Hi can some help me with this question as i don't understand it at all, and just do not know were to begin.
I can get you started.
Part A:
There are a lot choices. Simply pick a measurement in units of customary and metric. Example: pounds vs. kilograms, miles vs. kilometers, Fahrenheit vs. Celsius, etc.
For instance, take a temperature conversion. We know that the relationship between F and C is: $\displaystyle F = 9/5 C + 32$.
So if $\displaystyle y_i$ represented Fahrenheit and $\displaystyle x_i$ represented Celsius, then $\displaystyle a$ is $\displaystyle 9/5$ and $\displaystyle b$ is 32.
Part B:
(i)
$\displaystyle \bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i = \frac{1}{n} \sum_{i=1}^{n} (a x_i + b) = \frac{a}{n} \sum_{i=1}^{n} x_i + \frac{1}{n} \sum_{i=1}^{n} b = a \bar{x_i} + b$
So the claim is true, as proven by algebra above.
(ii)
For $\displaystyle \bar{z}$, the claim is false. It is up to you to find a counterexample. Pick any arbitrary set of values for $\displaystyle x_i$ and see whether or not $\displaystyle \bar{z} = \frac{1}{n} \sum_{i=1}^{n} \frac{c}{x_i}$ is equal to $\displaystyle \frac{c}{\bar{x}}$. Most likely not, if you picked your values strategically.
(iii)
Since the relationship between $\displaystyle y_i,x_i$ is linear (and thus bijective), the ordering of the values of $\displaystyle x_i$ are preserved during a mapping to $\displaystyle y_i$. Thus, the smallest value of $\displaystyle x_i$ gets sent to the smallest value of $\displaystyle y_i$. Same with the largest value and the median, etc.
(iv)
Again, find a counterexample.