Exercise · ZW-CENTRAL-LIMIT-THEOREM-007 · ★★☆ Intermediate
What does the central limit theorem require about the distribution of each \(X_i\)?
Choose one option.
Show answer
- D. That it have finite mean and variance
Explanation
Independence, identical distribution and finite variance suffice; the shape of the original distribution does not matter, which is why the theorem is so useful.
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-002 With mean 50, standard deviation 10 and n=100, what is the approximate value of P(bar X_n≤51)? (Phi…
- 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-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?
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