Question

A random sample of n=900 observations is selected from a population with $\mathbf{\mu }\mathbf{=}\mathbf{100}\mathbf{}\mathbf{and}\mathbf{}\mathbf{\sigma }\mathbf{=}\mathbf{10}$

a. What are the largest and smallest values of$\overline{\mathbf{x}}$ that you would expect to see?

b. How far, at the most, would you expect $\overline{\mathbf{x}}$to deviate from $\mathbf{\mu }$?

c. Did you have to know $\mathbf{\mu }$to answer part b? Explain.

Short answer

$$\begin{gathered} \mathrm{a}.\mathrm{The}\mathrm{smallest}\mathrm{value}\mathrm{of}\overline{\mathrm{x}}\mathrm{is}99,\mathrm{and}\mathrm{the}\mathrm{largest}\mathrm{value}\mathrm{of}\overline{\mathrm{x}}\mathrm{is}101. \\ \mathrm{b}.\mathrm{It}\mathrm{would}\mathrm{not}\mathrm{be}\mathrm{expected}\mathrm{that}\overline{\mathrm{x}}\mathrm{deviates}\mathrm{from}\mathrm{\mu } \\ \mathrm{c}.\mathrm{No},\mathrm{because}\mathrm{previous}\mathrm{answer}\mathrm{only}\mathrm{dependent}\mathrm{on}\mathrm{the}\mathrm{standard}\mathrm{deviation}\mathrm{of}\mathrm{the} \\ \mathrm{sampling}\mathrm{distribution}\mathrm{of}\mathrm{the}\mathrm{sample}\mathrm{mean},\mathrm{but}\mathrm{not}\mathrm{the}\mathrm{mean}\mathrm{itself}. \end{gathered}$$

Step-by-step solution

Given information

$$\begin{gathered} \mathrm{A}\mathrm{random}\mathrm{sample}\mathrm{of}\mathrm{n}=900\mathrm{observations}\mathrm{is}\mathrm{selected}\mathrm{from}\mathrm{a}\mathrm{population}\mathrm{with}\mathrm{\mu }=100 \\ \mathrm{and}\mathrm{\sigma }=10 \end{gathered}$$

Finding the largest and smallest values of x

a.

Here, the sample mean is ${\mathbf{\mu }}_{\overline{\mathbf{x}}}\mathbf{=}\mathbf{\mu }$and the sample standard deviation is ${\mathbf{\sigma }}_{\overline{\mathbf{x}}}\mathbf{=}\frac{\mathbf{\sigma }}{\sqrt{\sqrt{\mathbf{n}}}}$

Therefore,

$m_{\overline{x}}=100,$

$$\begin{gathered} {\sigma }_{\overline{x}}=\frac{10}{\sqrt{\sqrt{900}}} \\ =\frac{10}{30} \\ =0.33 \end{gathered}$$

Since, almost all of the time, the sample mean will be within three standard deviations of the mean.

So,

$$\begin{gathered} \left(\mu \pm 3\sigma \right)=\left(100\pm 3\left(0.33\right)\right) \\ =\left(100-0.99,100+0.99\right) \\ \approx \left(99,101\right) \end{gathered}$$

Therefore, the smallest value of $\overline{x}$is 99, and the largest value of $\overline{x}$is 101.

Checking whether deviation of X from μ expect or not.

b.

No, because if more than three standard deviation then

$$\begin{gathered} 3{\sigma }_{\overline{x}}=3\left(\frac{1}{3}\right) \\ =1 \end{gathered}$$

Therefore, It would not be expected that that $\overline{x}$deviates from$\mu$.

Checking whether μ is necessary to answer part (b) or not.

c.

No, because previous answer only dependent on the standard deviation of the sampling distribution of the sample mean, but not the mean itself.