Question

Suppose a random sample of n = 25 measurements are selected from a population with mean $\mathbf{\mu }$and standard deviation s. For each of the following values of $\mathbf{\mu }$and role="math" localid="1651468116840" $\mathbf{\sigma }$, give the values of $\mathbf{\mu }\overline{\mathbf{\chi }}$ and $\mathbf{\sigma }\overline{\mathbf{\chi }}$.

1. $\mathbf{\mu }\mathbf{=}\mathbf{100}\mathbf{,}\mathbf{\sigma }\mathbf{=}\mathbf{3}$
2. $\mathbf{\mu }\mathbf{=}\mathbf{100}\mathbf{,}\mathbf{\sigma }\mathbf{=}\mathbf{25}$
3. $\mathbf{\mu }\mathbf{=}\mathbf{20}\mathbf{,}\mathbf{\sigma }\mathbf{=}\mathbf{40}$
4. $\mathbf{\mu }\mathbf{=}\mathbf{10}\mathbf{,}\mathbf{\sigma }\mathbf{=}\mathbf{100}$

Short answer

Random sampling is a sampling strategy in which every sample has an equal chance to be selected. A basic random sample is intended to reflect a group in an unbiased manner.

Step-by-step solution

 Step 1: (a) The data is given below

The calculation is given below:

$\begin{array}{l}{\mu }_{}=\text{1}0\\ \frac{\sigma }{\text{χ}}=\frac{\text{3}}{\sqrt{25}}\\ \text{=}\frac{\text{3}}{5}\\ \text{=0.6}\end{array}$

(b) The data is given below

The calculation is given below:

$\begin{array}{l}{\mu }_{\stackrel{-}{\chi }}=\text{1}00\\ \frac{\sigma }{\text{χ}}=\frac{\text{25}}{5}\\ \text{=5}\end{array}$

(c) The data is given below

The calculation is given below:

$\begin{array}{l}{\mu }_{\stackrel{-}{\chi }}=20\\ \frac{\sigma }{\text{χ}}=\frac{40}{5}\\ \text{=8}\end{array}$

(d) The data is given below

The calculation is given below:

$\begin{array}{l}{\mu }_{\stackrel{-}{\chi }}=10\\ \frac{\sigma }{\text{χ}}=\frac{100}{5}\\ \text{=20}\end{array}$