Say we have a sample of size n from a geometric random variable with rate p, that \(P(X=x)=p(1-p)^{x-1}\); \(x=1,2,...\).

As an example we have the data set

x Freq
1 49
2 23
3 14
4 8
5 4
6 2

In this homework you can use R for numerical calculations but you can not use any of the probability routines such as pgeom, qbinom etc.

Problem 1

Find the mle of \(p\)

Problem 2

Derive an exact hypothesis test for

\[H_0: p=p_0\text{ vs. }H_a:p>p_0\] based on the mle.

Problem 3

Find the power of this test. Use it to draw the power curve for the case \(n=100, p_0=0.5,\alpha=0.05\) and \(p_1\in [0.5,0.65]\).

Problem 4

If in fact \(p=0.54\), what sample size n would be needed so that the test has a power of 80%?

Problem 5

Find another test, again based on the mle but now using a normal approximation (aka the central limit theorem). For the case \(n=100,\alpha=0.05,p_1=0.6\), which of these tests is more powerful?