Answer to Question #312084 in Statistics and Probability for Cali

Question

Suppose X₁,X₂,...,X₂₅ is a random sample from P (λ = 320). Use the central limit theorem to approximate P(X̄ >542)

Answer 1

We have

$EX_1=\lambda =320\\DX_1=\lambda =320\\Z=\sqrt{n}\frac{\bar{X}-EX_1}{\sqrt{DX_1}}=5\frac{\bar{X}-320}{\sqrt{320}}\rightarrow N\left( 0,1 \right) \\P\left( \bar{X}>542 \right) =P\left( 5\frac{\bar{X}-320}{\sqrt{320}}>5\frac{542-320}{\sqrt{320}} \right) =\\=P\left( Z>62.0509 \right) \approx 1-\varPhi \left( 62.0509 \right) =\\=\varPhi \left( -62.0509 \right) =1.2\cdot 10^{-837}$