Are there normal tables for a random variable which is the sum of N standard normal random vars?
https://youtu.be/rYefUsYuEp0?t=776
The lecture here says we can find $\xi'$ such that this probability $\alpha$ is $5$%.
But how do we do this really?
$H_0$ is the hypothesis that the $n$ variables are $X_i \sim N(0,1)$.
$H_1$ is the hypothesis that the $n$ variables are $X_i \sim N(1,1)$.
The $N$ variables are independent and identically distributed.
So if e.g. $N=5$ how do I find such $\xi'$ so that the tail (of the distribution of the sum of 5 standard normal random variables) has probability 5%. I thought only the standard normal distribution is tabulated really.
I mean, is there some way to do this without (too much) computer help?
Say I want to do this in an old-fashioned way with paper, pencil,
and some books and tables that I have in my hands. How?
It is a matter of verifying a system of simple hypothesis (both simple) thus Neyman Pearson's lemma can be applied.
At the end you evaluate
$$P(\overline{X}_n\geq k|\mu=0)=0.05$$
$k$ can be found using Z-table
If $\overline{X}_n\geq k$ you decide for $H_1: \mu=1$ otherwise you decide for the mean to be zero
Speaking about numbers, using a random sample with size $n=5$, you set
$$P(\overline{X}_n\sqrt{5}\geq k)=0.05$$
$k=1.64$ is given by the z-table thus you reject $H_0$, say you decide for $H_1$ if and only if
$$\overline{X}_n\geq 0.7356$$