58  Mean Difference - One sample

Testing to see if a mean is equal to a reference value

\[ n=\left(\sigma\frac{z_{1-\alpha/2}+z_{1-\beta}}{\mu-\mu_0}\right)^2\] \[1-\beta= \Phi\left(z-z_{1-\alpha/2}\right)+\Phi\left(-z-z_{1-\alpha/2}\right) \quad ,\quad z=\frac{\mu-\mu_0}{\sigma/\sqrt{n}} \]

• \(n\) is sample size

• \(\sigma\) is standard deviation

• \(\Phi\) is the standard Normal distribution function

• \(\Phi^{-1}\) is the standard Normal quantile function

• \(\alpha\) is Type I error

• \(\beta\) is Type II error, meaning \(1-\beta\) is power

mu=2
mu0=1.5
sd=1
alpha=0.05
beta=0.20
(n=(sd*(qnorm(1-alpha/2)+qnorm(1-beta))/(mu-mu0))^2)

[1] 31.39552

ceiling(n)# 32

[1] 32

z=(mu-mu0)/sd*sqrt(n)
(Power=pnorm(z-qnorm(1-alpha/2))+pnorm(-z-qnorm(1-alpha/2)))

[1] 0.800001