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I have read a lot of explanations about the t-test and how it could be used. I read one statement about it and wasn't able to comprehend it.

It tests whether the observed mean of a sample is likely to be just a random fluctuation around the mean of another sample. It can be transformed into a probability of that event.

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Let's say you have two groups, males and females, and you are comparing mean exam scores. The mean exam score for males is 4 and the mean exam score for females is 4.2. The mean for females is greater than the mean for males but it's possible that if you were to take another samples of females, the mean would be lower (e.g 3.9) and perhaps in another sample it would be the same and so on and so forth. This is what they are referring to as random fluctuation and you need to be able to discern your effect from this random fluctuation. The t test tests for this by computing a t statistic which is essentially effect over noise/error. If the effect (the difference between means) becomes greater than the noise (essentially the standard error), then you'll have a larger t statistic. Using a t distribution, you can obtain the probability value (p-value) by "transforming" the t statistic to a p-value.

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  • $\begingroup$ Good explanation, however, are you referring p-value to the probability of random fluctuation? I would certainly accept the answer if you could clear out some facts $\endgroup$ Apr 13, 2020 at 14:29
  • $\begingroup$ Yes. Essentially, the p value tells you the probability of having received your result simply by chance (the sampling variation I mentioned). For instance, if the p value is 0.04, that means that there is a 4 percent probability that the difference you had between your means were just due to chance. Because that is a low probability (relative to the 0.05 threshold), then we can conclude that it's very likely that our difference was genuine. This article explains p-values well. blog.minitab.com/blog/adventures-in-statistics-2/… $\endgroup$
    – Neal
    Apr 13, 2020 at 14:32
  • $\begingroup$ Oh by random fluctuation you are referring to the probability of observing the difference due to chance? But I don't get one thing that hereby random fluctuation don't we mean the fluctuation between the mean of two distributions? I dont get the gist of it $\endgroup$ Apr 13, 2020 at 14:37

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