Question 14.9: Table 14.7 shows the results obtained by four analysts deter...
Table 14.7 shows the results obtained by four analysts determining the purity of a pharmaceutical preparation of sulfanilamide. Determine if the difference in their results is significant at \alpha=0.05. If such a difference exists, estimate values for \sigma_{\text {rand }}^{2} and \sigma_{\text {sys. }}^{2}.
Table 14.7 Results of Four Analysts for the %Purity of a Preparation of Sulfanilamide
| Replicate | Analyst A | Analyst B | Analyst C | Analyst D |
| 1 | 94.09 | 99.55 | 95.14 | 93.88 |
| 2 | 94.64 | 98.24 | 94.62 | 94.23 |
| 3 | 95.08 | 101.1 | 95.28 | 96.05 |
| 4 | 94.54 | 100.4 | 94.59 | 93.89 |
| 5 | 95.38 | 100.1 | 94.24 | 94.95 |
| 6 | 93.62 | 95.49 |
To begin, we calculate the global mean and variance and the means for each analyst. These results are
\overline{\bar{X}}=95.87 \quad \overline{\overline{s^{2}}}=5.506 \quad \bar{X}_{\mathrm{A}}=94.56 \quad \bar{X}_{\mathrm{B}}=99.88 \quad \bar{X}_{\mathrm{C}}=94.77 \quad \bar{X}_{\mathrm{D}}=94.75
Using these values, we then calculate the total sum of squares
S S_{\mathrm{t}}=\overline{\overline{s^{2}}}(N-1)=(5.506)(22-1)=115.63
the between-sample sum of squares
\begin{aligned} S S_{\mathrm{b}}=\sum_{i=1}^{4} n_{i} \bar{X}_{i}-\overline{\bar{X}}^{2}= & 6(94.56-95.87)^{2}+5(99.88-95.87)^{2} \\ & +5(94.77-95.87)^{2}+6(94.75-95.87)^{2}=104.27 \end{aligned}
and the within-sample sum of squares
S S_{\mathrm{w}}=S S_{\mathrm{t}}-S S_{\mathrm{b}}=115.63-104.27=11.36
The remainder of the necessary calculations are summarized in the following table.
\begin{array}{cccc} \text{Source }& \text{Sum of Squares} & \text{Degrees of Freedom} & \text{Variance }\\ \hline \text{between }& 104.27 & h-1=4-1=3 & 34.76 \\ \text{within }& 11.36 & N-h=22-4=18 & 0.631 \end{array}
The value of the test statistic is
F_{\exp }=\frac{s_{\mathrm{b}}^{2}}{s_{\mathrm{W}}^{2}}=\frac{34.76}{0.631}=55.08
for which the critical value of F(0.05,3,18) is 3.16 . Because F_{\text {exp }} is greater than F(0.05,3,18), the null hypothesis is rejected, and the results for at least one analyst are significantly different from those for the other analysts. The value for \sigma_{\text {rand }}^{2} is estimated by the within-sample variance
\sigma_{\text {rand }}^{2} \approx s_{\mathrm{w}}^{2}=0.631
whereas, from equation 14.24,
s_{\mathrm{b}}^{2}=\sigma_{\mathrm{rand}}^{2}+\overline{{{n}}}\sigma_{\mathrm{sys}}^{2} (14.24)
\sigma_{\text {sys }}^{2} is
\sigma_{\text {sys }}^{2} \approx \frac{s_{\mathrm{b}}^{2}-s_{\mathrm{w}}^{2}}{\bar{n}}=\frac{34.76-0.631}{22 / 4}=6.205
In this example the variance due to systematic differences between the analysts is almost an order of magnitude greater than the variance due to the method’s precision.