Answer to Question #258159 in Statistics and Probability for @1111mae

Question #258159
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.
1
Expert's answer
2021-11-15T18:17:47-0500

The Central Limit Theorem

Let "X_1, X_2, . . . , X_n" be a random sample from a distribution with mean "\\mu" and variance "\\sigma^2." Then if "n" is sufficiently large, "\\bar{X}" has approximately a normal distribution with "\\mu_{\\bar{X}}=\\mu" and "\\sigma_{\\bar{X}}^2=\\sigma^2\/n." The larger the value of "n," the better the approximation.

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


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



"\\mu_{\\bar{X}}=\\mu=20""\\sigma^2_{\\bar{X}}=\\sigma^2\/n=400\/65"

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

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