Table 14.7 shows the results obtained by four analysts determining the purity of a

Question

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 [latex]\alpha=0.05[/latex]. If such a difference exists, estimate values for [latex]\sigma_{ ext {rand }}^{2}[/latex] and [latex]\sigma_{ ext {sys. }}^{2}[/latex].

Accepted Answer

To begin, we calculate the global mean and variance and the means for each analyst. These results are

[latex]\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[/latex]

Using these values, we then calculate the total sum of squares

[latex]S S_{\mathrm{t}}=\overline{\overline{s^{2}}}(N-1)=(5.506)(22-1)=115.63[/latex]

the between-sample sum of squares

[latex] \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} [/latex]

and the within-sample sum of squares

[latex]S S_{\mathrm{w}}=S S_{\mathrm{t}}-S S_{\mathrm{b}}=115.63-104.27=11.36[/latex]

The remainder of the necessary calculations are summarized in the following table.

[latex]\begin{array}{cccc} ext{Source }& ext{Sum of Squares} & ext{Degrees of Freedom} & ext{Variance }\ \hline ext{between }& 104.27 & h-1=4-1=3 & 34.76 \ ext{within }& 11.36 & N-h=22-4=18 & 0.631 \end{array}[/latex]

The value of the test statistic is

[latex]F_{\exp }=\frac{s_{\mathrm{b}}^{2}}{s_{\mathrm{W}}^{2}}=\frac{34.76}{0.631}=55.08[/latex]

for which the critical value of [latex]F(0.05,3,18)[/latex] is 3.16 . Because [latex]F_{ ext {exp }}[/latex] is greater than [latex]F(0.05,3,18)[/latex], 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 [latex]\sigma_{ ext {rand }}^{2}[/latex] is estimated by the within-sample variance

[latex]\sigma_{ ext {rand }}^{2} \approx s_{\mathrm{w}}^{2}=0.631[/latex]

whereas, from equation 14.24,

[latex]s_{\mathrm{b}}^{2}=\sigma_{\mathrm{rand}}^{2}+\overline{{{n}}}\sigma_{\mathrm{sys}}^{2}[/latex] (14.24)

[latex]\sigma_{ ext {sys }}^{2}[/latex] is

[latex]\sigma_{ ext {sys }}^{2} \approx \frac{s_{\mathrm{b}}^{2}-s_{\mathrm{w}}^{2}}{\bar{n}}=\frac{34.76-0.631}{22 / 4}=6.205[/latex]

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.

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

Question 14.9: Table 14.7 shows the results obtained by four analysts deter...

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