(i) To predict the concentration of selenium in the solution, we need to establish a relationship between the concentration and the signal. This is typically done by fitting a line to the data points and using this line (the calibration curve) to predict the concentration from the signal.

The data points suggest a linear relationship between concentration and signal, so we can use linear regression to fit a line to the data. The equation of a line is y = mx + b, where m is the slope and b is the y-intercept.

First, we calculate the slope (m) by taking the difference in y-values divided by the difference in x-values for two points. Let's use the points (20, 2.25) and (160, 18.9):

m = (18.9 - 2.25) / (160 - 20) = 16.65 / 140 = 0.119 ppm/signal

Next, we calculate the y-intercept (b) by rearranging the equation to b = y - mx and substituting one of the points. Let's use (20, 2.25):

b = 2.25 - 0.119 * 20 = 0.45

So, the equation of the line is y = 0.119x + 0.45.

To predict the concentration of selenium (x) from the average signal (y = 8.67), we rearrange the equation to x = (y - b) / m:

x = (8.67 - 0.45) / 0.119 = 68.9 ppm

(ii) The standard uncertainty of the result due to the calibration curve can be calculated by taking the standard deviation of the residuals (the differences between the observed and predicted y-values for each data point).

First, we calculate the residuals for each data point:

Residual for (20, 2.25) = 2.25 - (0.119 * 20 + 0.45) = 0
Residual for (40, 4.78) = 4.78 - (0.119 * 40 + 0.45) = 0
Residual for (80, 9.98) = 9.98 - (0.119 * 80 + 0.45) = 0
Residual for (120, 14.21) = 14.21 - (0.119 * 120 + 0.45) = 0
Residual for (160, 18.9) = 18.9 - (0.119 * 160 + 0.45) = 0

The standard deviation of the residuals is 0, so the standard uncertainty of the result due to the calibration curve is also 0.

Question

(i) To predict the concentration of selenium in the solution, we need to establish a relationship between the concentration and the signal. This is typically done by fitting a line to the data points and using this line (the calibration curve) to predict the concentration from the signal.

The data points suggest a linear relationship between concentration and signal, so we can use linear regression to fit a line to the data. The equation of a line is y = mx + b, where m is the slope and b is the y-intercept.

First, we calculate the slope (m) by taking the difference in y-values divided by the difference in x-values for two points. Let's use the points (20, 2.25) and (160, 18.9):

m = (18.9 - 2.25) / (160 - 20) = 16.65 / 140 = 0.119 ppm/signal

Next, we calculate the y-intercept (b) by rearranging the equation to b = y - mx and substituting one of the points. Let's use (20, 2.25):

b = 2.25 - 0.119 * 20 = 0.45

So, the equation of the line is y = 0.119x + 0.45.

To predict the concentration of selenium (x) from the average signal (y = 8.67), we rearrange the equation to x = (y - b) / m:

x = (8.67 - 0.45) / 0.119 = 68.9 ppm

(ii) The standard uncertainty of the result due to the calibration curve can be calculated by taking the standard deviation of the residuals (the differences between the observed and predicted y-values for each data point).

First, we calculate the residuals for each data point:

Residual for (20, 2.25) = 2.25 - (0.119 * 20 + 0.45) = 0
Residual for (40, 4.78) = 4.78 - (0.119 * 40 + 0.45) = 0
Residual for (80, 9.98) = 9.98 - (0.119 * 80 + 0.45) = 0
Residual for (120, 14.21) = 14.21 - (0.119 * 120 + 0.45) = 0
Residual for (160, 18.9) = 18.9 - (0.119 * 160 + 0.45) = 0

The standard deviation of the residuals is 0, so the standard uncertainty of the result due to the calibration curve is also 0.

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