Exercise · ZW-CENTRAL-LIMIT-THEOREM-005 · ★★☆ Intermediate
With \(\mu=100\), \(\sigma=15\) and \(n=225\), what is the approximate value of \(P(98<\bar X_n<102)\)? (\(\Phi(2)\approx0.9772\))
Choose one option.
Show answer
- C. \(0.9545\)
Explanation
The standard error is \(15/15=1\), so the interval is \(\mu\pm2\) standard errors: \(2\Phi(2)-1\approx0.9545\).
Review the concept: Central limit theorem
The central limit theorem states that the average \(\bar X_n\) of \(n\) independent, identically distributed variables \(X\), with mean \(\mu\) and deviation \(\sigma\), once standardized, resembles a normal \(N\).
\[\dfrac{\bar X_n-\mu}{\sigma/\sqrt n}\ \xrightarrow{d}\ N(0,1)\]
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