Answer to Question #335010 in Statistics and Probability for Kevin

Question #335010

Random samples with size 5 are drawn from the population containing the values 26, 32, 41, 50, 58, and 63.


Determine the standard error of the sample means

1
Expert's answer
2022-05-04T14:33:03-0400

We have population values 26, 32, 41, 50, 58, and 63 population size N=6 and sample size n=5.

Mean of population (μ)(\mu) = 

26+32+41+50+58+636=45\dfrac{26+32+41+50+58+63}{6}=45


Variance of population 


σ2=Σ(xixˉ)2N=10646\sigma^2=\dfrac{\Sigma(x_i-\bar{x})^2}{N}=\dfrac{1064}{6}


σ=σ2=10646\sigma=\sqrt{\sigma^2}=\sqrt{\dfrac{1064}{6}}

The number of possible samples which can be drawn without replacement is NCn=6C5=6.^{N}C_n=^{6}C_5=6.

noSampleSamplemean (xˉ)126,32,41,50,58207/5226,32,41,50,63212/5326,32,41,63,58220/5426,32,63,50,58229/5526,63,41,50,58238/5663,32,41,50,58244/5\def\arraystretch{1.5} \begin{array}{c:c:c:c:c} no & Sample & Sample \\ & & mean\ (\bar{x}) \\ \hline 1 & 26, 32, 41, 50, 58 & 207/5 \\ \hdashline 2 & 26, 32, 41, 50, 63 & 212/5 \\ \hdashline 3 & 26, 32, 41, 63, 58 & 220/5 \\ \hdashline 4 & 26, 32, 63, 50, 58 & 229/5 \\ \hdashline 5 & 26, 63, 41, 50, 58 & 238/5 \\ \hdashline 6 & 63, 32, 41, 50, 58 & 244/5 \\ \hdashline \end{array}




Xˉf(Xˉ)Xˉf(Xˉ)Xˉ2f(Xˉ)207/51/6207/3042849/150212/51/6212/3044944/150220/51/6220/3048400/150229/51/6229/3052441/150238/51/6238/3056644/150244/51/6244/3059536/150\def\arraystretch{1.5} \begin{array}{c:c:c:c:c} \bar{X} & f(\bar{X}) &\bar{X} f(\bar{X}) &\bar{X}^2 f(\bar{X})\\ \hline 207/5 & 1/6 & 207/30 & 42849/150\\ \hdashline 212/5 & 1/6 & 212/30 & 44944/150\\ \hdashline 220/5 & 1/6 & 220/30 & 48400/150\\ \hdashline 229/5 & 1/6 & 229/30 & 52441/150 \\ \hdashline 238/5 & 1/6 & 238/30 & 56644/150\\ \hdashline 244/5 & 1/6 & 244/30 & 59536/150 \\ \hdashline \end{array}


Mean of sampling distribution 

μXˉ=E(Xˉ)=Xˉif(Xˉi)=45=μ\mu_{\bar{X}}=E(\bar{X})=\sum\bar{X}_if(\bar{X}_i)=45=\mu


The variance of sampling distribution 

Var(Xˉ)=σXˉ2=Xˉi2f(Xˉi)[Xˉif(Xˉi)]2Var(\bar{X})=\sigma^2_{\bar{X}}=\sum\bar{X}_i^2f(\bar{X}_i)-\big[\sum\bar{X}_if(\bar{X}_i)\big]^2=304814150(45)2=1064150=σ2n(NnN1)=\dfrac{304814}{150}-(45)^2=\dfrac{1064}{150}= \dfrac{\sigma^2}{n}(\dfrac{N-n}{N-1})

The standard error of the sample means is

σXˉ=10641502.6633\sigma_{\bar{X}}=\sqrt{\dfrac{1064}{150}}\approx2.6633


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment