Critical regions - Hypothesis testing

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maggiehodgson
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#1
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#1
The question asks:

X~B(20,p) and Ho: p=0.4 H1<>0.4
Find the critical region for X if the significance level is 10%

So I did when X = 5 p = 0.1255 which is > 0.1
X= 4 p = 0.0509 which is < 0.1

cr => X<= 4

Then when X = 11 p = 0.943 which is > 0.9
X=10 p = 0.8724 which is <0.9
cr => X >=10

The book answer is X<=3 or x>= 13

Could someone pleas explain where I'm going wrong
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Kevin De Bruyne
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#2
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#2
(Original post by maggiehodgson)
The question asks:

X~B(20,p) and Ho: p=0.4 H1<>0.4
Find the critical region for X if the significance level is 10%

So I did when X = 5 p = 0.1255 which is > 0.1
X= 4 p = 0.0509 which is < 0.1

cr => X<= 4

Then when X = 11 p = 0.943 which is > 0.9
X=10 p = 0.8724 which is <0.9
cr => X >=10

The book answer is X<=3 or x>= 13

Could someone pleas explain where I'm going wrong
<> means it's two tailed.

When it's two tailed, each side makes half of the significance level, or as close as possible (so the 5% on each side whereas you've done 10%, which combines to give 20%)
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maggiehodgson
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(Original post by Kevin De Bruyne)
<> means it's two tailed.

When it's two tailed, each side makes half of the significance level, or as close as possible (so the 5% on each side whereas you've done 10%, which combines to give 20%)
There is a two-tailed example in the book for !0% and they don't split it.

Here it is:

Q.
You are told that X~B(40,p) and Ho=0.25 H1: p <> 0.25
For each of the following values i) state the p value to 4dp and ii) state with clear reason your conclusion for the hypothesis test at 10%

a) X=7 b) X=15

A.
0.25x40 = 10 so 10 is the most likely outcome. This is a two-tailed test so you must consider values much higher and much lower than 10.

a) P(x <= 7) = 0.1820
the p value is greater than 0.1, so accept Ho

b) P(X>= 15) = 1-P(X<=14)
= 1 - 0.9456
= 0.0544
the p value is less than 0.1 so reject Ho
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Kevin De Bruyne
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#4
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#4
(Original post by maggiehodgson)
There is a two-tailed example in the book for !0% and they don't split it.

Here it is:

Q.
You are told that X~B(40,p) and Ho=0.25 H1: p <> 0.25
For each of the following values i) state the p value to 4dp and ii) state with clear reason your conclusion for the hypothesis test at 10%

a) X=7 b) X=15

A.
0.25x40 = 10 so 10 is the most likely outcome. This is a two-tailed test so you must consider values much higher and much lower than 10.

a) P(x <= 7) = 0.1820
the p value is greater than 0.1, so accept Ho

b) P(X>= 15) = 1-P(X<=14)
= 1 - 0.9456
= 0.0544
the p value is less than 0.1 so reject Ho
To me that example is wrong, or whoever wrote it got lost midway through - if for example p(x<=6) = 0.075, then in a two tailed test it would be accepted as it is not less than 0.05, which is what should happen in a two tailed test - but the answer seems to apply a one tailed test for no particular reason.

And as you can see, there is a contradiction as a result between your two questions. Having done hundreds of hypothesis testing questions, my experience is that they are split and this question is utter rubbish - someone will correctly if I'm missing something obvious but I'm not convinced that this example is what it's meant to be.
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maggiehodgson
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#5
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#5
(Original post by Kevin De Bruyne)
To me that example is wrong, or whoever wrote it got lost midway through - if for example p(x<=6) = 0.075, then in a two tailed test it would be accepted as it is not less than 0.05, which is what should happen in a two tailed test - but the answer seems to apply a one tailed test for no particular reason.

And as you can see, there is a contradiction as a result between your two questions. Having done hundreds of hypothesis testing questions, my experience is that they are split and this question is utter rubbish - someone will correctly if I'm missing something obvious but I'm not convinced that this example is what it's meant to be.
Thank you so much. The book I am using is new on the market and the statistics section is full of errors that even I can spot. I've never done the HT stuff before so was completely lost with the example and the answers.

If I do split it then I can get X<=3 but at the other end I get X >=12 so in as much as X<=11 gives p= 0.943 and X <=12 p=0.9789.
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Kevin De Bruyne
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#6
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#6
(Original post by maggiehodgson)
Thank you so much. The book I am using is new on the market and the statistics section is full of errors that even I can spot. I've never done the HT stuff before so was completely lost with the example and the answers.

If I do split it then I can get X<=3 but at the other end I get X >=12 so in as much as X<=11 gives p= 0.943 and X <=12 p=0.9789.
I see. How useful for someone trying to learn stuff!

I'd agree with x>=12 being the upper region based on your numbers - sometimes I've seen it stipulated that the significance level has to be strictly less than ... so your 0.943 would have not been enough, but as this is not mentioned in the question it's safe to pick whichever is closer to 0.95, which is indeed x>=12 rather than x>=13.
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maggiehodgson
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#7
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#7
(Original post by Kevin De Bruyne)
I see. How useful for someone trying to learn stuff!

I'd agree with x>=12 being the upper region based on your numbers - sometimes I've seen it stipulated that the significance level has to be strictly less than ... so your 0.943 would have not been enough, but as this is not mentioned in the question it's safe to pick whichever is closer to 0.95, which is indeed x>=12 rather than x>=13.
You have been a great help. The questions that I have done since also don't measure up to the answers in the book. I will be in touch with the publishers yet again.

I'll not be bothering you with any more questions this evening. Good night.
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username1732133
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#8
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#8
(Original post by maggiehodgson)
You have been a great help. The questions that I have done since also don't measure up to the answers in the book. I will be in touch with the publishers yet again.

I'll not be bothering you with any more questions this evening. Good night.
If the book is unreliable you could check your answers here: http://burymathstutor.co.uk/binom.html
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maggiehodgson
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#9
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#9
(Original post by BuryMathsTutor)
If the book is unreliable you could check your answers here: http://burymathstutor.co.uk/binom.html
What a star you are

Thanks
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