Exercise · ZW-CENTRAL-LIMIT-THEOREM-002 · ★☆☆ Basic
With mean 50, standard deviation 10 and \(n=100\), what is the approximate value of \(P(\bar X_n\le51)\)? (\(\Phi(1)\approx0.8413\))
Choose one option.
Show answer
- A. \(0.8413\)
Explanation
Standardize: \(z=(51-50)/1=1\), and \(P(Z\le1)\approx0.8413\).
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 \(N\).
\[\dfrac{\bar X_n-\mu}{\sigma/\sqrt n}\ \xrightarrow{d}\ N(0,1)\]
More practice problems: central limit theorem
- ZW-CENTRAL-LIMIT-THEOREM-001 The X_i are independent with mean 50 and standard deviation 10. What is the standard deviation of b…
- ZW-CENTRAL-LIMIT-THEOREM-003 36 independent variables, each with mean 2 and variance 9, are added. What is the mean of the sum S?
- ZW-CENTRAL-LIMIT-THEOREM-004 With the same data (36 variables, mean 2, variance 9), what is the variance of the sum S?
- ZW-CENTRAL-LIMIT-THEOREM-005 With μ=100, σ=15 and n=225, what is the approximate value of P(98<bar X_n<102)? (Phi(2)approx0.9772)
- ZW-CENTRAL-LIMIT-THEOREM-006 With μ=50, σ=8 and n=64, what is Z=(bar X_n-μ)/(σ/√ n) if bar X_n=52?
- ZW-CENTRAL-LIMIT-THEOREM-007 What does the central limit theorem require about the distribution of each X_i?
- ZW-CENTRAL-LIMIT-THEOREM-008 400 independent variables with mean 0 and standard deviation 2 are added. What is the approximate v…
- ZW-CENTRAL-LIMIT-THEOREM-009 If the sample size is multiplied by four, what happens to the standard error σ/√ n?
← ZW-CENTRAL-LIMIT-THEOREM-001 ZW-CENTRAL-LIMIT-THEOREM-003 →