S2 Two tailed Hypothesis Test

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#1
I am a bit confused on how to perform two tailed tests. For example on question 7 on the jan 2006 paper, for part (a) they test P(X>9) and part (b) P(x<18), why don't they do P(X<9) or P(X>18)?? Basically how do you know which tail to test when you have two?

I understand hypothesis testing I don't understand how to perform a test with two tails to pick from, please can someone help me as the exam is soon and i'm panicking

(I have uploaded pics of the question and mark scheme)

Thank you!
0
5 years ago
#2
(Original post by Katiee224)
I am a bit confused on how to perform two tailed tests. For example on question 7 on the jan 2006 paper, for part (a) they test P(X>9) and part (b) P(x<18), why don't they do P(X<9) or P(X>18)?? Basically how do you know which tail to test when you have two?

I understand hypothesis testing I don't understand how to perform a test with two tails to pick from, please can someone help me as the exam is soon and i'm panicking

(I have uploaded pics of the question and mark scheme)

Thank you!
You could try http://burymathstutor.co.uk/binom.html to get a picture of what's going on.

Try it with n=25 and you'll see that x=9 is one of the sufficiently unlikely outcomes on the high side. You can try it with n=100 too although the question asks for an approximation of course.
1
5 years ago
#3
(Original post by Katiee224)
I am a bit confused on how to perform two tailed tests. For example on question 7 on the jan 2006 paper, for part (a) they test P(X>9) and part (b) P(x<18), why don't they do P(X<9) or P(X>18)?? Basically how do you know which tail to test when you have two?

I understand hypothesis testing I don't understand how to perform a test with two tails to pick from, please can someone help me as the exam is soon and i'm panicking

(I have uploaded pics of the question and mark scheme)

Thank you!
In part (a), if the percentage of pupils that read the Deano was in fact 20%, then you would expect to find 4 students from the sample who read the Deano. However, in the sample there are 9 students who read the book, which exceeds the expected number (i.e 4 students). Hence they test P(X>9), or in other words, they test how 'far' the data exceed the expectation. If it is too far from what we expect (probability is less than 5% sig. level in this case) then the hypothesis 20% is statistically incorrect

Same thing applied for part (b), only in this case the number of students is lower than the expected number (i.e 20 students are expected to read the Deano from these samples in total) thus they test P(X<18). Hope this makes sense
3
#4
(Original post by NDVA)
In part (a), if the percentage of pupils that read the Deano was in fact 20%, then you would expect to find 4 students from the sample who read the Deano. However, in the sample there are 9 students who read the book, which exceeds the expected number (i.e 4 students). Hence they test P(X>9), or in other words, they test how 'far' the data exceed the expectation. If it is too far from what we expect (probability is less than 5% sig. level in this case) then the hypothesis 20% is statistically incorrect

Same thing applied for part (b), only in this case the number of students is lower than the expected number (i.e 20 students are expected to read the Deano from these samples in total) thus they test P(X<18). Hope this makes sense
Ohh ok, I think I see where I was getting confused. So for part (a) my hypotheses are null hyp: p=0.2, alt hyp: p>0.2 .... and for part (b) my hypotheses would be null hyp: p=0.2, alt hyp p<0.2 ??

And I solve it like a one tailed test?
0
5 years ago
#5
(Original post by Katiee224)
Ohh ok, I think I see where I was getting confused. So for part (a) my hypotheses are null hyp: p=0.2, alt hyp: p>0.2 .... and for part (b) my hypotheses would be null hyp: p=0.2, alt hyp p<0.2 ??

And I solve it like a one tailed test?
No, for both parts it would be:

and they are both two-tailed tests

EDIT: Give me 5 mins, I will type out a detailed solution for this question so that you can understand it better.
0
5 years ago
#6
(Original post by Katiee224)
Ohh ok, I think I see where I was getting confused. So for part (a) my hypotheses are null hyp: p=0.2, alt hyp: p>0.2 .... and for part (b) my hypotheses would be null hyp: p=0.2, alt hyp p<0.2 ??

And I solve it like a one tailed test?
No. In both cases the alternative hypothesis is .

The question has the phrase "...is different from 0.2."

If you use a one tailed test instead then, roughly speaking, your critical region will be too big.
0
5 years ago
#7
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1
#8
(Original post by NDVA)
No, for both parts it would be:

and they are both two-tailed tests

EDIT: Give me 5 mins, I will type out a detailed solution for this question so that you can understand it better.
(Original post by BuryMathsTutor)
No. In both cases the alternative hypothesis is .

The question has the phrase "...is different from 0.2."

If you use a one tailed test instead then, roughly speaking, your critical region will be too big.
Oh ok, I understand what you guys mean with the hypotheses

So I could essentially solve it like a one-tailed test once I have halfed the significance level?
1
5 years ago
#9
(Original post by Katiee224)
Oh ok, I understand what you guys mean with the hypotheses

So I could essentially solve it like a one-tailed test once I have halfed the significance level?
Yes this is correct
0
#10
Thank you for the help both of you, if I get an A on monday it will be down to your help
1
5 years ago
#11
(Original post by Katiee224)
I am a bit confused on how to perform two tailed tests. For example on question 7 on the jan 2006 paper, for part (a) they test P(X>9) and part (b) P(x<18), why don't they do P(X<9) or P(X>18)?? Basically how do you know which tail to test when you have two?

I understand hypothesis testing I don't understand how to perform a test with two tails to pick from, please can someone help me as the exam is soon and i'm panicking

(I have uploaded pics of the question and mark scheme)

Thank you!
Part (a)

at 5% sig. level

Expected number of pupils who read the Deano .

Observed number is thus we need to
1/ calculate the probability that the number of pupils who read Deano is 9 or greater under
2/ check if this probability is smaller than 2.5% or not (as this is a two-tailed test). If it is smaller than 2.5%, then we reject our null hypothesis. (we are looking at the upper tail here)

Perform the test
This means the probability of is too small for to be true at this level of significance. Hence we reject our null hypothesis, there is evidence at 5% significant level that the percentage of pupils that read Deano is not 20%.

Part (b)

at 5% sig. level

Expected number of pupils who read the Deano .

Observed number is thus we need to
1/ calculate the probability that the number of pupils who read Deano is 18 or smaller under
2/ check if this probability is smaller than 2.5% or not (as this is a two-tailed test). If it is smaller than 2.5%, then we reject our null hypothesis. (we are looking at the other tail here)

Use normal approximation to make the calculation easier -> perform the test -> Conclusion as in the mark scheme.

Hope this helps
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