$P(\mu \in (\bar{X} - 1/2, \bar{X} + 1/2))$ for Gauss distribution

Question

Let $\bar{X}$ be the mean of a random sample of size $n$ from $N(\mu=u, \sigma^2 = 10)$ Find $n$ so that the probability is a approximately $0.954$ that the random interval $(\bar{X} - 1/2, \bar{X} + 1/2)$ includes $\mu$.

I began with $P(\bar{X} - 1/2 < u < \bar{X} + 1/2) = 0.954$ then I'm not sure as to where to go from there.

Answer 1

Some hints:

• $\bar X = \frac{1}{n}\sum_{k=1}^nX_i$ with (most probably you forgot to mention) $X_i \stackrel{i.i.d}{\sim}N(u,10)$
• $\Rightarrow \bar X \sim N(u,10/n)$
• $\Rightarrow Z = \frac{\bar X - u}{\sqrt{10/n}} \sim N(0,1)$
• $\bar X - \frac{1}{2} < u < \bar X + \frac{1}{2} \Leftrightarrow |\bar X - u | < \frac{1}{2}$
• $P(|\bar X - u |< \frac{1}{2}) \Leftrightarrow P(|Z| < \frac{\sqrt{n}}{2\sqrt{10}})$
• $P(|Z| < 2) \approx 0.9545$
• $\Rightarrow 2 = \frac{\sqrt{n}}{2\sqrt{10}}$