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I need to have one understanding on statistical hypothesis testing. In a typical hypothesis test, we have 2 opposite hypotheses; namely Null and Alternative. Here my textbook says that "those 2 hypotheses are not symmetrical in the sense that if we swap the hypotheses then the result will alter".

Here I am unable to grasp the point which that textbook wants to say. Can somebody explain to me in detail? It would be helpful if someone can give some example of that asymmetry as well.

Appreciate your help.

Thanks,

Shantanu
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Bogaso
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2 Answers2

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Suppose you have a random sample of uncirculated US double eagle gold coins which you have assayed for gold content. Although manufacturing variations precluded the US Mint from guaranteeing that every double eagle had exactly 0.9675 Troy ounces of gold, the Mint tried to achieve this as an average value. If they succeeded, the distribution of weights in your sample should be centered close to this value. A sample average that is sufficiently far from 0.9675 (compared to the spread of your measured values) would be evidence that these coins are not honest double eagles.

In this example the null hypothesis is that the population of double eagles averages 0.9675 ounces and the alternative is that the average differs from 0.9675. Suppose you were to swap these two statements and instead tried to test whether the population mean differs from 0.9675. You cannot test this hypothesis with data because (literally) any set of values would be consistent with it. (If you always obtained 0.9675 in every assay, your measurement procedure itself would be called into question because the results would be too consistent!) There is an inherent, profound, asymmetry between the two hypotheses because one of them makes a specific quantitative prediction about how the data might be distributed but the other does not.

There's another asymmetry in hypothesis testing. In the same situation, you might be interested in assessing whether the population of US double eagles is underweight. A sample average that is sufficiently low would be good evidence of that. The average would have to be substantially lower than 0.9675, though: how much lower is the "critical value" for the test. In this situation you can switch the null and alternate hypotheses. The new null is that the population of US double eagles is overweight. A sample average that is sufficiently high would be good evidence of that. That average would have to be substantially higher than 0.9675, though: how much higher is the critical value for this reversed test. In each case, the set of possible sample averages is partitioned into two parts: those less than the critical value and those greater than the critical value. Because the two critical values are not the same, the partitions differ, too. For instance, a sufficiently low sample average in the first case lets you conclude that the population mean is underweight, but in the second case it is consistent with the null hypothesis that the population mean is not overweight. Notice the distinction between evidence that is consistent with a hypothesis and evidence that falsifies a hypothesis. That asymmetry is inherent in the logic of hypothesis testing.

whuber
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  • To continue with your last example, why can't you say the following: there are 2 critical values, c1 less than 0.9675, c2 greater than 0.9675; if the sample average is less than c1, then accept H0. If the sample average is greater then c2, then accept H1. If the sample average is between c1 and c2, we decide nothing? – ericj Nov 25 '16 at 08:25
  • @Ericj, there is no such thing as "deciding nothing.". That is not one of the options. – whuber Nov 25 '16 at 10:28
  • I meant: not accept H0. Or as I would say, because I do not understand the asymmetry, and not accept H1. – ericj Nov 25 '16 at 11:40
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I suspect it means that if you perform a test for H1 with null H0 and are not able to reject the null hypothesis, that does not imply that if you performed a test for H0 with H1 as the null that you would be able to reject H1.

The reason is that failing to reject the null hypothesis does not mean that the null hypothesis is true, it could just mean that there isn't enough data to be confident that the null hypothesis is false.

Dikran Marsupial
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  • Yes, this is the obvious way hypothesis testing of this sort is assymetric - the burden of proof is all with the alternative hypothesis. What do you think is meant though by "in the sense that if we change the hypotheses then result will alter" - which seems to be a) so obvious as to not need saying and b) not really relevant to the question of symmetry? – Peter Ellis Jan 30 '12 at 20:28
  • I suspect they meant "exchange the hypotheses", otherwise it is rather vacuous as you suggest. That seems the smallest edit required for the statement to make sense! – Dikran Marsupial Jan 30 '12 at 20:48
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    BTW a nice feature of the Bayes factor, which is symmetric. Exchanging the hypothesis just gives the reciprocal of the Bayes factor. – Dikran Marsupial Jan 30 '12 at 20:50