S2 - Type II Errors

A pharmaceutical company is developing a new drug to treat a certain disease. The company will continue to develop the drug if the proportion of thos who have the disease show a substantial improvement after treatment is greater than 0.7. The company carries out a test at the 5% signif level, on a random sample of 14 patients who suffer from the disease.

a) find the critical region for the test

b) Given that 12 out of 14 patients show an improvemnt, carry out the test

c) find the prob that the test results in a type II error if in fact p=0.8


So far I have done this:

a) z>1.645

b) Xobserved = 12, X~B(14, 0.7), E(X) 12>14
so P(x>12)=1-p(x<11) = 1-0.8392 = 1.608
1.608<1.645 so accept H1.

c) P(type II error | p=0.8)
Now I did start doing this but got really confused as only done one question like this in class!! (We did lots on continuous data but only one for discrete!) Could someone explain how to do it??

Also I am unsure if (a) and (b) are correct as well, so if you can confirm this/find errors?

Thankyouu

(a) H0: p=0.7 H1: p>0.7
Under H0, X~B(14, 0.7)
The critical region will be of the form X>=c where P(X>=c) <0.05
Solve this to find c.

---NOTE: I haven't done stats for a while. I'm only trying to help here but if someone jumps in and corrects me, they're probably right ---

Not sure where z came from in part a) - you've recognised this is binomial, so why did you make it normal?

Also for part b), the bit in bold makes no sense -

so P(x>12)=1-p(x<11) = 1-0.8392 = 1.608
1.608<1.645 so accept H1.

I think for this sort of question you should be looking at your binomial tables... hint there

Hmm.. god my brain is fuzzled. Sorry my teacher really rushed the whole bit of hypothesis testing so I keep getting bits muddled!!

Am I right in thinking then that I don't do the 'normal' critical region as this is binomial? Is that why you are suggesting the P(x>c) bit? Also, I do not quite understand what you mean by this, could you explain it a bit fuller? Thanks!

Hmm.. god my brain is fuzzled. Sorry my teacher really rushed the whole bit of hypothesis testing so I keep getting bits muddled!!

Am I right in thinking then that I don't do the 'normal' critical region as this is binomial? Is that why you are suggesting the P(x>c) bit? Also, I do not quite understand what you mean by this, could you explain it a bit fuller? Thanks!

Critical region are the X values that lead to rejection of H0. You reject H0 when
P(X>=c) <0.05
i.e. 1 - P(X <= c-1) <0.05
P(X <= c-1) >0.95

Use binomial tables to find the range of values for c-1 and hence for c.

I think that is what I just did in the post above Is that right?