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I have two data sets $M$ and $N$ that are equally large. They have different averages and medians and I want to prove that, with 5% significance, the values don't differ solely out of chance.

I'm trying to understand what the two-tailed paired t-test really tells me. I've performed such a test using Google Sheets built-in function TTEST() and it returns the value 0.03. Does this mean that it's a 3% chance that the values differed as they did solely out of chance and that I've successfully proved (at a 5% significance level) that they in fact differ?

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    $\begingroup$ Yes. This is the so-called $p$-value. And if you get $0.03<0.05$ then the hypothesis that means are equal should be rejected at 5% significance level since there is only 3% of chances that data sets with the same means show given difference of avarages. $\endgroup$ – NCh Apr 12 '17 at 15:22
  • $\begingroup$ That is an excellent answer, thank you so much. If you want to post it as a separate answer I'll gladly mark it as accepted. $\endgroup$ – Nyfiken Gul Apr 12 '17 at 15:28
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Yes. This is the so-called $p$-value. And if you get $0.03<0.05$ then the hypothesis that the means are equal should be rejected at $5\%$ significance level since there is only $3\%$ of chances that data sets with the same means show such difference of averages.

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