How do I determine the likelihood of a single measurement coming from a known distribution? [duplicate]

Question

I have a set of 2000 numbers that form an approximately normal distribution, with a mean of about 0.24, and an SD of 0.06. These form an estimate of the population. I now have a single sample measurement of 0.38 and would like to determine the chance (p value) that this single sample measurement comes from the same population distribution. (Unfortunately, in my work the nature of the "test" means that it is unfortunately theoretically impossible to obtain another measurement).

It must be true that the further that the single sample is from the mean of the population, the more unlikely that single event is. Furthermore, of course one needs to take into account the distribution of the population - had the SD of the population been 6 rather than 0.6 the chance of obtaining the single measurement at 0.38 would be much higher, and hence the p value lower. I am happy to assume for my purposes that the distribution from which the sample has been obtained would have the same SD as the population.

The question is how I calculate the likelihood (p-value) that the observed single sample comes from a distribution with the same mean as the population?

I have thought about carrying out a one sample t-test, testing whether the observed population is different from the single sample value, but that doesn't seem right to me as that test assumes that the single sample value is the mean of the distribution being tested against, rather than a single sample.

What is the best approach to this analysis?

Thanks in advance for any advice and help.

Answer 1

The question for a p-Value of one measurment is an interesting topic. However, you flips the meaning of the p-Value. You are asking for a p-Value that the sample measurement is drawn from the previously measured distribution. But you can only ask for a p-Value that you can reject your starting hypothesis.

For calculating the p-value commonly you are looking for the frequency that experiments result in a result more extreme (or more of the expected) than your actual result, when you repeat the experiment a lot,

Comparing one measurement to the distribution, which is a gaussian means in this sense, you have to get the area of the probability density function below 0.10 and above 0.34. Why I am looking for both sides is as when thinking in extremes lower and higher values are equal extreme.

Finally this gives a p-Value