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I am unsure how to approach this problem:

Consider a random variable, X. Our hypothesis is that X~N(0,1) or X is standard normal distribution.

If we observe that X = 2.2. What is the likelihood of this hypothesis. Would the hypothesis be rejeted if we wanted to maintain a specificity of .95?

Thanks, RH

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    @MichaelHardy I disagree. The question is not clear at all. First, the null is$X$normal; what is the alternative? X is chi-square? My guess was that Rich had in mind something along the lines of Dilip's answer. That's why I wrote my 1st comment. Rich then said that he believes$X=2.2$is an observation on X, which made me think that he is dealing with samples, in which case you need a sample larger than one to get estimates of the mean and the variance of X. This is why I wrote my 2nd comment. Sorry if you think this is obtuse.2012-09-22

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You have specified the null hypothesis that $X$ is a standard normal random variable. The alternative hypothesis is sometimes not specified explicitly, though in simple examples of this kind, the alternative could be that $X$ is a unit-variance normal random variable with mean $\mu \neq 0$. A typical question that needs to be resolved is:

Given that we observed that $X$ has value $\alpha$, is this observation consistent with the null hypothesis?

The idea here is that a standard normal random variable $X$ is quite unlikely to take on large positive or large negative values. With high probability, $X$ lies in the interval $[-3,+3]$. So if we had observed $X = 10$, say, we could quite confidently reject the null hypothesis since the alternative, that the observation came from a distribution with mean $\mu$ closer to $10$ looks to a more reasonable assumption. But even in the absence of a specified (or vaguely specified or unspecified) alternative hypothesis, the observation $X=10$ seems not very consistent with the null hypothesis. This observation could occur by chance even when the null hypothesis is true, but it is our fondest hope we hope that we have not been so unlucky when we confidently reject the null hypothesis.

On the other hand, if $X = 0.1$, we would not be inclined to reject the null hypothesis. It is perfectly consistent with $X$ being a standard normal variable. But understand that

not rejecting the null hypothesis is not the same as a whole-hearted embrace or acceptance of the null hypothesis.

All you are saying when you fail to reject the null is that the available evidence is not strong enough to force you into consideration of alternatives. Notice, for example, that the observation $X=0.1$ is also quite consistent with the hypothesis that $X$ is a unit-variance normal random variable with mean $0.00000001$, say, rather than the mean $0$ insisted upon in the null hypothesis.

Now, turning to your specific problem, $P\{|X| > 1.96\} = 0.05$ and so if you observe that the observed value $\alpha$ of $X$ is outside the interval $[-1.96,+1.96]$, you reject the null hypothesis, while if $\alpha \in [-1.96,+1.96]$, you do not reject the null hypothesis. Your confidence level in this choice is $0.95$.

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Hint: With one observation, you should ask if there is at least a 5% chance you would see something this far from zero. Look up your normal distribution table and see if 5% of the samples are this far from the center.