Question

Consider the population described by the probability distribution shown below.

The random variable x is observed twice. If these observations are independent, verify that the different samples of size 2 and their probabilities are as shown below.

a. Find the sampling distribution of the sample mean$\overline{\mathbf{x}}$.

b. Construct a probability histogram for the sampling distribution of$\overline{\mathbf{x}}$.

c. What is the probability that$\overline{\mathbf{x}}$is 4.5 or larger?

d. Would you expect to observe a value of$\overline{\mathbf{x}}$equal to 4.5 or larger? Explain.

Short answer

a.

Mean	Probability
1	0.04
1.5	0.12
2	0.17
2.5	0.20
3	0.20
3.5	0.14
4	0.08
4.5	0.04
5	0.01

b.

c. The required answer is 0.05.

d. The value of$\overline{\mathbf{x}}$isequal to 4.5.

Step-by-step solution

Calculation of the value of the mean

a.

The meanof the respective samples has been calculated by summing up the numbers and then dividing the same by 2.The final values are shown below.

Sample	Mean
1,1	1
1,2	1.5
1,3	2
1,4	2.5
1,5	3
2,1	1.5
2,2	2
2,3	2.5
2,4	3
2,5	3.5
3,1	2
3,2	2.5
3,3	3
3,4	3.5
3,5	4
4,1	2.5
4,2	3
4,3	3.5
4,4	4
4,5	4.5
5,1	3
5,2	3.5
5,3	4
5,4	4.5
5,5	5

Calculation of the probabilities

The respective probabilities of the samples are added to get the final probabilities of the mean values, as shown below.

Mean	Probability
1	0.04
1.5	$\mathbf{0}\mathbf{.}\mathbf{06}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{06}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{12}$
2	$\mathbf{0}\mathbf{.}\mathbf{04}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{09}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{04}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{17}$
2.5	$\mathbf{0}\mathbf{.}\mathbf{04}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{06}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{06}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{04}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{20}$
3	$\mathbf{0}\mathbf{.}\mathbf{02}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{06}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{06}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{04}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{20}$
3.5	$\mathbf{0}\mathbf{.}\mathbf{03}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{04}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{04}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{06}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{02}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{20}$
4	$\mathbf{0}\mathbf{.}\mathbf{02}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{04}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{04}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{03}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{14}$
4.5	$\mathbf{0}\mathbf{.}\mathbf{02}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{02}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{04}$
5	0.01

The probabilities of the respective samples are greater than 0 but less than 1.

List of the probabilities of the values of the means

b.

The means of the respective samples have been calculated by summing up the numbers and then dividing the same by 2.The final values are shown below.

Mean	Probability
1	0.04
1.5	0.12
2	0.17
2.5	0.20
3	0.20
3.5	0.14
4	0.08
4.5	0.04
5	0.01

Elucidation of the graph

The graph contains probabilities on the y-axis and the values of the means of x from 1 to 5 on the x-axis.

From the graph, it can be deduced that 2.5 and 3 show the highest probability, which is 0.20.

List of the probabilities of the values of the means

c.

The list of all the probabilities of the mean values is shown below.

Mean	Probability
1	0.04
1.5	0.12
2	0.17
2.5	0.20
3	0.20
3.5	0.14
4	0.08
4.5	0.04
5	0.01

Computation of the probabilities

The calculation of the probability of $\overline{\mathbf{x}}$to be 4.5 and above is shown below.

localid="1658118837019" $$\begin{gathered} \mathit{P}\mathbf{(}\overline{\mathbf{x}}\mathbf{\ge }\mathbf{4}\mathbf{.}\mathbf{5}\mathbf{)}\mathbf{=}\mathit{P}\mathbf{(}\overline{\mathbf{x}}\mathbf{=}\mathbf{5}\mathbf{)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{}\mathbf{0}\mathbf{.}\mathbf{04}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{01} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{5} \end{gathered}$$

The final value of the probability of$\overline{\mathbf{x}}$to be 4.5 and above is 0.05.

Determination of the probabilities of the means 

d.

From Part (a), it has been found that the probability of $\overline{\mathbf{x}}$ is 0.04 when it is equal to 4.5, and when it is above 4.5, the probability is 0.01. Therefore, the probability of being equal to 4.5 is larger than that of$\overline{\mathbf{x}}$beinglarger than 4.5.

Reason for x being equal to 4.5

It has been observed that in Part (a),$\overline{\mathbf{x}}$ has only one value, that is, 5 (above 4.5). On the other hand, in the table, 4.5 has appeared two times.So, it can be deduced that$\overline{\mathbf{x}}$ being equal to 4.5 has a greater chance.