Answer:
(a)
[tex]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)\}[/tex]
(b) Sampling Distribution (Table)
[tex]\begin{array}{cccccccc}{\bar x} & {1} & {1.5} & {2} & {2.5} & {3} & {3.5} & {4} & {Pr}& {\frac{1}{16}} & {\frac{1}{8}} & {\frac{3}{16}} & {\frac{1}{4}} & {\frac{3}{16}} & {\frac{1}{8}} & {\frac{1}{16}} \ \end{array}[/tex]
(b) Sampling Distribution (Histogram)
See attachment
Step-by-step explanation:
Given
[tex]Set = \{1,2,3,4\}[/tex]
[tex]n =4[/tex]
Solving (a): A list of sample size 2
We have:
[tex]n =4[/tex]
[tex]r = 2[/tex] --- the sample size
First, we calculate the number of list using permutation (orders matter)
[tex]n(List) = n^r[/tex]
So, we have:
[tex]n(List) = 4^2[/tex]
[tex]n(List) = 16[/tex]
And the list is:
[tex]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)\}[/tex]
Solving (b): Sample distribution of sample means of (a)
First, calculate the mean of each set using:
[tex]Mean = \frac{Sum}{2}[/tex]
So, we have:
[tex](1,1) \to \frac{1+1}{2} \to 1[/tex] [tex](1,2) \to \frac{1+2}{2} \to 1.5[/tex] [tex](1,3) \to \frac{1+3}{2} \to 2[/tex] [tex](1,4) \to \frac{1+4}{2} \to 2.5[/tex]
[tex](2,1) \to \frac{2+1}{2} \to 1.5[/tex] [tex](2,2) \to \frac{2+2}{2} \to 2[/tex] [tex](2,3) \to \frac{2+3}{2} \to 2.5[/tex] [tex](2,4) \to \frac{2+4}{2} \to 3[/tex]
[tex](3,1) \to \frac{3+1}{2} \to 2[/tex] [tex](3,2) \to \frac{3+2}{2} \to 2.5[/tex] [tex](3,3) \to \frac{3+3}{2} \to 3[/tex] [tex](3,4) \to \frac{3+4}{2} \to 3.5[/tex]
[tex](4,1) \to \frac{4+1}{2} \to 2.5[/tex] [tex](4,2) \to \frac{4+2}{2} \to 3[/tex] [tex](4,3) \to \frac{4+3}{2} \to 3.5[/tex] [tex](4,4) \to \frac{4+4}{2} \to 4[/tex]
Write out the sample means (sorted)
[tex]\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\}[/tex]
Construct a frequency table
[tex]\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}[/tex]
Construct the sampling distribution where the probability is calculated using: [tex]\frac{f}{Total}[/tex]
So, we have:
[tex]\begin{array}{cccccccc}{\bar x} & {1} & {1.5} & {2} & {2.5} & {3} & {3.5} & {4} & {Pr}& {\frac{1}{16}} & {\frac{1}{8}} & {\frac{3}{16}} & {\frac{1}{4}} & {\frac{3}{16}} & {\frac{1}{8}} & {\frac{1}{16}} \ \end{array}[/tex]
