Problem 1. A scientist has two light objects to weigh. She decides to use an old fashioned pan balance scale in the lab since she heard about a design from a colleague that is supposed to increase the accuracy of her measurements and take less time. The scientist decides to obtain weight measurements using the following design (DESIGN I): weigh the two objects together in one pan; weigh one object in one pan, and the other object in the other pan; pick one of two objects, and weigh it. In a pan balance scale when one object is in one pan and another object is in another pan the measurement obtained is the difference in weight between the two objects. Let y1, y2, y3 be the readings from the scale, and 1 the weight of the object that has been on the scale through all three weighings and 2 the other object. The standard deviation of each weighing is denoted by . Answer the following questions. 1. Write three equations relating the observed weights y1, y2, y3 to the unknown weights 1, 2. Make sure to include an appropriate error term and any necessary assumptions about the error term. 2. Find the least-squares estimates of 1, 2. 3. Find the standard error of the least-squares estimates of 1, 2. 4. If the scientist measured each object three times could she achieve the same precision (standard error) as this design? Explain. 5. (DESIGN II) Suppose that instead of the design above the scientist uses the following design. weigh both objects in one pan together twice; weigh the objects in opposite pans. Question: Find the least-squares estimates of the weights and standard error of the weights using this design. 6. Does DESIGN II determine the weights of the objects with equal precison compared to DESIGN I? Explain your reasoning.