Exercise · ZW-CENTRAL-LIMIT-THEOREM-010 · ★★★ Advanced
With \(\mu=10\), \(\sigma=5\) and \(n=25\), what is the approximate value of \(P(\bar X_n>11)\)? (\(\Phi(1)\approx0.8413\))
Choose one option.
Show answer
- D. \(0.1587\)
Explanation
The standard error is \(5/5=1\), \(z=1\) and \(P(Z>1)=1-0.8413=0.1587\).
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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