Example R programs and commands
Student-t test power demo
# All lines preceded by the "#" character are my comments.
# All other left-justified lines are my input.
# All other indented lines are the R program output.
# Truth: X is a normal random variable with mean 0.34
# and standard deviation 2
# Hypothesis: X is a normal random variable with mean 0
# and unknown standard deviation
#
# How likely are we to reject the (false) hypothesis if we
# can only afford to take 100 samples of X?
### Preliminary investigation:
# Simulate sampling X 100 times:
x <- rnorm(100, mean=0.34, sd=2 )
# Student-t test of "H0:mu=0" from first principles:
s <- sd(x); # estimated standard deviation of X from samples
se <- s/sqrt(100); # estimated standard error for a 100-sample average
t <- abs(mean(x)-0)/se; # t-statistic for null hypothesis "H0:mu=0"
2*pt(t,df=99,lower.tail=FALSE) # p-value for the test.
# ...so given a significance level alpha (such as alpha=0.05), we
# Do not reject H0 if p>alpha ;
# Reject H0 if p