Question 14.10: Determine the source of the significant difference for the d...
Determine the source of the significant difference for the data in Example 14.9.
Individual comparisons using Fisher’s least significant difference test are based on the following null hypothesis and one-tailed alternative hypothesis
H_{0}: \bar{X}_{i}=\bar{X}_{j} \quad H_{\mathrm{A}}: \bar{X}_{i}>\bar{X}_{j} \quad \text { or } \quad \bar{X}_{i}<\bar{X}_{j}
Using equation 14.25 , we can calculate values of t_{\exp } for each possible comparison. These values can then be compared with the one-tailed critical value of 1.73 for t(0.05,18), as found in Appendix 1B. For example, t_{\exp } when comparing the results for analysts \mathrm{A} and \mathrm{B} is
\left(t_{\exp }\right)_{\mathrm{AB}}=\frac{\left|\bar{X}_{\mathrm{A}}-\bar{X}_{\mathrm{B}}\right|}{\sqrt{s_{\mathrm{w}}^{2}\left[\left(1 / n_{\mathrm{A}}\right)+\left(1 / n_{\mathrm{B}}\right)\right]}}=\frac{|94.56-99.88|}{\sqrt{(0.631)[(1 / 6)+(1 / 5)]}}=11.06
Because \left(t_{\exp }\right)_{\mathrm{AB}} is greater than t(0.05,18), we reject the null hypothesis and accept the alternative hypothesis that the results for analyst \mathrm{B} are significantly greater than those for analyst A. Working in the same fashion, it is easy to show that
\begin{array}{ll} \left(t_{\exp }\right)_{\mathrm{AC}}=0.437 & H_{0} \text { is retained } \\ \left(t_{\exp }\right)_{\mathrm{AD}}=0.414 & H_{0} \text { is retained } \\ \left(t_{\exp }\right)_{\mathrm{BC}}=10.17 & H_{0} \text { is rejected and } H_{\mathrm{A}} \text { is accepted } \\ \left(t_{\exp }\right)_{\mathrm{BD}}=10.67 & H_{0} \text { is rejected and } H_{\mathrm{A}} \text { is accepted } \\ \left(t_{\exp }\right)_{\mathrm{CD}}=0.04 & H_{0} \text { is accepted } \end{array}
Taken together, these results suggest that there is a significant systematic difference between the work of analyst \mathrm{B} and that of the other analysts. There is no way to decide, however, whether any of the four analysts has done accurate work.
| Appendix 1B t-Table^a |
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| Value of t for confidence interval of : Critical value of |t| for α values of : Degrees of Freedom |
90% 0.10 |
95 % 0.05 |
98 % 0.02 |
99 % 0.01 |
| 1 | 6.31 | 12.71 | 31.82 | 63.66 |
| 2 | 2.92 | 4.30 | 6.96 | 9.92 |
| 3 | 2.35 | 3.18 | 4.54 | 5.84 |
| 4 | 2.13 | 2.78 | 3.75 | 4.60 |
| 5 | 2.02 | 2.57 | 3.36 | 4.03 |
| 6 | 1.94 | 2.45 | 3.14 | 3.71 |
| 7 | 1.89 | 2.36 | 3.00 | 3.50 |
| 8 | 1.86 | 2.31 | 2.90 | 3.36 |
| 9 | 1.83 | 2.26 | 2.82 | 3.25 |
| 10 | 1.81 | 2.23 | 2.76 | 3.17 |
| 12 | 1.78 | 2.18 | 2.68 | 3.05 |
| 14 | 1.76 | 2.14 | 2.62 | 2.98 |
| 16 | 1.75 | 2.12 | 2.58 | 2.92 |
| 18 | 1.73 | 2.10 | 2.55 | 2.88 |
| 20 | 1.72 | 2.09 | 2.53 | 2.85 |
| 30 | 1.70 | 2.04 | 2.46 | 2.75 |
| 50 | 1.68 | 2.01 | 2.40 | 2.68 |
| \infty | 1.64 | 1.96 | 2.33 | 2.58 |
| ^aThe t-values in this table are for a two-tailed test. For a one-tailed test, the α values for each column are half of the stated value. For example, the first column for a one-tailed test is for the 95% confidence level, α = 0.05. |
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