# S2: hypothesis testing

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Thread starter 5 years ago
#1
I don't understand why when you have a two tailed test with 5% significance level, you make the critical region 2.5% at each end? Why isn't it 5% at each end? Surely this means that values are being accepted which wouldn't be if the test was one-tailed?

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5 years ago
#2
(Original post by anoymous1111)
I don't understand why when you have a two tailed test with 5% significance level, you make the critical region 2.5% at each end? Why isn't it 5% at each end? Surely this means that values are being accepted which wouldn't be if the test was one-tailed?

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A significance level is a probability. The probability of the null hypothesis being rejected when it is true, to be precise. In a one-tailed test if we have a 5% SL then since the only way for the null hypothesis to be rejected is if the test statistic falls into the least-likely 5% at which ever tail you are testing. For a two-tailed test however it could be unexpectedly low or unexpectedly high, so we need to "share it" between the upper and lower tails. This way the overall probability of a test statistic falling in one of the critical regions is 5%. If we had put 5% at both ends then it would be a 10% probability of the test statistic falling into one of the critical regions.
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Thread starter 5 years ago
#3
(Original post by 16Characters....)
A significance level is a probability. The probability of the null hypothesis being rejected when it is true, to be precise. In a one-tailed test if we have a 5% SL then since the only way for the null hypothesis to be rejected is if the test statistic falls into the least-likely 5% at which ever tail you are testing. For a two-tailed test however it could be unexpectedly low or unexpectedly high, so we need to "share it" between the upper and lower tails. This way the overall probability of a test statistic falling in one of the critical regions is 5%. If we had put 5% at both ends then it would be a 10% probability of the test statistic falling into one of the critical regions.
If you had a one tailed test and were testing whether something had increased, would you assume that it definitely hadn't decreased therefore there would be no critical region at the lower end so the whole 5% must be at the upper end?

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5 years ago
#4
(Original post by anoymous1111)
If you had a one tailed test and were testing whether something had increased, would you assume that it definitely hadn't decreased therefore there would be no critical region at the lower end so the whole 5% must be at the upper end?

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In such a situation you are testing only at the upper end yes.
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Thread starter 5 years ago
#5
(Original post by 16Characters....)
In such a situation you are testing only at the upper end yes.
But shouldn't it be the any value that has a less than 5% probability or occurring (that value or lower/higher depending on the tail) should lead to the hypothesis being rejected? Then surely it should still be 5% at each end? Sorry I'm still missing the point.

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Thread starter 5 years ago
#6
(Original post by 16Characters....)
In such a situation you are testing only at the upper end yes.
A one failed test is more powerful in one direction than a two tailed test. Why don't they just make the significance level of 5% mean that you take 5% at each tail so that the two-tailed test is equally powerful at detecting a change as the one-tailed test?

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5 years ago
#7
(Original post by anoymous1111)
A one failed test is more powerful in one direction than a two tailed test. Why don't they just make the significance level of 5% mean that you take 5% at each tail so that the two-tailed test is equally powerful at detecting a change as the one-tailed test?

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The significance level means the probability of the null hypothesis being rejected when it is true. That is the definition of a significance level and it won't be changed :-)

I understand what you are saying but this is just custom on what is meant by sig. level. When we use the normal system for hypothesis testing with a P% sig level then:
- For a one-tailed test the probability of a test statistic falling in the critical region by chance even though the null hypothesis is true is P%
- For a two-tailed test the probability of a test statistic falling in the critical region by chance even though the null hypothesis is true is 0.5P% at each tail, hence P% overall.

You are correct that this means that a test statistic at a given tail is more likely to fall into the critical region at that tail, but if you want to carry out a two tailed test which gives a T% chance of falling into the critical region at a particular region then just set your significance level to 2T%.
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Thread starter 5 years ago
#8
(Original post by 16Characters....)
The significance level means the probability of the null hypothesis being rejected when it is false. That is the definition of a significance level and it won't be changed :-)

I understand what you are saying but this is just custom on what is meant by sig. level. When we use the normal system for hypothesis testing with a P% sig level then:
- For a one-tailed test the probability of a test statistic falling in the critical region by chance even though the null hypothesis is true is P%
- For a two-tailed test the probability of a test statistic falling in the critical region by chance even though the null hypothesis is true is 0.5P% at each tail, hence P% overall.

You are correct that this means that a test statistic at a given tail is more likely to fall into the critical region at that tail, but if you want to carry out a two tailed test which gives a T% chance of falling into the critical region at a particular region then just set your significance level to 2T%.
That makes sense thank you. Just about your definition of significance level....in your first post you said it was the probability of the null hypothesis being rejected when it is true. In the post your said 'false' instead. Which is correct? I think it's when it is true isn't it?

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5 years ago
#9
(Original post by anoymous1111)
That makes sense thank you. Just about your definition of significance level....in your first post you said it was the probability of the null hypothesis being rejected when it is true. In the post your said 'false' instead. Which is correct? I think it's when it is true isn't it?

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Yes you are right. It is should be "true". I shall correct my post now, apologies.
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Thread starter 5 years ago
#10
(Original post by 16Characters....)
Yes you are right. It is should be "true". I shall correct my post now, apologies.
Great thank you so much!

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