# For a normally distributed random variable, find a value from given tail probability

### Problem

Let $X \sim N(65,20)$. Find correct to $3$ Decimal Place the value of $x$ such that $Pr(X>x) = 0.43$.

### Progress

I've gotten to $\frac {x-65}{2(5)^{1/2}} =0.1764$ and hence $x = 67.789$? I'm not 100% sure where I've gone wrong with my working out?

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What have you tried? Which tools are you allowed to use? –  Stefan Hansen Feb 6 '13 at 9:34
Down-voted because the OP didn't indicate any interaction with their question. –  Rustyn Feb 6 '13 at 9:47
@StefanHansen i've gotten to $\frac {x-65}{2(5)^{1/2}} =0.1764$ and hence $x = 67.789$ ?? –  Sam Feb 6 '13 at 10:05
Note that $P(X>x)=0.43$ if and only $P(X\leq x)=0.57$. So you need to find the $57\%$ quantile/percentile of an $\mathcal{N}(65,20)$ distribution. See e.g. this explanation. –  Stefan Hansen Feb 6 '13 at 10:09
im not 100% sure where i've gone wrong with my working out? –  Sam Feb 6 '13 at 10:15

Making my comment as an answer. The equation $$\frac{x-65}{\sqrt{20}}=0.1764$$ you proposed is correct. However, the solution to this is $$0.1764\cdot \sqrt{20}+65\approx 65.789$$ and not $67.789$.