Hard Statisitics question

**nerdo** · January 17th 2009, 11:52 AM

Hi can some help me with this question as i don't understand it at all, and just do not know were to begin.

**Last_Singularity** · January 17th 2009, 03:09 PM

Originally Posted by nerdo

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: $F = 9/5 C + 32$ .

So if $y_i$ represented Fahrenheit and $x_i$ represented Celsius, then $a$ is $9/5$ and $b$ is 32.

Part B:
(i)
$\bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i = \frac{1}{n} \sum_{i=1}^{n} (a x_i + b) = \frac{a}{n} \sum_{i=1}^{n} x_i + \frac{1}{n} \sum_{i=1}^{n} b = a \bar{x_i} + b$

So the claim is true, as proven by algebra above.

(ii)
For $\bar{z}$ , the claim is false. It is up to you to find a counterexample. Pick any arbitrary set of values for $x_i$ and see whether or not $\bar{z} = \frac{1}{n} \sum_{i=1}^{n} \frac{c}{x_i}$ is equal to $\frac{c}{\bar{x}}$ . Most likely not, if you picked your values strategically.

(iii)
Since the relationship between $y_i,x_i$ is linear (and thus bijective), the ordering of the values of $x_i$ are preserved during a mapping to $y_i$ . Thus, the smallest value of $x_i$ gets sent to the smallest value of $y_i$ . Same with the largest value and the median, etc.

(iv)
Again, find a counterexample.

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