Using the data from Table 5.1, determine the relationship between Smeas and CS by an unweighted linear regression.

Question

Using the data from Table 5.1, determine the relationship between [latex]S_{meas} ext{and} C_S[/latex] by an unweighted linear regression.

Table 5.1 Data for Hypothetical Multiple- Point External Standardization
[latex]C_S[/latex]	[latex]S_{meas} [/latex]
0.000	0.00
0.100	12.36
0.200	24.83
0.300	35.91
0.400	45.79
0.500	60.42

Accepted Answer

Equations 5.13 and 5.14 are written in terms of the general variables x and y. As you work through this example, remember that x represents the concentration of analyte in the standards ([latex]C_S[/latex]), and that y corresponds to the signal ([latex]S_{meas}[/latex]). We begin by setting up a table to help in the calculation of the summation terms [latex]\sum x_i, \sum y_i, \sum x_i², and \sum x_iy_i[/latex] which are needed for the calculation of [latex]b_0 ext{and} b_1[/latex]

[latex]b_1=\frac{n\sum{x_iy_i}-\sum x_i\sum y_i }{n\sum x_i^2-(\sum x_i)^2}[/latex] (5.13)

[latex]b_0=\frac{\sum y_i-b_1\sum x_i }{n} [/latex] (5.14)

[latex]x_i[/latex]	[latex]y_i[/latex]	[latex]x_i^2[/latex]	[latex]x_iy_i[/latex]
0.000	0.00	0.000	0.000
0.100	12.36	0.010	1.236
0.200	24.83	0.040	4.966
0.300	35.91	0.090	10.773
0.400	48.79	0.160	19.516
0.500	60.42	0.250	30.210

Adding the values in each column gives

[latex] \sum x_i =1.500 \sum y_i=182.31 \sum x_i^2=0.550 \sum x_iy_i=66.701 [/latex]

Substituting these values into equations 5.12 and 5.13 gives the estimated slope

[latex]y=\beta _0 + \beta _1 x[/latex] (5.12)

[latex]b_1=\frac{(6)(66.701)-(1.500)(182.31)}{(6)(0.550)-(1.500)^2} =120.706[/latex]

and the estimated y-intercept

[latex]b_0=\frac{182.31-(120.706)(1.500)}{6}=0.209 [/latex]

The relationship between the signal and the analyte, therefore, is

[latex]S_{meas} = 120.70 × C_S + 0.21[/latex]

Note that for now we keep enough significant figures to match the number of decimal places to which the signal was measured. The resulting calibration curve is shown in Figure 5.10.

Question 5.10: Using the data from Table 5.1, determine the relationship be...

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