Suppose that in Problem 13-16 the bar stock may be purchased in many sizes and that the three sizes actually used in the experiment were selected randomly. Obtain the expected mean squares for this situation and modify the previous analysis appropriately. Use the restricted form of the mixed model.

Problem 13-16

A structural engineer is studying the strength of aluminum alloy purchased from three vendors. Each vendor submits the alloy in standard-sized bars of 1.0, 1.5, or 2.0 inches. The processing of different sizes of bar stock from a common ingot involves different forging techniques, and so this factor may be important. Furthermore, the bar stock is forged from ingots made in different heats. Each vendor submits two test specimens of each size bar stock from three heats. The resulting strength data follow. Analyze the data, assuming that vendors and bar size are fixed and heats are random. Use the restricted form of the mixed model.

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Vendor 1 Vendor 2 Vendor 3 Heat 1 2 3 1 2 3 1 2 Bar size: 1 inch 1.230 1.346 1.235 1.301 1.346 1.315 1.247 1.275 1.324 1.259 1.400 1.206 1.263 1.392 1.320 1.296 1.268 1.315 1 inch 1.316 1.329 1.250 1.274 1.384 1.346 1.273 1.260 1.392 1.300 1.362 1.239 1.268 1.375 1.357 1.264 1.265 1.364 2 inch 1.287 1.346 1.273 1.247 1.362 1.336 1.301 1.280 1.319 1.292 1.382 1.215 1.215 1.328 1.342 1.262 1.271 1.323