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Book question: The concentration of salt in a cell, $X$, can be modelled by a normal distribution with mean $\mu$ and standard deviation $2$%. Find the value of $\alpha$ such that $P(\mu-\alpha< X < \mu+\alpha) = 0.9 $.

What does it mean, standard deviation $2$%? I assumed it meant $X \sim N(\mu, 0.02^2)$, and I then did:

$P(X<\mu-\alpha) = 0.05$. So the $Z$-value is: $z=\frac{x - \mu}{\sigma} = \frac{(\mu-\alpha) - \mu}{\sigma} = \frac{-\alpha}{0.02} = -50\alpha$, where $Z \sim N(0,1)$ is the standard normal deviation. And using the inverse normal distribution function, I get: $-50-\alpha = -1.6448... \implies \alpha =0.0329\ (3sf)$.

But the answer in the back is $3.29$.

Ah but this is $100$ times more than my answer, so perhaps the % sign is just a typo in the question? Standard deviation $2$ instead of $0.02$ would get me the correct answer I think.

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The textbook answer is correct. There's just a misunderstanding because the concentration is always misured in %. Thus the result is 3.29 (%, obviously).

Your error is due to the fact that you transformed in % a number that originally was already expressed in %

The correct solution was

$$\mathbb{P}[Z<\frac{a}{2}]=0.95 \rightarrow a\approx3.29$$

Anyway also your reasoning is correct. It is enough that you realize that your result is a % espressed with respect to 1 and not to 100

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  • $\begingroup$ But then I would have thought they would write, "$\mu$ % and standard deviation $2$ %" in the question, which they didn't. Maybe just an English language thing... $\endgroup$ Commented Nov 11, 2020 at 17:46
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It means the standard deviation is $0.02 \mu$. So you should be looking at $N(\mu, (0.02 \mu)^2)$.

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  • $\begingroup$ I was thinking that, but then I thought that would give me $\alpha$ in terms of $\mu$. Yeah, surely $\alpha$ depends on $\mu$ there... $\endgroup$ Commented Nov 2, 2020 at 23:37
  • $\begingroup$ So the book answer is wrong? $\endgroup$ Commented Nov 5, 2020 at 16:13

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