Using the number 1,2,3,4,5 and 6 as the element of the populations A.construct the sampling distribution of the sample mean and the probability histogram

The number of possible samples is $\binom{N}{n}=\binom{6}{3}=20$.

The possible samples with their means are given below.

The sample means are derived from the formula,

$\bar x_i={\sum(x_i)\over3}$

Sample mean

(1,2,3) 2

(1,2,4) 2.33

(1,2,5) 2.67

(1,2,6) 3

(1,3,4) 2.67

(1,3,5) 3

(1,3,6) 3.33

(1,4,5) 3.33

(1,4,6) 3.67

(1,5,6) 4

(2,3,4) 3

(2,3,5) 3.33

(2,3,6) 3.67

(2,4,5) 3.67

(2,4,6) 4

(2,5,6) 4.33

(3,4,5) 4

(3,4,6) 4.33

(3,5,6) 4.67

(4,5,6) 5

$a)$

The sampling distribution is,

$\bar x_i$ 2 2.33 2.67 3 3.33 3.67 4 4.33 4.67 5

$p(\bar x _i)$ ${0.05}$ ${0.05}$ $0.1$ $0.15$ $0.15$ $0.15$ $0.15$ $0.1$ $0.05$ $0.05$

$b)$

The probability Histogram is given below.