Exercise · ZW-CENTRAL-LIMIT-THEOREM-008 · ★★☆ Intermediate
400 independent variables with mean 0 and standard deviation 2 are added. What is the approximate value of \(P(S\le40)\)?
Choose one option.
Show answer
- B. \(0.8413\)
Explanation
The standard deviation of \(S\) is \(2\sqrt{400}=40\), so \(z=40/40=1\) and \(P\approx0.8413\).
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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