Exercise 1.2. Suppose that a population consists of 5 units given by: 10, 15, 20

Question

Exercise 1.2. Suppose that a population consists of 5 units given by: 10, 15, 20, 25, and 30. Select all possible samples of 3 units using SRSWR and SRSWOR sampling a) Show that the sample mean is an unbiased estimator of population mean in each case. (b) The sample variance is unbiased estimator of the population variance under SRSWR sampling, and for population mean squared error under SRSWOR sampling ( c) Also plot the sampling distribution of sample mean and sample variance in each situation. d) Find the variance of sample mean under SRSWOR sampling using the definition of variance? Show all steps. (e) Also compute( )s and comment on it.

Explanation / Answer

Rcode

> pop.mean=mean(pop)
> pop.mean
[1] 20
> x1=sample(pop,3,replace =FALSE)
> x1
[1] 15 30 25
> x2=sample(pop,3,replace =TRUE)
> x2
[1] 25 30 15
> x1bar=mean(x1)
> x1bar
[1] 23.33333
> x2bar=mean(x2)
> x2bar
[1] 23.33333
> pop.var=var(pop)
> pop.var
[1] 62.5
> sam.var=var(x2)
> sam.var
[1] 58.33333
> mse=mean( (var(x1) - pop.var)^2, na.rm = TRUE)
> mse
[1] 17.36111

for part c,d,e please post same question again

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