Question

A balanced scale You have two scales for measuring weights in a chemistry lab. Both scales give answers that vary a bit in repeated weighings of the same item. If the true weight of a compound is $2.00$grams ($g$), the first scale produces readings $X$that have mean $2.000$$g$and standard deviation $0.002g$. The second scale’s readings $Y$have mean $2.001g$and standard deviation . $0.001g$The readings$XandY$are independent. Find the mean and standard deviation of the difference $Y-X$between the readings. Interpret each value in context.

Short answer

The difference between the readings is on average $\mu Y-X=0.001g$,which varies on about $\sigma Y-X\approx 0.002236g$.

Step-by-step solution

Given Information

Given in the question that:

$\mu X=2.000$

$\sigma X=0.002$

$\mu Y=2.001$

$\sigma Y=0.001$

We have to find out that the mean and standard deviation of the difference $Y-X$between the readings.

Explanation

The property mean and variance (if$X$and $Y$are independent):

$$\begin{gathered} \mu X+Y=\mu X+\mu Y \\ {\sigma }^{2}X+Y={\mu }^{2}X+{\mu }^{2}Y \end{gathered}$$

Then we obtain:

$$\begin{gathered} \mu Y-X=\mu Y-\mu X \\ =2.001-2.000 \\ =0.001 \end{gathered}$$

$$\begin{gathered} {\sigma }^{2}Y-X={\mu }^{2}Y+{\mu }^{2}X \\ =0.{001}^2+0.{002}^2 \\ =0.000005 \end{gathered}$$

The standard deviation is the square root of the variance:

$$\begin{gathered} \sigma Y-X=\sqrt{{\sigma }^{2}Y-X} \\ =\sqrt{0.000005} \\ \approx 0.002236 \end{gathered}$$