# How do I find the probability of a type II error?

I know that a Type II error is where H1 is true, but H0 is not rejected.

### Question

How do I calculate the probability of a Type II error involving a normal distribution, where the standard deviation is known?

• Feb 19, 2011 at 21:01
• I would rephrase this question as "how do I find the power of a general test, such as $H_{0}:\mu=\mu_{0}$ versus $H_{1}:\mu > \mu_{0}$?" This is often the more frequently performed test. I don't know how one would calculate the power of such a test. Feb 20, 2011 at 0:24

In addition to specifying $$\alpha$$ (probability of a type I error), you need a fully specified hypothesis pair, i.e., $$\mu_{0}$$, $$\mu_{1}$$ and $$\sigma$$ need to be known. $$\beta$$ (probability of type II error) is $$1 - \textrm{power}$$. I assume a one-sided $$H_{1}: \mu_{1} > \mu_{0}$$. In R:

> sigma <- 15    # theoretical standard deviation
> mu0   <- 100   # expected value under H0
> mu1   <- 130   # expected value under H1
> alpha <- 0.05  # probability of type I error

# critical value for a level alpha test
> crit <- qnorm(1-alpha, mu0, sigma)

# power: probability for values > critical value under H1
> (pow <- pnorm(crit, mu1, sigma, lower.tail=FALSE))
[1] 0.63876

# probability for type II error: 1 - power
> (beta <- 1-pow)
[1] 0.36124


Edit: visualization

xLims <- c(50, 180)
left  <- seq(xLims[1],   crit, length.out=100)
right <- seq(crit, xLims[2],   length.out=100)
yH0r  <- dnorm(right, mu0, sigma)
yH1l  <- dnorm(left,  mu1, sigma)
yH1r  <- dnorm(right, mu1, sigma)

curve(dnorm(x, mu0, sigma), xlim=xLims, lwd=2, col="red", xlab="x", ylab="density",
main="Normal distribution under H0 and H1", ylim=c(0, 0.03), xaxs="i")
curve(dnorm(x, mu1, sigma), lwd=2, col="blue", add=TRUE)
polygon(c(right, rev(right)), c(yH0r, numeric(length(right))), border=NA,
col=rgb(1, 0.3, 0.3, 0.6))
polygon(c(left,  rev(left)),  c(yH1l, numeric(length(left))),  border=NA,
col=rgb(0.3, 0.3, 1, 0.6))
polygon(c(right, rev(right)), c(yH1r, numeric(length(right))), border=NA,
density=5, lty=2, lwd=2, angle=45, col="darkgray")
abline(v=crit, lty=1, lwd=3, col="red")

• Is there a typo in this answer? I think what is called $\texttt{pow}$ is actually $\beta$ and vice versa. Either way, this is an excellent graph and example R code! Nov 19, 2018 at 2:30
• @jdods Indeed, there was a lower.tail=FALSE missing. Thank you very much! Nov 19, 2018 at 8:56