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I have an exercise with the following information; $ 20\% $ of the set are youth, my sample of $ 139 $ persons have $ 39 $ youths. The exercise is to test if this sample is a representative sample of the set and to find the level of significance of the result.

I have two hypothesis $ H_0 = 0.20 $ and $ H_1 \ne 0.20 $, sample statistics $ \hat{p} = 0.2955 $ and the test statistic $ z = 2.74 $ the p-value: $ 2 * P(z\geqslant 2.74) = 0.0062 $

How do I go from this to finding the level of significance?

Help is much appreciated.

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    $\begingroup$ Forgive my confusion; what level of significance are you looking for if not the $p$ value itself? $\endgroup$ – Nick Stauner May 14 '14 at 22:17
  • $\begingroup$ Hi, and sorry for the confusion. I think the level of significance of the p-value? The exercise is diffuse about this. I have the solution, but not how to get there; level of significance = 1%. $\endgroup$ – oyvindym May 15 '14 at 7:36
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This is sounding like a homework problem, so my answer will invoke the homework policy and provide hints.

Statistical software often marks the significance of results with different symbols to represent "levels" of significance. Statistical significance is continuous, but can be grouped into bins to simplify the distinctions among results of varying significance. Here's a common line in summary output:

Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

If you were looking for results at the significance level of .01, you'd be looking for p values marked with **. This would include anything not marked with ***, which would be any p values below .001, and anything not marked with *, ., or without any mark at all.

Personally, I think this binning procedure reflects the ambivalence the scientific community feels toward the question, "Is the exact value of a 'p-value' meaningless?" Another sign of that appears in the way p values are often reported: "< .05" or "< .01", for instance. Of course, software often outputs results with many more digits than we want to report for any statistic, and we often round our results to a reasonably brief degree of precision like two or three digits.

It sounds like you've done most of your work already, and you know what result your exercise is looking for. Do you see any results in your output that could be rounded to the value you're looking for? Put another way, levels generally represent upper bounds on the bins for p values. Thus if you had a p = .0011, wouldn't mark that with ***, it would use **, because it's not quite low enough to reach the next level below .001.

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Before looking at P values and significance levels, step back and try to articulate the question you are trying to answer. If you know your sample was randomly drawn from a population where 20% are youth, then there is nothing to do. Statistical calculations can help you when you aren't sure which population your data came from. But if you already know the population, then there really is nothing to test with statistics.

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This may be a source of possible confusion: "the significance level" is the type I error rate, $\alpha$, which you choose before you start. (See here, here and here)

Sometimes that (the significance level) will get called "the level of significance", but confusingly, some people call the p-value "the level of significance" (in the sense of 'how significant is it', but in a Neyman-Pearson framework it's not really appropriate to do so)

Your question should make clear which kind of quantity you mean, but in either case, you should already know the quantity by the time you see the mention of a p-value (in the first because whoever carries out the test should specify their significance level before finding the p-value, and in the second case because it is the p-value we've just seen mention of).

[Sometimes in papers people only give the p-value; you are able to compare it with your own choice of significance level.]

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