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Population 1,2,4,5,8 · Draw all possible sample of size 2 W.O.R · Sampling distribution of Proportion of even No. · Verify the results

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Question:

A population consists 1, 2, 4, 5, 8. Draw all possible samples of size 2 without replacement from this population.

Verify that the sample mean is an unbiased estimate of the population mean.

Answer:


Samples: \{(1,2),(1,4),(1,5),(1,8),(2,4),(2,5),(2,8),(4,5),(4,8),(5,8)\}


\hat p = (3)/(5) --- proportion of evens

The sample mean is an unbiased estimate of the population mean.

Explanation:

Given


Numbers: 1, 2, 4, 5, 8

Solving (a): All possible samples of 2 (W.O.R)

W.O.R means without replacement

So, we have:


Samples: \{(1,2),(1,4),(1,5),(1,8),(2,4),(2,5),(2,8),(4,5),(4,8),(5,8)\}

Solving (b): The sampling distribution of the proportion of even numbers

This is calculated as:


\hat p = (n(Even))/(Total)

The even samples are:


Even = \{2,4,8\}


n(Even) = 3

So, we have:


\hat p = (3)/(5)

Solving (c): To verify


Samples: \{(1,2),(1,4),(1,5),(1,8),(2,4),(2,5),(2,8),(4,5),(4,8),(5,8)\}

Calculate the mean of each samples


Sample\ means = \{1.5,2.5,3,4.5,3,3.5,5,4.5,6,6.5\}

Calculate the mean of the sample means


\bar x = (1.5 + 2.5 +3 + 4.5 + 4 + 3.5 + 5 + 4.5 + 6 + 6.5)/(10)


\bar x = (40)/(10)


\bar x = 4

Calculate the population mean:


Numbers: 1, 2, 4, 5, 8


\mu = (1 +2+4+5+8)/(5)


\mu = (20)/(5)


\mu = 4


\bar x = \mu = 4

This implies that
\bar x is an unbiased estimate of the
\mu

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