Gillbert got a new job in a shoe store. The number of pairs of shoes he was able to sell for three days are 2, 4, and 5. Assume that samples of size 2 are randomly selected with replacement from this population of three values. a. List down the 9 different possible samples. b. Find the mean of each sample. c. Find the mean of the sampling distribution of means. d. Identify the probability of each sample. e. Find the population mean. f. Compare the population mean with the mean of the sampling distribution of means.

$a)$

The 9 different possible samples are, (2,2),(2,4),(2,5),(4,4),(4,5),(5,5),(5,4),(5,2),(4,2)

$b)$

The sample means are derived from the formula,

$\bar x_i={\sum x_i\over 2}$

The possible samples are with their corresponding means are:

Sample Mean

(2,2) 2

(2,4) 3

(2,5) 3.5

(4,4) 4

(4,5) 4.5

(5,5) 5

(5,4) 4.5

(5,2) 3.5

(4,2) 3

$c)$

The mean of the sampling distribution of means is the mean of the sample means given as,

$\bar x={\sum\bar x_i\over 9}$ where $\bar x_i$ are the means for the listed samples above and $9$ represent the different possible samples.

Therefore,

$\bar x={2+3+3.5+4+4.5+5+4.5+3.5+3\over9}=3.6666667$

$d)$

The probability distribution for the sample means is,

$\bar x_i$ 2 3 3.5 4 4.5 5

$p(\bar x_i)$ ${1\over9}$ $2\over9$ $2\over9$ $1\over9$ $2\over9$ $1\over9$

$e)$

The population mean $\mu={2+4+5\over3}={11\over3}=3.6666667$

$f)$

Here, $\bar x=\mu=3.6666667$

This affirms the property that the mean of the sampling distribution of the means $(\bar x)$ is the population mean $(\mu)$.