Prestige <-
  read.table('http://math.uttyler.edu/nathan/classes/statistics/data/prestige.data',header=TRUE)

attach(Prestige)

m1 <- lm(prestige ~ income + education + census + women)
summary(m1)

#prestige_i = b0+ b1 income_i + b2 education_i + b3 census_i + b4 women_i + e_i

#H0: b_1  =  0
#H1: b_1 =/= 0

#H0: all other terms in model, income has no extra effect on prestige
#H1: all other terms in model, income has an extra effect on prestige

# p-value = 1.15e-05 
# really small, so reject H0, conclude b_1 =/= 0
#

#H0: b_2  =  0
#H1: b_2 =/= 0

# p-value = 7.38e-12
# really small, so reject H0, concluce b_2 =/= 0

# book calles this one the ``omnibus F test''
#H0: b_1 = b_2 = b_3 = b_4 = 0
#H0: at least one b_i (i > 0) is not zero

#H0: none of the variables contributes (linearly) to an understanding
#    of prestige
#H1: at least some of our variables contribute to an understanding of
#    the variability in prestige

# p-value: 2.2e-16  (i.e. 0.0000000000000002)
# reject H0



#### Let's look at the above in terms of comparing models:

#H0: b_1  =  0
#H1: b_1 =/= 0

# this test compares the models:

#H0: prestige_i = b0+ b2 education_i + b3 census_i + b4 women_i + e_i
#H1: prestige_i = b0+ b1 income_i+b2 education_i+b3 census_i+b4 women_i + e_i


#H0: b_1 = b_2 = b_3 = b_4 = 0
#H0: at least one b_i (i > 0) is not zero

#this test compares the models:

#H0: prestige_i = b0 + e_i
#H1: prestige_i = b0+ b1 income_i+b2 education_i+b3 census_i+b4 women_i + e_i

# looking at the regression output, we'd like to compare these two models:

#H0: prestige_i = b0+ b1 income_i+b2 education_i+ e_i
#H1: prestige_i = b0+ b1 income_i+b2 education_i+b3 census_i+b4 women_i + e_i

# here's how to do that.  First you need to build both models:
m1 <- lm(prestige ~ income + education + census + women)
m2 <- lm(prestige ~ income + education)

anova(m1,m2)
# this tests:
#H0: b_3 = b_4 = 0
#H1: at least one of b_3, b_4 non-zero

# p-value: .562 -- fail to reject H0 -- safe to use smaller model