##### Demonstration Script #3a
##### MATH322
#####
##### Power Curves
#####


##### This covers power curves when
##### the functional form is known. 


### The Rayleigh Distribution
#   This distribution is used to model magnitudes of
#   two-dimensional vectors. This means it is used to
#   model wind speeds and complex numbers.
#
#   As expected, it is based on the Normal distribution. In
#   fact, it is defined as the cartesian distance between the
#   origin and the point whose x- and y-coordinates are Normal

# Define X and Y
x = rnorm(1e6)
y = rnorm(1e6)

# Calculate distance
r = sqrt(x^2 + y^2)

# The standard Rayleigh pdf
hist(r)





### The Beta curve for Height example

# The Beta curve is known exactly. So, this is just an
# example of using R to plot a function.

mu = seq(50, 68, length=1e4)
beta = 1-pnorm(66.355-mu)

plot(mu,beta)



### Power Curve

# The Power curve is known exactly here. So, this is 
# another example of using R to plot a function.

PWR = 1-beta
plot(mu,PWR)







##### Sample Size
# How do we determine the minimum sample size needed to attain
# a necessary power level?



## See notes

mu0 = 68         # H0
mu  = 67.5       # Reality
alpha = 0.05     # Selected
sig2 = 3         # From trial experiment

n = seq(1, 100)  # Possible n values

# Power function:
pwr = pnorm( (mu0-mu)/(sqrt(sig2/n)) + qnorm(alpha) )

# Plot
plot(n,pwr)

# Which power belongs to which sample size?
cbind(n,pwr)

# So, if reality is mu = 67.5, then we would need a sample size
# of at least 75 to have an 80% chance of determining that the
# null hypothesis is not correct.