# Finding a type 1 error

Suppose that $$T_1,…,T_{10}$$ are iid $$Exp(λ)$$ and the goal is to test if $$H_0:λ≤1$$ versus $$H_a:λ≥2$$. Suppose that the test statistic is $$S=\sum_{i=1}^{10}T_i$$, and rejection of the null occurs when $$S≤7$$.

I realize that $$T_1...T_{10}$$ is a gamma distribution of $$S =Gamma(10, \lambda$$), but I am not sure how to use this information to find the p-value for the type 1 error.

• Does this answer your question? Finding bound for type 1 and type 2 error – B.Liu Apr 1 at 0:30
• No, it doesn't answer the question – rabito Apr 1 at 0:35
• Apologies, that is an artefact of flagging it as a duplicate. Please consider editing your original question, as they are quite similar as written and we prefer keeping answers for similar questions in the same page. – B.Liu Apr 1 at 0:39
• Do not cross-post please. – StubbornAtom Apr 1 at 11:22

When computing the Type I error, you assume $$H_0$$ is true. So you are finding the $$P(S \le 7 | \lambda \le 1)$$. As $$\lambda$$ gets smaller, we expect $$S$$ to get larger, so $$P(S \le 7 | \lambda \le 1)$$ will be the largest when $$\lambda = 1$$. $$P(S \le 7 | \lambda \le 1) \le P(S \le 7 | \lambda = 1) = P(S \sim Gamma(10, 1) \le 7) = 0.1695$$

One other note, the null and alternate hypothesis normally divide the parameter space without holes. Did you mean to say $$H_a: \lambda > 1$$? You can compute this in R with pgamma(7, 10, 1)

Nsims <- 10000

lambda0 <- 1
n <- 10
S <- numeric(Nsims)
for (i in 1:Nsims) {
Tn <- rexp(n, rate = lambda0) # lambda is E(exp(lambda)) = 1 / lambda
S[i] <- sum(Tn)
}

hist(S, freq = FALSE)
lines(seq(0,30,length=1000), dgamma(seq(0,30,length=1000), n, lambda0)) #P(S < 7 | lambda0 = 1)

length(which(S <= 7)) / length(S)
#>  0.1723
pgamma(7, 10, 1)
#>  0.1695041

• thank you! this makes so much sense. The problem actually gives me $H_a \geq 2$, but I agree, it normal should be $H_a >1$ – rabito Apr 1 at 1:12

You reject when $$S \le 7$$ where, under $$H_0; \lambda \le 1,$$ $$S \sim \mathsf{Gamma}(10, 1),$$ so the probability of Type I Error (rejecting $$H_0$$ when $$H_0$$ is true) is $$\alpha = P(S \le 7\, |\, H_0) = 0.1695.$$ [Computation in R, where pgamma is a gamma CDF.]

pgamma(7, 10, 1)
 0.1695041


Then $$\beta = P(x \ge 7\, |\, H_a) = 1 - P(S \le 7\, |\, H_a) = 0.109,$$ whee $$H_a: \lambda \ge 2.$$

1 - pgamma(7, 10, 2)
 0.1093994


In the figure below $$\alpha$$ is the area under the blue density curve to the left of $$s = 7$$ and $$\beta$$ is the area under the brown density curve to the right of $$s=7.$$ R code for figure:

hdr = "Density functions of GAMMA(10,1) [null, blue] and GAMMA(10,2)"
curve(dgamma(x, 10, 1), 0, 20, ylim=c(0,.3), col="blue", lwd=2,
ylab="Density", xlab="s", main=hdr)
curve(dgamma(x, 10, 2), add=T, col="brown", lwd=2)
abline(v=0, col="green2");  abline(h=0, col="green2")
abline(v=7, col="red", lwd=2, lty="dashed")

• Small detail: The alternative hypothesis is a composite hypothesis. The computation for $\beta$ is the computation of the suppremum of $\beta$. The actual type II error might be smaller when $\lambda>2$. (But anyway the type II error was not asked) – Sextus Empiricus Apr 1 at 6:47
• @SextusEmpiricus: All valid points. // Seemed incomplete without type II error and graphs. – BruceET Apr 1 at 7:39