Reference no: EM132857927

The construction of a foundation system for a building calls for the excavation of poor soil to be replaced with soil (fill) that is properly compacted in each of three lifts (or what some might call layers). The total depth of the fill (D) is the sum of the three individual layers (Xi), or D = X1 + X2 + X3. The specifications require that the total depth to be 9 meters with a permitted variation of ± 0.03 meters. The contractor assumes that the depth of each lift when placed is independent and normally distributed with the same mean (m = 3 meters) and the same but unknown standard deviation (s).

(a) The contractor seeks a 95% chance that the total depth of fill meets the specification (D = 8.97 m to 9.03 m). Find the standard deviation of each lift (s) required to meet the contractor's desires. Use the summation model for the total depth: D = X1 + X2 + X3. Assume each layer's depth is independent.

(b) The contractor suggests an alternative approach to construct each layer, that is to place each at 3± 0.01 meters. Here, the error is simply divided among the three layers. If the standard deviation for each layer is what you found in part (a), find the chance that the total depth of fill meets the specifications. [Suggestion: find the chance that each layer meets the layer's specification and then find a way to combine those probabilities.]

(c) As a third construction approach, the contractor decides to place the first layers, measure their depth, and then determine what the specifications need to be for the final layer. Suppose that the same total depth specification applies (D = 8.97 m to 9.03 m) and that the measured mean and standard deviation for each of the first two lifts is 3 meters with a standard deviation of 0.009 m (not necessarily the answer to part (a)). Find now the mean and standard deviation of the final lift in order to meet the 95% desired reliability for the total depth. X3 = D - (X1 + X2)