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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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I don't understand your question. What do you mean by $X=2.2$? $X$ is a continuous random variable and hence the prob that takes on a particular value is zero...perhaps I am missing your point in the question...could you clarify it please? –  Cristian Sep 22 '12 at 1:51
    
I think the question is terribly worded. I believe that X = 2.2 means that an observation on X was recorded to be 2.2. –  Rich H Sep 22 '12 at 1:57
    
Ok. But even so, what do you plan to do with a single observation? –  Cristian Sep 22 '12 at 2:01
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You should try to explain better your problem! In hypothesis testing we usually have two hypothesis, the null and the alternative. Then you can compute things such as the likelihood ratio. But you did not give any alternative hypothesis, so we cannot do that. This leaves us with very little to work on, so pleas explain better. –  kjetil b halvorsen Sep 22 '12 at 3:24
    
I think there are several imperfections in the way the question is phrased, but I completely disagree with the comments from Cristian and Rich H. That particular detail was clearly expressed. And I find Cristian's second comment somewhat obtuse. –  Michael Hardy Sep 22 '12 at 3:59

2 Answers 2

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.

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