Determine the source of the significant difference for the data in Example 14.9.

Question

Accepted Answer

Individual comparisons using Fisher's least significant difference test are based on the following null hypothesis and one-tailed alternative hypothesis

[latex]H_{0}: \bar{X}_{i}=\bar{X}_{j} \quad H_{\mathrm{A}}: \bar{X}_{i}>\bar{X}_{j} \quad ext { or } \quad \bar{X}_{i}<\bar{X}_{j}[/latex]

Using equation 14.25 , we can calculate values of [latex]t_{\exp }[/latex] for each possible comparison. These values can then be compared with the one-tailed critical value of 1.73 for [latex]t(0.05,18)[/latex], as found in Appendix 1B. For example, [latex]t_{\exp }[/latex] when comparing the results for analysts [latex]\mathrm{A}[/latex] and [latex]\mathrm{B}[/latex] is

[latex]\left(t_{\exp } ight)_{\mathrm{AB}}=\frac{\left|\bar{X}_{\mathrm{A}}-\bar{X}_{\mathrm{B}} ight|}{\sqrt{s_{\mathrm{w}}^{2}\left[\left(1 / n_{\mathrm{A}} ight)+\left(1 / n_{\mathrm{B}} ight) ight]}}=\frac{|94.56-99.88|}{\sqrt{(0.631)[(1 / 6)+(1 / 5)]}}=11.06[/latex]

Because [latex]\left(t_{\exp } ight)_{\mathrm{AB}}[/latex] is greater than [latex]t(0.05,18)[/latex], we reject the null hypothesis and accept the alternative hypothesis that the results for analyst [latex]\mathrm{B}[/latex] are significantly greater than those for analyst A. Working in the same fashion, it is easy to show that

[latex] \begin{array}{ll} \left(t_{\exp } ight)_{\mathrm{AC}}=0.437 & H_{0} ext { is retained } \ \left(t_{\exp } ight)_{\mathrm{AD}}=0.414 & H_{0} ext { is retained } \ \left(t_{\exp } ight)_{\mathrm{BC}}=10.17 & H_{0} ext { is rejected and } H_{\mathrm{A}} ext { is accepted } \ \left(t_{\exp } ight)_{\mathrm{BD}}=10.67 & H_{0} ext { is rejected and } H_{\mathrm{A}} ext { is accepted } \ \left(t_{\exp } ight)_{\mathrm{CD}}=0.04 & H_{0} ext { is accepted } \end{array} [/latex]

Taken together, these results suggest that there is a significant systematic difference between the work of analyst [latex]\mathrm{B}[/latex] and that of the other analysts. There is no way to decide, however, whether any of the four analysts has done accurate work.

Question 14.10: Determine the source of the significant difference for the d...

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