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I was recently at a statistics competition and a question came up as follows:

They drew a normal distribution with $\mu=7$ and the area between the values $7.75$ and $8.25$ equal to $0.12$. No other information was provided.

In order to solve the rest of the question you had to figure out the standard deviation. In the solutions guide they said 'Well if you look at it its probably $\sigma=1.1$' Evidently, if you do pnorm(7.75,8.25,7,1.1) you get 0.1179... which is definitely not 0.12.

I later solved this question using wolfram alpha and the integral form of the normal CDF function (and got $\sigma=0.904...$), but figured there has to be a better way.

NOTE: This will be used on a TI-NSpire CAS

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This may have been a simple trial and error kind of task, since they gave you a very nice round number. Anyway, their solution is not wrong, both of your solutions are correct (to a degree of acuracy). If you plot the area given a range of values for $\sigma$ you will see why.

enter image description here

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I do not see a resolution without the call to the Normal cdf, either on a table or via a computer code: the equation determining $\sigma$ being $$\mathbb{P}_{\sigma,\mu}(7.75\le X\le 8.25)=\Phi(1.25/\sigma)-\Phi(0.75/\sigma)=0.12$$ the value $\sigma=1.1$ brings the coverage pretty close to 0.12, since it is $0.1197748$. (I do not understand the connection in the title with the infinite lower bound.)

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