Chapter 12, pg 32, Q 12.19

I do not understand how to interpret this question.

Well it was obvious that we need to estimate theta first. The solution says that the MLE is Xbar/6. How come?

And why is n=6? As far I understand, n is the number of trials and I do not think only 6 trials were conducted.

I cant get my head around this problem. What am I missing? Help me with this.

 

Well John I still do not understand. The Solution says "If you cant spot immediately that the MLE here is just theta(hat) = Xbar/6, you can derive it as follows..."

How do I spot that IMMEDIATELY? I havnt spotted that yet! Can you just guide me conceptually with this problem? And how is n=6? There were 200 crates and all the crates were tested for impure bottles. Then why cant n be 200?

And theta(hat) = 0.25083, so Xbar should be 6 x 0.25083 = 1.50498

I tried finding the mean from the first principles ie summation(fi*xi)/summation(fi) = 351/200 = 1.755. I do not know if I am making sense. But what is going wrong here?

 

Oh! Sorry about that mistake.

So theta here is the probability of a bottle being impure in a crate of 6 bottles and hence X ~ Bin(6,theta). Is this the right interpretation?

If yes, can I say that in this problem there are 200 such independent Bin(6,theta) random variables since there are 200 crates. And thats the reason I multiply the binomial probabilities of X=0,1,2,3,4 and multiply each of them by 200 giving me the expected number of impure bottles(ei's)? Is this the right interpretation?

 

Thanks John