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Consider the set {1, 2, 3, 4}.

a. Make a list of all samples of size 2 that can be drawn from this set of integers.
b. Construct the sampling distribution of sample means for samples of size 2 selected from this set.
c. Provide the distribution both in the form of a table and histogram.

1 Answer

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Answer:

(a)


List = \{(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(2,4),(3,1),(3,2),(3,3),(3,4),\\(4,1),(4,2),(4,3).(4,4)\}

(b) Sampling Distribution (Table)


\begin{array}{cccccccc}{\bar x} & {1} & {1.5} & {2} & {2.5} & {3} & {3.5} & {4} & {Pr}& {(1)/(16)} & {(1)/(8)} & {(3)/(16)} & {(1)/(4)} & {(3)/(16)} & {(1)/(8)} & {(1)/(16)} \ \end{array}

(b) Sampling Distribution (Histogram)

See attachment

Explanation:

Given


Set = \{1,2,3,4\}


n =4

Solving (a): A list of sample size 2

We have:


n =4


r = 2 --- the sample size

First, we calculate the number of list using permutation (orders matter)


n(List) = n^r

So, we have:


n(List) = 4^2


n(List) = 16

And the list is:


List = \{(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(2,4),(3,1),(3,2),(3,3),(3,4),\\(4,1),(4,2),(4,3).(4,4)\}

Solving (b): Sample distribution of sample means of (a)

First, calculate the mean of each set using:


Mean = (Sum)/(2)

So, we have:


(1,1) \to (1+1)/(2) \to 1
(1,2) \to (1+2)/(2) \to 1.5
(1,3) \to (1+3)/(2) \to 2
(1,4) \to (1+4)/(2) \to 2.5


(2,1) \to (2+1)/(2) \to 1.5
(2,2) \to (2+2)/(2) \to 2
(2,3) \to (2+3)/(2) \to 2.5
(2,4) \to (2+4)/(2) \to 3


(3,1) \to (3+1)/(2) \to 2
(3,2) \to (3+2)/(2) \to 2.5
(3,3) \to (3+3)/(2) \to 3
(3,4) \to (3+4)/(2) \to 3.5


(4,1) \to (4+1)/(2) \to 2.5
(4,2) \to (4+2)/(2) \to 3
(4,3) \to (4+3)/(2) \to 3.5
(4,4) \to (4+4)/(2) \to 4

Write out the sample means (sorted)


\bar x =\{1,1.5,1.5,2,2,2,2.5,2.5,2.5,2.5,3,3,3,3.5,3.5,4\}

Construct a frequency table


\begin{array}{cc}{\bar x} & {f} & {1} & {1} & {1.5} & {2} & {2} & {3} & {2.5} & {4} & {3} & {3} & {3.5} &{2} & {4} & {1} & Total & 16\ \end{array}

Construct the sampling distribution where the probability is calculated using:
(f)/(Total)

So, we have:


\begin{array}{cccccccc}{\bar x} & {1} & {1.5} & {2} & {2.5} & {3} & {3.5} & {4} & {Pr}& {(1)/(16)} & {(1)/(8)} & {(3)/(16)} & {(1)/(4)} & {(3)/(16)} & {(1)/(8)} & {(1)/(16)} \ \end{array}

Consider the set {1, 2, 3, 4}. a. Make a list of all samples of size 2 that can be-example-1
User Claes Wikner
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