Answer to Question #193710 in Statistics and Probability for Raizel Villasenior

Question #193710

A random sample of n=70 measurements is obtained from a population with a mean 200 and standard deviation of 50.describe the sampling for the sample means by computing

1
Expert's answer
2021-05-18T12:24:05-0400

The Central Limit Theorem

For samples of size 30 or more, the sample mean is approximately normally distributed, with mean μXˉ=μ\mu_{\bar{X}}=\mu and standard deviation σXˉ=σ/n,\sigma_{\bar{X}}=\sigma/\sqrt{n}, where nn  is the sample size.

The larger the sample size, the better the approximation.


XˉN(μ,σ2/n)\bar{X}\sim N(\mu, \sigma^2/n)

Given μ=200,σ=50,n=70.\mu=200, \sigma=50, n=70.


The distribution of the sample mean is normal with mean μXˉ=μ=200\mu_{\bar{X}}=\mu=200 and standard deviation σXˉ=σ/n=50/70\sigma_{\bar{X}}=\sigma/\sqrt{n}=50/\sqrt{70}


XˉN(200,502/70)\bar{X}\sim N(200, 50^2/70)


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