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 probability statement.
Given
\[ 2\overline{X}-1<2\mu<2\overline{X}+1 \]
Subtract \(2\overline{X}\) throughout:
\[ -1<2\mu-2\overline{X}<1 \] \[ -1<2(\mu-\overline{X})<1 \] Dividing by \(2\),
\[ -\frac12<\mu-\overline{X}<\frac12 \] Thus,
\[ |\overline{X}-\mu|<\frac12 \]

Step 2: Use the distribution of \(\overline{X}\).
Since
\[ X_i\sim N(\mu,1), \] we have
\[ \overline{X}\sim N\left(\mu,\frac{1}{n}\right) \] Therefore,
\[ \sqrt{n}(\overline{X}-\mu)\sim N(0,1) \]

Step 3: Standardize the probability.
\[ P\left(|\overline{X}-\mu|<\frac12\right) = P\left(\left|\sqrt{n}(\overline{X}-\mu)\right|<\frac{\sqrt{n}}{2}\right) \] Let
\[ Z=\sqrt{n}(\overline{X}-\mu)\sim N(0,1) \] Then,
\[ P\left(|Z|<\frac{\sqrt{n}}{2}\right)\geq 0.99 \] So,
\[ 2\Phi\left(\frac{\sqrt{n}}{2}\right)-1\geq 0.99 \] \[ \Phi\left(\frac{\sqrt{n}}{2}\right)\geq 0.995 \]

Step 4: Use the given normal table value.
Given
\[ \Phi(2.58)=0.9951 \] So, we need
\[ \frac{\sqrt{n}}{2}\geq 2.58 \] \[ \sqrt{n}\geq 5.16 \] \[ n\geq 26.6256 \] Since \(n\) must be an integer, the minimum value is
\[ n=27 \]

Step 5: Final conclusion.
Hence, the minimum value of \(n\) is
\[ \boxed{27} \]