Question

Let \(X_1,X_2,\ldots\) be a sequence of independent and identically distributed random variables with mean \(4\) and variance \(9\). If \(T_n=\frac{1}{n}\sum_{i=1}^{n}X_i\), then

Hint:

For sample mean problems, use the Central Limit Theorem: \[ \frac{\sqrt{n}(\bar{X}-\mu)}{\sigma}\xrightarrow{d}N(0,1). \]

Answer 1

Correct Answer: 0.82

Step 1: Centre the inequality.
With mean $4$, variance $9$, $\sigma=3$. The event $4\sqrt n-3<\sqrt n T_n<6+4\sqrt n$ becomes $-3<\sqrt n(T_n-4)<6$ after subtracting $4\sqrt n$.

Step 2: Scale by $\sigma$.
Dividing by $3$ gives $-1<\frac{\sqrt n(T_n-4)}3<2$.

Step 3: Apply the CLT.
The middle quantity tends to $N(0,1)$, so the limit is $P(-1<Z<2)$.

Step 4: Use table values.
$\Phi(2)+\Phi(1)-1=0.9772+0.8413-1=0.8185$.

Step 5: Round.
\[ \boxed{0.82} \]