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Test 1 Mean: 1-Sample, 2-Sided Equality

This calculator is useful for tests concerning whether a mean, μ\mu, is equal to a reference value, μ0\mu_0. The Null and Alternative hypotheses are
H0:μ=μ0H_0: \mu = \mu_0
H1:μ≠μ0H_1: \mu \neq \mu_0

Formulas

This calculator uses the following formulas to compute sample size and power, respectively:
n=(σz1−α/2+z1−βμ−μ0)2n = \left(\sigma\frac{z_{1-\alpha/2} + z_{1-\beta}}{\mu - \mu_0}\right)^2
1−β=Φ(z−z1−α/2)+Φ(−z−z1−α/2),z=μ−μ0σ/n1-\beta = \Phi(z - z_{1-\alpha/2}) + \Phi(-z - z_{1-\alpha/2}), \quad z = \frac{\mu - \mu_0}{\sigma / \sqrt{n}}
where: nn is sample size σ\sigma is standard deviation Φ\Phi is the standard Normal distribution function Φ−1\Phi^{-1} is the standard Normal quantile function α\alpha is Type I error β\beta is Type II error, meaning 1−β1-\beta is power

R Code

1mu=2
2mu0=1.5
3sd=1
4alpha=0.05
5beta=0.20
6(n=(sd*(qnorm(1-alpha/2)+qnorm(1-beta))/(mu-mu0))^2)
7ceiling(n)# 32
8z=(mu-mu0)/sd*sqrt(n)(Power=pnorm(z-qnorm(1-alpha/2))+pnorm(-z-qnorm(1-alpha/2)))

References

Chow S, Shao J, Wang H. 2008. Sample Size Calculations in Clinical Research. 2nd Ed. Chapman & Hall/CRC Biostatistics Series. page 51.

Rosner B. 2010. Fundamentals of Biostatistics. 7th Ed. Brooks/Cole. page 232.