#  look at a number of different data sets
#  to see situations where the regression equation
#  might allow us to
#     predict, or 
#     have a narrow range on which to predict, or
#     have some idea (but not a great idea) of a range in which to predict, or
#     really does not help us to predict
#  a dependent value, y, given an independent value, x.
#
# 
source("../gnrnd4.R")

# This is an example where the correlation is so high
# that we could, for all practical purposes, say that
# we can use the regression equation to predict the 
# dependent variable if we are given the independent 
# variable.
#
#  generate the values
gnrnd4(641292306, 0140080302, 18000070)
#  look at the values
L1
L2
#  make a scatter plot of the values
plot(L1,L2)
# compute the correlation coefficient
cor(L1,L2)
#  a correlation coefficient of 0.9997968 is absurdly 
#  close to 1.

#  find our regression equation
hold_our_model <- lm(L2~L1)
hold_our_model
#   so our equation is  y = 4.001 + 1.501*x
#   plot that line
abline( hold_our_model )

# use the equation to predict the value of y when
#   x = 175
y <- 4.001 + 1.501*175
y
# plot that point
points( 175, y, pch=19, col="red")


###############################################

# This is an example where the correlation is high
# but still allows for some variability so we can say that
# we can use the regression equation to get a value within 
# a narrow range for the dependent variable if we 
# are given the independent variable.
#
#  generate the values
gnrnd4(641292306, 3140080302, 18000070)
#  look at the values
L1
L2
#  make a scatter plot of the values
plot(L1,L2)
# compute the correlation coefficient
cor(L1,L2)
#  a correlation coefficient of 0.9795289 is  
#  high.  Find the percent of the variation in
#  the data that is explained by a linear regression.
0.9795289^2

#  find our regression equation
hold_our_model <- lm(L2~L1)
hold_our_model
#   so our equation is  y = 3.084 + 1.515*x
#   plot that line
abline( hold_our_model )

# use the equation to predict the value of y when
#   x = 175
y <- 3.084 + 1.515*175
y
# plot that point
points( 175, y, pch=19, col="red")



###############################################

# This is an example where the correlation is not high
# so there is a good deal of variability. Then we can say that
# we can use the regression equation to get a value within 
# a wide range for the dependent variable if we 
# are given the independent variable.
#
#  generate the values
gnrnd4(641292306, 6140080302, 18000070)
#  look at the values
L1
L2
#  make a scatter plot of the values
plot(L1,L2)
# compute the correlation coefficient
cor(L1,L2)
#  a correlation coefficient of 0.8710001 is good, but not 
#  high.  Find the percent of the variation in
#  the data that is explained by a linear regression.
0.8710001^2

#  find our regression equation
hold_our_model <- lm(L2~L1)
hold_our_model
#   so our equation is  y = 1.551 + 1.541*x
#   plot that line
abline( hold_our_model )

# use the equation to predict the value of y when
#   x = 175
y <- 1.551 + 1.541*175
y
# plot that point
points( 175, y, pch=19, col="red")



###############################################

# This is an example where the correlation is so low that
# there is a huge deal of variability. Then we can say that
# we can use the regression equation but it will not give
# us a good predicted value for the dependent variable if we 
# are given the independent variable.
#
#  generate the values
gnrnd4(641292301,  43000000)
L2<-L1
gnrnd4(567472301,18000070)
#  look at the values
L1
L2
#  make a scatter plot of the values
plot(L1,L2)
# compute the correlation coefficient
cor(L1,L2)
#  a correlation coefficient of 0.01674324 is terrible. 
#  There is no expectation that the linear regression 
#  line will tell us anything.

#  find our regression equation
hold_our_model <- lm(L2~L1)
hold_our_model
#   so our equation is  y = 199.55392 + 0.04151*x
#   plot that line
abline( hold_our_model )

# use the equation to predict the value of y when
#   x = 175
y <- 199.55392 + 0.04151*175
y
# plot that point
points( 175, y, pch=19, col="red")