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I have been reading up on probability a bit. Most of the time the type of distribution used is provided but sometimes the question seems rather ambiguous. In such cases how do you go about determining which distribution to use?

A soft drink vending machine is set in such a way that the amount of drink dispensed is a random variable with a mean of 200 millilitres and a standard deviation of 15 millilitres. What is the probability that the average amount dispensed in a random sample of size 36 is at least 205 millilitres?

My intuition says I should go with normal distribution.

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  • $\begingroup$ In the real world you just have to figure it out. In elementary probability it will usually either be directly specified ("use an exponential distribution with this parameter"), indirectly specified ("18 flips of a fair coin"), or normal. In particular, here they tell you the sample size is $36$, which suggests normality as well. $\endgroup$ – Ian Mar 19 '17 at 19:16
  • $\begingroup$ @Ian In elementary probability students are also taught the uniform, binomial, Poisson, etc. $\endgroup$ – Anna SdTC Mar 19 '17 at 19:21
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    $\begingroup$ @AnnaSdTC Yes, but when they are relevant it is usually either directly or indirectly specified. Of course, "indirectly specified" is a bit subjective; for example, is it obvious that "waiting times of mean 5 minutes" should be exponentially distributed? $\endgroup$ – Ian Mar 19 '17 at 19:22
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In general, there is no way of knowing what distribution to use apart from the context of the problem (arrivals are usually Poisson, time to failure is usually exponential, etc.).

Now, notice that you are interested in the distribution of the SAMPLE MEAN of a sample of size 36. By the Central Limit Theorem, no matter what the distribution of one shot of the machine is, the sample mean will be distributed as a normal (approximately), if $n$ is "large enough".

If the individual machine shots (observations) have mean $\mu$ and standard deviation $\sigma$, then the sample mean of a sample of size $n$ will have mean $\mu$ and standard deviation $\sigma/\sqrt{n}$.

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