91Ó°ÊÓ

To find the slope and intercept, we can use a new worksheet, a scientific calculator, or do it ourselves by using the following formulas:

$$\begin{gathered} \mathit{m}\mathbf{=}\frac{\mathbf{n}\underset{\mathbf{}}{\mathbf{∑}}\left(x_i{y}_i\right)\mathbf{-}\underset{\mathbf{}}{\mathbf{∑}}{\mathbf{x}}_{\mathbf{i}}\underset{\mathbf{}}{\mathbf{∑}}{\mathbf{y}}_{\mathbf{i}}}{\mathbf{n}\underset{\mathbf{}}{\mathbf{∑}}\left(x_i^2\right)\mathbf{-}{\left(\underset{}{∑}{x}_i\right)}^{\mathbf{2}}}\mathbf{.} \\ \mathit{b}\mathbf{=}\frac{\underset{\mathbf{}}{\mathbf{∑}}\left(x_i^2\right)\mathbf{-}\underset{\mathbf{}}{\mathbf{∑}}{\mathbf{y}}_{\mathbf{i}}\mathbf{-}\underset{\mathbf{}}{\mathbf{∑}}\left(x_i{y}_i\right)\underset{\mathbf{}}{\mathbf{∑}}{\mathbf{x}}_{\mathbf{i}}}{\mathbf{n}\underset{\mathbf{}}{\mathbf{∑}}\left(x_i^2\right)\mathbf{-}{\left(\underset{}{∑}{x}_i\right)}^{\mathbf{2}}}\mathbf{.} \end{gathered}$$

To find the slope and the intercept using a scientific calculator, go to Menu $→$Statistics $→y=a+bx→$enter the given data $→$OPTN $→$Regression calc.

In this case, we must plot a calibration curve; it is easiest to use a worksheet. First, enter the given experimental data in a new worksheet. Then, click on Insert $→$Insert chart $→$Scatter. Then, choose x and y values and go to Chart Tools, then Layout $→$Trendline $→$Linear Trendline. It should fit a straight line through the dots on the graph. To show the data on the graph, choose More Trendline Options.

(a)

After we've completed all the steps, we will get something like this as shown in the image below.

*Value 9.1 was subtracted from the calculation.

So, the slope and the intercept are:

$$\begin{gathered} m=874.67. \\ b=-20.177. \end{gathered}$$

To calculate their uncertainties, we can also use a worksheet. So, choose a random empty cell, write =LINEST and click on the offered formula. Then, choose the x and y values. For constants, we can put in 1 or simply leave it empty, and for stat, write TRUE. We willget something like this as shown in the image below.

Now, we must know what each of these values present. We notice that the first cell is a slope and the one next to it is the y -intercept.

The values can be described as shown above.

So, the uncertainties of the slope and intercept are given in the second row.

(b)

Average of an unknown:

$\stackrel{¯}{a}=\frac{152.1+154.9+153.9+155.1}{4}$

Average of a blank:

$\stackrel{¯}{b}=\frac{8.2+9.4+10.6+7.8}{4}$

Average corrected signal for the unknown:

$154.0-9.0=145.0$

(c)

To find the concentration of the unknown, we must use the equation obtained in step (A), where the values for signal (in mV) are y values and the concentrations are X values.

$$\begin{gathered} y=874.67x-20.177. \\ x=\frac{y+20.177}{874.67}. \end{gathered}$$

Again, we can use a worksheet or a simple calculator.

The values of concentration are shown in the picture below, as well as the formula on how to calculate in a worksheet.

To find the 95% confidence interval, we must calculate the standard deviation.

confidence interval $=Â\pm ts/\sqrt{n}$

role="math" localid="1663329165416" $$\begin{gathered} \mathrm{SD}=\sqrt{\frac{\underset{}{∑}{\left|\mathrm{x}-\stackrel{¯}{\mathrm{x}}\right|}^2}{\mathrm{n}}} \\ \stackrel{¯}{\mathrm{x}}=0.19913 \\ \mathrm{SD}=1.357×{10}^{-3} \\ \mathrm{confidence}\mathrm{interval}=\left(Â\pm 3.182×1.357×{10}^{-3}\right)/\sqrt{4} \\ =2.159×{10}^{-3} \end{gathered}$$