Question

Let \(X_1,X_2,\ldots,X_n\) be a random sample of size \(n\;(n>1)\) from a \(N(\mu,1)\) distribution, where \(\mu\in\mathbb{R}\) is an unknown parameter. If \(\overline{X}=\frac{1}{n}\sum_{i=1}^{n}X_i\), then the minimum value of \(n\) such that

Hint:

For normal sample mean problems, use \(\overline{X}\sim N\left(\mu,\frac{\sigma^2}{n}\right)\) and standardize using \(Z=\frac{\overline{X}-\mu}{\sigma/\sqrt{n}}\).

Answer 1

Correct Answer: 27

Step 1: Simplify the event.
$2\bar X-1<2\mu<2\bar X+1$ reduces to $|\bar X-\mu|<\frac12$.

Step 2: Distribution of $\bar X$.
Since $X_i\sim N(\mu,1)$, $\sqrt n(\bar X-\mu)\sim N(0,1)$.

Step 3: Standardise the requirement.
We need $P(|Z|<\frac{\sqrt n}2)\ge0.99$, i.e. $\Phi(\frac{\sqrt n}2)\ge0.995$.

Step 4: Use the cutoff.
Since $\Phi(2.58)=0.995$, we need $\frac{\sqrt n}2\ge2.58$, so $\sqrt n\ge5.16$, $n\ge26.63$.

Step 5: Smallest integer.
\[ \boxed{27} \]