following Studypool's honor code & terms of service.