Given the data 6, 9, 12, 15, 16, 19, 22. Construct a sampling distribution of the sample mean without replacement and repetition by selecting 2 samples at a time.

The size of the population is $N=7$. If samples of size $n=2$ are to be selected from this population then, the number of possible samples is $\binom{N}{n}=\binom{7}{2}=21$. The possible samples with their corresponding means are listed below.

The sample means are obtained using the formula, $\bar x_i={\sum x_i\over 2}$

Sample Mean

(6,9) 7.5

(6,12) 9

(6,15) 10.5

(6,16) 11

(6,19) 12.5

(6,22) 14

(9,12) 10.5

(9,15) 12

(9,16) 12.5

(9,19) 14

(9,22) 15.5

(12,15) 13.5

(12,16) 14

(12,19) 15.5

(12,22) 17

(15,16) 15.5

(15,19) 17

(15,22) 18.5

(16,19) 17.5

(16,22) 19

(19,22) 20.5

The sampling distribution.

 $\bar x _i$  7.5 9   10.5  11 12 12.5   13.5  14    15.5  17 17.5    18.5   19 20.5

$p(\bar x_i)$ $1\over21$ $1\over21$ $2\over21$ $1\over21$ $1\over21$ $2\over21$        $1\over21$     $3\over21$ $3\over21$ $2\over21$ $1\over21$ $1\over21$ $1\over21$    $1\over21$