# Hyphothesis Test For A Binomial Proportion, p

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Thread starter 1 month ago
#1
Hi,

I cant understand the following thing in my book. According, to my own understanding, the critical region boundary for 5% { i.e. P(X>=c) = 5% where c is between 8 and 9 ) cannot exist as X cannot be equal to c { i.e. X is Discrete }. I also cannot understand the diagram on the last page showing the critical region boundary of 5% and the last two lines by the author

" Since { P(X>=8) is approximately 10% and P(X>=9) is approximately 4% }, the boundary comes between 8 and 9. Note that with discrete distributions you will probably not get a perfect 5% in your calculations. "

Thanks for help.

Regards,
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1 month ago
#2
In the question, given how many suits Sid guessed correctly, you're trying to investigate whether they are psychic. Since by calculation the chance of them guessing 7 correctly is decently high (21%), you conclude there's not enough evidence to say they're psychic, since one out of 5 times that would happen. You can change the significance level you use according to how accurate you want to be. The last 2 lines say since P(X>=8) is greater than 5%, there's still not enough certainty, but for 9, since it's only 4%, we're confident enough to make the alternative conclusion.
In this case neither is exactly 5%, so we have to settle for the one below 5%, ie P(X>=9). This is usually the case for discrete distributions since it's rare to get exactly 0.05 in the calculation. For continuous distributions eg normal, you can precisely work out the z value for above which the probability is below 5%, which is the critical region.
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Thread starter 1 month ago
#3
(Original post by *****deadness)
In the question, given how many suits Sid guessed correctly, you're trying to investigate whether they are psychic. Since by calculation the chance of them guessing 7 correctly is decently high (21%), you conclude there's not enough evidence to say they're psychic, since one out of 5 times that would happen. You can change the significance level you use according to how accurate you want to be. The last 2 lines say since P(X>=8) is greater than 5%, there's still not enough certainty, but for 9, since it's only 4%, we're confident enough to make the alternative conclusion.
In this case neither is exactly 5%, so we have to settle for the one below 5%, ie P(X>=9). This is usually the case for discrete distributions since it's rare to get exactly 0.05 in the calculation. For continuous distributions eg normal, you can precisely work out the z value for above which the probability is below 5%, which is the critical region.
So, why is it showing in the diagram of a discrete distribution, the critical region boundary of 5% between 8 and 9, although it does not exists?
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1 month ago
#4
(Original post by Tesla3)
So, why is it showing in the diagram of a discrete distribution, the critical region boundary of 5% between 8 and 9, although it does not exists?
Yeah you can't have 8.5 guesses, but the point is 8 is outside the critical region while everything >=9 is
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Thread starter 1 month ago
#5
(Original post by *****deadness)
Yeah you can't have 8.5 guesses, but the point is 8 is outside the critical region while everything >=9 is
I know that everything less than 9 is outside the critical region but why is the critical region boundary not at X=9?
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Thread starter 1 month ago
#6
Hey, can't we say that we are performing the significance test at a significance level of 4% and draw the critical region boundary at X=9? That would make it easier to grasp.. *****deadness
Last edited by Tesla3; 1 month ago
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1 month ago
#7
(Original post by Tesla3)
I know that everything less than 9 is outside the critical region but why is the critical region boundary not at X=9?
In the exam for discrete distributions you do say the critical region is X>=9, but theoretically the 5% boundary is between 8-9. In the exam you'll be given the significance level and you can't just do it at whichever significance level you want, and even then P(X>=9) won't exactly be 4%, there'd be a string of decimals probably so you'd still have to know that you need to choose the closest one with a probability of less than 4%.
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Thread starter 1 month ago
#8
(Original post by *****deadness)
In the exam for discrete distributions you do say the critical region is X>=9, but theoretically the 5% boundary is between 8-9. In the exam you'll be given the significance level and you can't just do it at whichever significance level you want, and even then P(X>=9) won't exactly be 4%, there'd be a string of decimals probably so you'd still have to know that you need to choose the closest one with a probability of less than 4%.
When you say that " but theoretically the 5% boundary is between 8-9 ", what do you mean theoretically? ,means not mathematically!, can you elaborate?
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Thread starter 1 month ago
#9
Also, when you say that " The last 2 lines say since P(X>=8) is greater than 5%, there's still not enough certainty, but for 9, since it's only 4%, we're confident enough to make the alternative conclusion. " Whats the alternative conclusion? *****deadness
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1 month ago
#10
(Original post by Tesla3)
When you say that " but theoretically the 5% boundary is between 8-9 ", what do you mean theoretically? ,means not mathematically!, can you elaborate?
Well neither X>=8 or 9 gives exactly 0.05, one gives a probability below 0.05 while the other is above 0.05, and it's kind of pointless to argue where the boundary is because we can't calculate what P(X>=8.5) is, for example, and it's not meaningful even if we could, since Sid can't guess 8.5 correctly. That's the thing with discrete distributions and if we're only concerned about 8 or 9 on either side of the boundary, we know X>=9 is the critical region
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1 month ago
#11
(Original post by Tesla3)
Also, when you say that " The last 2 lines say since P(X>=8) is greater than 5%, there's still not enough certainty, but for 9, since it's only 4%, we're confident enough to make the alternative conclusion. " Whats the alternative conclusion? *****deadness
That Sid is psychic, the alternative hypothesis in this case. If they guessed so many correctly that it's highly unlikely you can suspect that the probability is greater than the 0.25 we based the calculations upon
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Thread starter 1 month ago
#12
(Original post by *****deadness)
Well neither X>=8 or 9 gives exactly 0.05, one gives a probability below 0.05 while the other is above 0.05, and it's kind of pointless to argue where the boundary is because we can't calculate what P(X>=8.5) is, for example, and it's not meaningful even if we could, since Sid can't guess 8.5 correctly. That's the thing with discrete distributions and if we're only concerned about 8 or 9 on either side of the boundary, we know X>=9 is the critical region
Then why did he show that boundary in the diagram if its pointless?
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1 month ago
#13
(Original post by Tesla3)
Then why did he show that boundary in the diagram if its pointless?
P(X>=9) is less than 0.05 so by definition isn't the exact boundary, and as the book says it lies between 8 and 9, but what's important is the critical region is X>=9
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Thread starter 1 month ago
#14
(Original post by *****deadness)
P(X>=9) is less than 0.05 so by definition isn't the exact boundary, and as the book says it lies between 8 and 9, but what's important is the critical region is X>=9
Can we say as the critical region boundary percentage is increasing as we move left and decreasing as we move right in the diagram, therefore it could be between 8 and 9?
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1 month ago
#15
(Original post by Tesla3)
Can we say as the critical region boundary percentage is increasing as we move left and decreasing as we move right in the diagram, therefore it could be between 8 and 9?
Yeah, since it's cumulative, going left from X>=9 means we include extra cases of X=8, 7 and so on. And since P(X>=9) is below 0.05 while P(X>=8) is above, the boundary lies somewhere in between although the exact value is unimportant
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Thread starter 1 month ago
#16
C

(Original post by *****deadness)
Yeah, since it's cumulative, going left from X>=9 means we include extra cases of X=8, 7 and so on. And since P(X>=9) is below 0.05 while P(X>=8) is above, the boundary lies somewhere in between although the exact value is unimportant
Can we say that the critical region boundary for 5% is imaginary, as in this particular case, it does not exist due to the fact that there is no such value of " c " for which P(X>=c) = 5%. ?
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