Statistics : Hypothesis Testing

Question

I'm back with another question I can't solve. For this question, I was able to get part of the answer; nk$^{n-1}$(1-k), but I can't seem to see where you'd get the k$^n$ part.

Questions

Answers

Answer 1

If the alternative hypothesis is $H_1: p>k$ then it is an right tailed test, where $c$ is the critical value. If the critical value is part of the critical region, then the critical region is $\{c,c+1,\ldots, n\}$.

In your case $c=n-1$. Therefore the critical region is $\{n-1, n\}$. The largest element is always $n$. Now you sum up these probabilities:

$$\sum_{x=n-1}^{n} \binom{n}{x}\cdot k^x\cdot (1-k)^{n-x}$$

$$=\binom{n}{n-1}\cdot k^{n-1}\cdot (1-k)^{1}+\binom{n}{n}\cdot k^{n}\cdot (1-k)^{0}=n\cdot k^{n-1}\cdot (1-k)+k^n$$

The source is german wiki (translated).

$\alpha$ is the significance level.