Answer to Question #331254 in Statistics and Probability for Mery

Question #331254

A population consists of the values (1, 3, 4). If the samples of size 2 will be drawn from the population, find the standard deviation of the sampling distribution of the means

1
Expert's answer
2022-04-21T08:30:30-0400

The population mean:

μ=1+3+43=83.\mu=\cfrac{1+3+4}{3}=\cfrac{8}{3}.


The population variance:

σ2=(xiμ)2P(xi),\sigma^2=\sum(x_i-\mu)^2\cdot P(x_i),

Xμ={183,383,483}=X-\mu=\begin{Bmatrix} 1-\cfrac{8}{3},3-\cfrac{8}{3},4-\cfrac{8}{3} \end{Bmatrix}=

={53,13,43},=\begin{Bmatrix} -\cfrac{5}{3}, \cfrac{1}{3},\cfrac{4}{3} \end{Bmatrix},

σ2=(53)213+(13)213+(43)213==4227=149=1.556.\sigma^2=\bigg(\cfrac{-5}{3}\bigg)^2\cdot \cfrac{1}{3}+\bigg(\cfrac{1}{3}\bigg)^2\cdot \cfrac{1}{3}+\bigg(\cfrac{4}{3}\bigg)^2\cdot \cfrac{1}{3}=\\ =\cfrac{42}{27}=\cfrac{14}{9}=1.556.


The population standard deviation:

σ=1.556=1.247.\sigma=\sqrt{1.556}=1.247.


The standard deviation of the sampling distribution of sample means:

σxˉ=σn=1.2472=0.882.\sigma_{\bar x}=\cfrac{\sigma}{\sqrt n}=\cfrac{1.247}{\sqrt 2}=0.882.

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