Let X 1 , , X n be an i i d sample from a population with unknown mean and standard deviation We take the sample mean X (X 1 X n ) n as an estimate for (a) According to Chebyshevs inequality, how large should the sample size n be so that with probability 0 99 the error X is less than 2 standard dev...

a Set 099 PX 2 1 PX 2 1 2 n4 2 1 14n Thus 14n 001 or n 25 ...

Let X 1 , . . . , X n be an i.i.d. sample from a population

Question:

Let X₁, . . . , X_n be an i.i.d. sample from a population with unknown mean μ and standard deviation σ. We take the sample mean X̅ = (X₁ +· · ·+X_n)/n as an estimate for μ.
(a) According to Chebyshev’s inequality, how large should the sample size n be so that with probability 0.99 the error |X̅ − μ| is less than 2 standard deviations?

(b) According to the central limit theorem, how large should n be so that with probability 0.99 the error |X̅ − μ| is less than 2 standard deviations?

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