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I apologize if this is too simple, but I just can't visualize how to get to the correct answer:

I have a normal distribution of $X$ with $\sigma=5$, given that $P[X<35]=0.015$, find the mean of this distribution.

I can only find it considering the mean is lower than 35, but this is obviously not true (I get to $\mu=24.15$, but the answer is $\mu=45.85$).

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Details depend on what you are using (tables, software). Using the most standard kind of table, we find that the $z$ such that $\Pr(Z\le z)=1-0.015$ is given approximately by $z=2.17$. (Here $Z$ is standard normal.)

So we have probability $0.015$ in the right tail above $z=2.17$, and therefore probability $0.015$ in the left tail below $z=-2.17$.

Thus $35$ is $2.17$ standard deviation units below the mean. It follows that the mean is $35+5(2.17)$.

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  • $\begingroup$ This is a totally new point of view!! Thank you for sharing this!!!! $\endgroup$ – Vinícius Lopes Simões Jun 12 '16 at 23:54
  • $\begingroup$ @ViníciusLopesSimões: You are welcome. One has to have a picture of the density function of a normal in mind, but then the "geometric" language can be very helpful, as is thinking in terms of units of standard deviation. $\endgroup$ – André Nicolas Jun 13 '16 at 0:01

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