Answer to Question #263316 in Statistics and Probability for Ruhi

Question #263316

Random sample of n=65 measurements is obtained from a population with µ=20 and σ²=400. Describe the sampling distribution for the sample means by computing µ˜x and σ2˜x. Consider a sampling without replacement.


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Expert's answer
2021-11-14T16:40:22-0500

The Central Limit Theorem

Let X1,X2,...,XnX_1, X_2, . . . , X_n be a random sample from a distribution with mean μ\mu and variance σ2.\sigma^2. Then if nn is sufficiently large, Xˉ\bar{X} has approximately a normal distribution with μXˉ=μ\mu_{\bar{X}}=\mu and σXˉ2=σ2/n.\sigma_{\bar{X}}^2=\sigma^2/n. The larger the value of n,n, the better the approximation.

If n>30,n > 30, the Central Limit Theorem can be used.


We have n=65>30.n=65>30. Then the Central Limit Theorem can be used


μXˉ=μ=20\mu_{\bar{X}}=\mu=20

σXˉ2=σ2/n=400/65\sigma^2_{\bar{X}}=\sigma^2/n=400/65

The sample mean XˉN(20,400/65)\bar{X}\sim N(20, 400/65)


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