Question

The following sample of n = 4 scores was obtainedfrom a population with unknown parameters. Scores:2, 2, 6, 2a. Compute the sample mean and standard deviation.(Note that these are descriptive values that sum-marize the sample data.)b. Compute the estimated standard error for M. (Notethat this is an inferential value that describes howaccurately the sample mean represents the un-known population mean.)

Answer 1

a) The sample mean is given by the formula below

$\begin{gathered} \mu_x=\frac{\sum_i^\text{ }x_i}{N} \\ x_i\rightarrow\text{ elements in the sample} \\ N\rightarrow\text{ sample size} \end{gathered}$

Therefore, in our case,

$\begin{gathered} \mu_x=\frac{2+2+6+2}{4}=\frac{12}{4}=3 \\ \Rightarrow\mu_x=3 \end{gathered}$

On the other hand, the sample standard deviation is

$s=\frac{\sqrt{\sum_i^{\text{ }}(x_i-\mu_x)^2}}{\sqrt{n-1}}$

Thus, in the case of the given sample,

$\begin{gathered} s=\frac{\sqrt{(2-3)^2+(2-3)^2+(6-3)^2+(2-3)^2}}{\sqrt{4-1}} \\ \Rightarrow s=\frac{\sqrt{1+1+9+1}}{\sqrt{3}}=\sqrt{\frac{12}{3}}=\sqrt{4}=2 \\ \Rightarrow s=2 \end{gathered}$

The sample mean is 3 and the sample standard deviation is 2.

b) The estimated standard error of the mean is given by the formula below

$SE=\frac{s}{\sqrt{n}}$

Hence, in our case,

$\Rightarrow SE=\frac{2}{\sqrt{4}}=1$

The estimated standard error for the mean is equal to 1.