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# S2 Significance Levels - very confused watch

1. Say the significance level required is 5%

In a hypothetical situation there are two combinations for the actual significance level.

One is 2.6% and the other one is 5.6%. In the book they say I have to choose one closer to 5%, but at the same time sometimes they reject values above 5%.

So my question is: should the actual significance level be:
a) As close to the required sig. level as possible
a) As close to the required sig. level as possible but not higher than it.

Example question:

A single observation of x is to be taken from a Poisson distribution with a parameter λ.
This observation is to be used to test H0:λ=7 against H1:λ≠7.

a) using 5% significance level, find the critical region for this test assuming that the probability of rejection in either tail is as close as possible to 2.5%.

So what I do is:
-look at the poisson tables, find λ=7
-I find out that P(X<=1)=0.0073 and P(X<=2)=0.0296
-I select X<=2 as a lower critical region because 0.0296 is closer to 0.0250 than 0.0073

-I look at the poisson table again to find the upper critical region.
-I find out that P(X<=13)=0.9872 and P(X<=12)=0.9730
-Because it is upper critical region I have to do this:
-> 1 - P(X<=13)=1-0.9872=0.0128=P(X>=14)
-> 1 - P(X<=12)=1-0.9730=0.0270=P(X>=13)
-I select the P(X>=13) as 0.270 is closer to 0.0250 than 0.0128.

So my critical region is:
X<=2 & X>=13

b) Write down the significance level of this test.

which gives the actual significance level of: P(X<=2) + P(X>=13)=0.0296+0.0270=0.0566 which is 5.66%

THE BOOK SAYS THAT MY BOUNDARIES ARE OK BY APPARENTLY THE ACTUAL SIGNIFICANCE LEVEL IS 4.56%. What the hell??

Also notice how boundaries which give me higher than specified significance level are allowed. At the same time in a different part of the book, in a different example it says:

H0:λ=7.5

At the 5% significance the one tailed test will reject H0 if X is 0, 1 or 2. There is a probability of 2% of rejecting H0 when it is in faact true which is as close to 5% as it is possible to get. If the value 3 was included, then this probability would rise to 5.9% which is greater than 5%.

BUT IT IS CLOSER TO 5% TOO!

tl;dr:
Please give me detailed answers, I have asked this several times and nobody was able to explain it to me. You have to read the examples to get a grasp of what I'm asking. If its too long to read, dont read at all
2. I think that your book is wrong - this looks like it was taken from the Jan 03 paper - and the answer is 0.0566 in the mark scheme.

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Updated: April 4, 2012
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