My textbook just says that the following test statistic is normal without actually going through the derivation. Here is the problem:
Suppose that $X_1...X_n$ are iid RV with each being $N(\mu,\sigma^2)$ where $\sigma^2$ is known. We want to test:
We then take the following test statistic:
Now what I don't understand is why did the book choose this test statistic? It cannot be arbitrary can it? But more importantly, the book says that this test statistic is standard normal, can anyone please show me mathematically why this is true?
When I take the expectation and variance of $\tau_n$ I do not get $0$ for the expectation and $1$ for the variance so how can that be?