Western Pygmy-Possum. The foraging behavior of the western pygmy-possum was investigated in the article "Strategies of a Small Nectarivorous Marsupial, the Western Pygmy-Possum, in Response to Seasonal Variation in Food Availability" (Journal of Mammalogy, Vol. 96, No. 6, pp. 1525-1535) by D. Morrant and S. Petit. The weights of adult male pygmy-possums in Australia are normally distributed with a mean of $8.5\mathrm{g}$ and a standard deviation of $0.3\mathrm{g}$

a. Sketch the normal curve for the pygmy-possum weights.

b. Find the sampling distribution of the sample mean for samples of size $4$Draw a graph of the normal curve associated with $\overline{x}$

c. Repeat part (b) for samples of size $9$

The weight of adult male pygmy-possums $\left(x\right)$is normally distributed with mean $\left(\mu \right)8.5\mathrm{g}$and standard deviation $\left(\sigma \right)0.3\mathrm{g}$

The formula used: Standard deviation ${\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}}$

To create the normal curve for the pygmy-possum weights, use MINITAB.

The weight of adult male pygmy-possums $\left(x\right)$is assumed to be regularly distributed, with a mean of $\left(\mu \right)8.5\mathrm{g}$and a standard deviation of $\left(\sigma \right)0.3\mathrm{g}$

Procedure for MINITAB:

Step (1): Select Graph$>$Probability Distribution Plot $>$View Single $>$Ok from the drop-down menu.

Step (2): Select a Normal from the Distribution menu.

Step (3): Enter $\mathbf{8}\mathbf{.}\mathbf{5}$in Mean and $\mathbf{0}\mathbf{.}\mathbf{3}$in Standard deviation.

Step (4): Click OK.

MINITAB output:

For samples of size $4$, obtain the sampling distribution of the sample mean.

The sample distribution's mean $\left({\mu }_x\right)$is $8.5\mathrm{g}$, and the standard deviation is

$$\begin{gathered} {\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}} \\ =\frac{0.3}{\sqrt{4}} \\ =\frac{0.3}{2} \\ =0.15 \end{gathered}$$

For samples of size $4$, the sampling distribution of the sample mean is mean $\left({\mu }_x\right)8.5\mathrm{g}$ and standard deviation $\left({\sigma }_{\overline{s}}\right)$ is equal to $0.15\mathrm{g}$

To create the normal curve for the sample distribution, use MINITAB.

Procedure for MINITAB:

Step (1): Select Graph$>$Probability Distribution Plot $>$View Single $>$Ok from the drop-down menu.

Step 2: Select a Normal from the Distribution menu.

Step 3: Enter $\mathbf{8}\mathbf{.}\mathbf{5}$in Mean and $\mathbf{0}\mathbf{.}\mathbf{15}$in Standard deviation.

Step 4: Click OK.

MINITAB output:

Procedure for MINITAB:

Step (1): Select Graph

For samples of size $9$, obtain the sampling distribution of the sample mean.

The sample distribution's mean $\left({\mu }_x\right)$is $8.5\mathrm{g}$, and the standard deviation is

$$\begin{gathered} {\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}} \\ =\frac{0.3}{\sqrt{9}} \\ =\frac{0.3}{3} \\ =0.1 \end{gathered}$$

For samples of size $4$, the sampling distribution of the sample mean is mean $\left({\mu }_x\right)8.5\mathrm{g}$and standard deviation$\left({\sigma }_x\right)$is$0.1\mathrm{g}$

Procedure for MINITAB:

Step (1): Select Graph$>$Probability Distribution Plot$>$View Single $>$Ok from the drop-down menu.

Step (2): Select a Normal from the Distribution menu.

Step (3): Enter$\mathbf{8}\mathbf{.}\mathbf{5}$in Mean and $\mathbf{0}\mathbf{.}\mathbf{1}$in Standard deviation.

Step (4): Click OK.

MINITAB output: