A study was made of the acidity levels in farmland on opposite sides of an island. The levels were measured at six randomly chosen points on the eastern side and at five randomly chosen points on the western side. The values obtained, in suitable units, are denoted by \(x _ { E }\) and \(x _ { W }\) respectively. The sample means \(\bar { x } _ { E }\) and \(\bar { x } _ { W }\), and unbiased estimates of the two population variances, \(s _ { E } ^ { 2 }\) and \(s _ { W } ^ { 2 }\), are as follows. $$\bar { x } _ { E } = 5.035 , s _ { E } ^ { 2 } = 0.0231 , \bar { x } _ { W } = 4.782 , s _ { W } ^ { 2 } = 0.0195 .$$ The population means on the eastern and western sides are denoted by \(\mu _ { E }\) and \(\mu _ { W }\) respectively. State suitable hypotheses for a test for a difference between the mean acidity levels on the two sides of the island. Stating any required assumptions, obtain the rejection region for a test at the \(5 \%\) significance level of whether the mean acidity levels differ on the two sides of the island. Give the conclusion of the test. Find the largest value of \(a\) for which the samples above provide evidence at the \(5 \%\) significance level that \(\mu _ { E } - \mu _ { W } > a\).