# Problem 9.11.1
# A coin is thrown independently 10 times with P(heads)=p
# To test null hypothesis p=.5 versus alternative that it is not,
# reject if 0 or 10 heads observed
#
# Let X be the number of heads observed.
# X ~ Binomial(size=10,prob=p)
#
# a). The significance level of the test is the
# probability of rejcting the null when it is true.
# (chance of getting 10 tails or getting 10 heads in a row)
sig.level=2*(.5^10)
print(sig.level)
## [1] 0.001953125
# b). What is the power of the test if P(heads)=.1
# The power is chance of a binomial(size=10,prob=.1) equalling 0 or 10
# This can be computing usin binomial pmf:
dbinom(0,size=10,prob=.1) + dbinom(10,size=10,prob=.1)
## [1] 0.3486784
# Or using the binomial cdf:
pbinom(0,size=10,prob=.1) + (1-pbinom(9,size=10,prob=.1))
## [1] 0.3486784
# Problem 9.11.6. Consider tossing the coin until a head comes up
# Define X to be the total number of tosses.
# The variable Y=(X-1) has a geometric distribution in R
# with pmf function dgeom(x, prob)
args(dgeom)
## function (x, prob, log = FALSE)
## NULL
x.grid=seq(1,15)
dgeom(0:10,prob=.5)
## [1] 0.5000000000 0.2500000000 0.1250000000 0.0625000000 0.0312500000
## [6] 0.0156250000 0.0078125000 0.0039062500 0.0019531250 0.0009765625
## [11] 0.0004882812
dgeom(0:10,prob=.1)
## [1] 0.10000000 0.09000000 0.08100000 0.07290000 0.06561000 0.05904900
## [7] 0.05314410 0.04782969 0.04304672 0.03874205 0.03486784
x.grid.probs.h0=dgeom(x.grid-1, prob=.5)
x.grid.probs.h1=dgeom(x.grid-1, prob=.1)
x.grid.likeratio=x.grid.probs.h0/x.grid.probs.h1
plot(x.grid, x.grid.likeratio,xlab="x", ylab="LikeRatio")
![](data:image/png;base64,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)
# a). If the prior probabilities are equal, which
# outcomes favor H0?
#
# These are the values of x for which the likelihood ratio exceeds 1.
x.grid[which(x.grid.likeratio>1)] # x=1,2, or 3
## [1] 1 2 3
# The values which favor H1 are the complement, values greater than 3.
# b). If the prior odds P(H0)/P(H1)=10,
# then the outcomes that favor H0 are those for which the
# posterior odds exceed 1, which are those for which the
# likelihood ratio exceeds 1/10:
x.grid[which(x.grid.likeratio>.1)]
## [1] 1 2 3 4 5 6 7
# c). What is the significance level of a test that
# rejects H0 if X >= 8
prob.h0=0.5
# Equals 1-Prob(accept H0 | H0)
sig.level=1-sum(dgeom(c(0:(7-1)), prob=prob.h0))
print(sig.level)
## [1] 0.0078125
# This should be close to
sum(dgeom(7:50,prob=prob.h0))
## [1] 0.0078125
# d) The power of the test is the probability of rejecting
# given prob=.1
prob.h1=.1
power.h1=1-sum(dgeom(c(0:(7-1)), prob=prob.h1))
print(power.h1)
## [1] 0.4782969
# This should be close to
sum(dgeom(7:50,prob=prob.h1))
## [1] 0.4736585