##### Problem One
Angell <- read.table('http://math.uttyler.edu/nathan/classes/statistics/data/angell.data',header=TRUE)

attach(Angell)

ineast <- ifelse(region=='E',1,0)

m1.1 <- lm(moral ~ hetero + ineast + hetero*ineast)

# model is
#  moral = b0 + b1 hetero + b2 ineast + b3 hetero*ineast + err
#
#  b2 gives difference in intercepts for the east/noneast lines
#  b3 gives difference in slopes for the east/noneast lines
#
# first question "Determine whether the effect of hetero on moral is
#                 the same for the East as for other regions"
# paraphrases to "Determine whether b3 is zero"

summary(m1.1)

# the p-value for H0: b3=0 is .6137, so we have no reason to reject
# this hypothesis, and can conclude that b3=0, or, in other words, that
# the effect of hetero on moral is the same for the East as for the
# other regions.

# second question "if so, whether there is, in general, a different
#                  level of moral for a specified level of hetero in
#                  the East when compared to the rest of the country."
#
# paraphrases to "if b3 = 0 determine whether b2 = 0 also"

m1.2 <- lm(moral ~ hetero + ineast)
summary(m1.2)

# model is
#  moral = b0 + b1 hetero + b2 ineast + err
#
#  p-value for H0: b2 = 0 is .0000135, so we reject this null hypothesis
#  and conclude b2 =/= 0, or, in other words, that there is a different
#  level of moral for a specified level of hetero in the East when
#  compared to the rest of the country.  Geometrically we have
#  distinct, but parallel, lines.


##### Problem Two

region
inmw <- ifelse(region=='MW',1,0)
inw <- ifelse(region=='W',1,0)

m2.1 <- lm(moral ~ hetero + mobility + ineast + inmw + inw)
m2.2 <- lm(moral ~ hetero + mobility)
anova(m2.1,m2.2)

# H0: slopes for ineast, inmw, and inw are all zero
# H1: at least one of these slopes is nonzero
#
# the p-value is .07562, indicating a marginal significance.  However,
# keeping things at the .05 level we'll fail to reject the null
# hypothesis.

summary(m2.1)
# we give the equation for each region:
#
# for the south:
#  moral = 13.40850 - 0.06647 hetero - 0.06729 mobility
#
# for the east:
#  moral = 13.40850+4.58939 - 0.06647 hetero - 0.06729 mobility
#        = 17.99789 - 0.06647 hetero - 0.06729 mobility
#
# for the midwest:
#  moral = 13.40850+2.06443 - 0.06647 hetero - 0.06729 mobility
#        = 15.47293 - 0.06647 hetero - 0.06729 mobility
#
# for the west:
#  moral = 13.40850+0.72487 - 0.06647 hetero - 0.06729 mobility
#        = 14.13337 - 0.06647 hetero - 0.06729 mobility


##### Problem Three

m3.1 <- lm(moral ~ hetero + mobility + ineast + inmw + inw +
                   ineast*hetero + ineast*mobility +
                   inmw*hetero + inmw*mobility +
                   inw*hetero + inw*mobility)
           

anova(m3.1,m2.1)

# H0: slopes for in(region)*hetero and in(region)*mobility are all zero
# H1: at least some of these are non-zero

# p-value = .6867, probably not a surprise that we fail to reject H0
# (given our results in problem 2)

m3.2 <- lm(moral ~ hetero + mobility,subset=region=='S')
Angell[region=='S',]

summary(m3.2)

#(Intercept) 13.75850  
#hetero      -0.05195 
#mobility    -0.10155

summary(m3.1)

#(Intercept)     13.758497   4.336680   3.173   0.0034 **
#hetero          -0.051954   0.035372  -1.469   0.1520   
#mobility        -0.101548   0.089959  -1.129   0.2676   
#ineast           8.739683   8.567938   1.020   0.3156   
#inmw             4.265281   6.224078   0.685   0.4983   
#inw             -2.407082   9.005628  -0.267   0.7910   
#hetero:ineast   -0.103960   0.097452  -1.067   0.2943   
#mobility:ineast -0.116646   0.378952  -0.308   0.7603   
#hetero:inmw     -0.086774   0.094983  -0.914   0.3680   
#mobility:inmw   -0.003521   0.144197  -0.024   0.9807   
#hetero:inw      -0.147150   0.245948  -0.598   0.5540   
#mobility:inw     0.167334   0.178624   0.937   0.3561   
#
# note that when we're in region 'S', ineast=inmw=inw = 0, so the line is
# moral =  13.758497 -0.05194 hetero - 0.101548 mobility + err
#
# exactly what we got when running the regression only for the south.