binomial distribution hypothesis testing question

Q: https://prnt.sc/q4ri78

For (ii), let p = the probability of getting heads on a given toss. Therefore, the null hypothesis is p = 0.5 (it is unbiased) & the alternative hypothesis is p < 0.5 (it is biased towards tails).

X = #. of heads in 50 trials, X~B(50, 0.5). Sig. level = 10%

The critical region is the region X ≤ k, where P(X ≤ k) ≤ 0.10. Based on the bionomial tables, the critical region is X ≤ 19. So then isn't the critical value 0? According to the textbook, for 29 tails, p = 0.8987 & for 30 tails p = 0.9405, so 30 is the critical value.

If it's a 1-tail distribution, how do I know which tail represents what is likely/unlikely? Wouldn't that depend on the p-value itself?

Why do you think the critical value is 0 if the critical region is X ≤ 19? You'll have to explain that one.

The textbook has used X = number of tails in 50 trials giving X~B(50, p), H_0 : p = 0.5, H_1 : p > 0.5. This is why their answer is different to yours. Since they talk about "tails" in the question it probably makes more sense to do the test using the book's method but I wouldn't say that your way is wrong.

It's 0 because that's the first x-value of the critical region. I don't see why I need to explain it.
The textbook didn't define the variable, so I should be able to use either one. Even if I did it by letting X = #. of tails in 50 trials, I would still get the same answer as I won't know which end of the distribution tail to focus on.

You're misunderstanding what the critical value is. It is the threshold between the acceptance region and the critical region e.g. if you look here:

https://miro.medium.com/max/1440/1*UKl3Fp9fjvsYvWySnMQiKA.png

The red region on the left is one of the critical regions and the LCV is the critical value.

If you used tails instead of heads in your random variable then your H_1 would be p > 0.5 which means that you only care about the upper tail.

It's 0 because that's the first x-value of the critical region. I don't see why I need to explain it.
The textbook didn't define the variable, so I should be able to use either one. Even if I did it by letting X = #. of tails in 50 trials, I would still get the same answer as I won't know which end of the distribution tail to focus on.


I understand now, but how do I know whether to focus on the upper tail or the lower tail? If it was heads instead of tails, then H_1 would be p < 0.5, but I would still need to find the UCV.

If X = #. of tails in 50 trials then X~B(50,p) and the "expected" number of tails is 25. If you're only testing if p>0.5 then when you do the experiment you may get large values of X if p>0.5 because it means that the coin is biased towards tails e.g. you may get 26 tails or 28 tails or 34 tails. When you do the hypothesis test you assume that p = 0.5 and you're checking if the number of tails you get is so extreme that it means that p is actually > 0.5. So the only critical region you care about is the upper critical region.

You may actually do the experiment and get only 2 tails but since you're only doing a 1-tailed test and checking if the coin is biased towards tails, this extreme low value is irrelevant. Does this make sense?

If instead you have X = #. of heads in 50 trials then X~B(50,p) and the expected number of heads if 25. Your H_1 would be p<0.5 so in this case you're likely to get low values and you only care about the lower critical region.

So to summarise, for a 1-tailed test checking p<a you only care about the lower critical region and for a 1-tailed test checking p>a you only care about the upper critical region.

That makes sense, but how do I know whether to let X = #. of heads in 50 trials or let X = #. of tails in 50 trials? From the way the question was worded, it seems like I could use either one.